text:
\section*{Introduction}

Let $G$ be a connected complex reductive algebraic group and let $B \subset G$ be a Borel subgroup. A subgroup $H \subset G$ is called {\em spherical} if it possesses finitely many orbits on the flag variety $G/B$. The best known example is that of $B$ itself, and more generally any parabolic subgroup of $G$. The $B$-orbits in $G/B$ are indeed the Schubert cells, and are finitely many thanks to the Bruhat decomposition. Another well studied case is that of the {\em symmetric subgroups} of $G$, i.e.\ when $H$ is the subgroup of fixed points of some algebraic involution of $G$. Especially in this case, the study of the $H$-orbits on $G/B$, their classification and the geometry of their closures are important in representation theory (see e.g.\ \cite{Sp1}, \cite{Wo}). An equivalent problem is the study of the $B$-orbits in $G/H$, and the geometry of their closures: they are fundamental objects to understand the topology of $G/H$ and of its embeddings (see \cite{FMSS}).

Spherical subgroups are classified in combinatorial terms, see \cite{Kr}, \cite{Br0}, \cite{Mi}, \cite{Lu1}, \cite{Lu2} where several particular classes of subgroups are considered, and the more recent papers \cite{Lo}, \cite{BP}, \cite{CF} where the classification is completed in full generality. Nevertheless the set $\mathscr B(G/H)$ of the $B$-orbits in $G/H$ is still far from being understood, essentially except for the cases of the parabolic subgroups and of the symmetric subgroups of $G$ (the latter especially thanks to the work of Richardson and Springer \cite{RS1}, \cite{RS2}).

The goal of the present paper is to explicitly understand the set $\mathscr B(G/H)$ in some other case, and to produce some combinatorial model for it. More precisely, we consider the case of the \textit{strongly solvable} spherical subgroups of $G$, that is, spherical subgroups of $G$ which are contained in a Borel subgroup. As a consequence of a theorem of Brion (see \cite[Theorem 6]{Br3}), under these assumptions the $H$-orbit closures in $G/B$ provide nice generalizations of the Schubert varieties: even though $H$ might be not connected, they are always irreducible, and they have rational singularities, so that in particular they are normal and Cohen-Macaulay.

When $H$ is a strongly solvable spherical subgroup, the set $\mathscr B(G/H)$ has already been studied in the literature in some special cases: Timashev \cite{Ti1} treated the case where $H = TU'$ (where $T$ denotes a maximal torus of $B$ and $U'$ the derived subgroup of the unipotent radical of $B$), and Hashimoto \cite{Ha} treated the case where $G = \mathrm{SL}_n$ and $H$ is a Borel subgroup of $\mathrm{SL}_{n-1}$, regarded as a subgroup of $\mathrm{SL}_n$.

Let $\Phi$ be the root system of $G$ and let $W$ be the its Weyl group. Given a strongly solvable spherical subgroup $H \subset G$, in our main result we give an explicit parametrization of the set $\mathscr B(G/H)$ by attaching to every $B$-orbit an element of $W$ and a root subsystem of $\Phi$. To do this, we build upon known results about the classification of strongly solvable subgroups, which is available in three different forms. The first one was given by Luna in 1993 (see \cite{Lu1}), the second one emerged in the framework of the general classification of spherical subgroups, and the third one, more explicit, has been given recently by Avdeev in \cite{Avd1} (see \cite{Avd2} for a comparison between the three approaches).

Our results on $\mathscr B(G/H)$ also provide a nice description of the action of $W$ on $\mathscr B(G/H)$, defined by Knop in \cite{Kn} for any spherical subgroup $H\subset G$. While the simple reflections of $W$ act in a rather explicit way, the resulting action of the entire $W$ is quite difficult to study. When $H$ is strongly solvable, we will see how this action becomes actually very simple: the fact that two $B$-orbits are in the same $W$-orbit will boil down to the fact that the associated root subsystems are $W$-conjugated. This will enable us to give a simple combinatorial model for $\mathscr B(G/H)$ as a finite set endowed with an action of $W$ in terms of ``generalized faces'' of weight polytopes.

We now explain our results in more detail. Fix a maximal torus $T \subset B$ and denote by $U$ the unipotent radical of $B$. We also denote by $\Phi$ the root system associated with $T$, by $W = N_G(T)/T$ its Weyl group, by $\Phi^+ \subset \Phi$ the set of positive roots associated with $B$ and by $\grD \subset \Phi^+$ the corresponding basis of $\Phi$. Let $H$ be a strongly solvable spherical subgroup of $G$, up to conjugation we may assume $H \subset B$. Up to conjugation by an element of $B$, we may assume as well that $T \cap H$ is a maximal diagonalizable subgroup of $H$. Given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$, let $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset U$ be the associated unipotent one dimensional subgroup.

Two natural sets which are attached to $H$ are the set of \textit{active roots}
$$
	\Psi = \{ \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \st U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \not \subset H\},
$$
introduced by Avdeev in \cite{Avd1}, and the corresponding set of restricted characters
$$\mathscr D^* = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\ristretto_{T \cap H} \st \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi\}.$$
From a geometrical point of view, $\mathscr D^*$ is canonically identified with the set of the $B$-stable prime divisors in $G/H$ which map dominantly to $G/B$ via the natural projection (see Section \ref{s:BmodH}). Given $I \subset \mathscr D^*$, let $\Psi_I \subset \Psi$ be the subset of those roots $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ such that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \ristretto_{T \cap H} \in I$ and set
$$\Phi_I= \mathbb Z \Psi_I \cap \Phi.$$
This is a parabolic subsystem of $\Phi$ which is explicitly described once $H$ is described in terms of active roots following Avdeev's classification, and the intersections $\Phi^\pm_I = \Phi_I \cap \Phi^\pm$ define a subdivision of $\Phi_I$ into positive and negative roots.

Given $w \in W$ and $I \subset \mathscr D^*$, we say that $(w,I)$ is a \textit{reduced pair} if $w(\Phi^+_I) \subset \Phi^-$. Our first main result is the following.

\begin{theorem-var}[Corollary~\ref{cor:parametrizzazione}]\label{thm-var:bij}
Let $H$ be a spherical subgroup of $G$ contained in $B$, there is a natural bijection between the set of reduced pairs and the set of $B$-orbits in $G/H$.
\end{theorem-var}

Given a reduced pair $(w,I)$ we denote by $\mathscr O_{w,I}$ the corresponding $B$-orbit in $G/H$. Reduced pairs encode important properties of the corresponding $B$-orbit, and they can be used to encode combinatorial properties of the entire set $\mathscr B(G/H)$ as well. Denote by $W_I$ the Weyl group of $\Phi_I$, canonically embedded in $W$. Our next main result describes in these terms the action of $W$ on $\mathscr B(G/H)$.

\begin{theorem-var}[Theorems~\ref{teo:stabilizzatori1} ~\ref{teo:stabilizzatori2} and Corollary~\ref{cor:numero-Borbite}]\label{thm-var:W}
Let $H$ be a spherical subgroup of $G$ contained in $B$ and let $(w,I)$ be a reduced pair, then $\Stab_W(\mathscr O_{w,I}) = w W_I w^{-1}$. Moreover two orbits $\mathscr O_{w,I}$ and $\mathscr O_{v,J}$ are in the same $W$-orbit if and only if $I = J$, in which case $\mathscr O_{w,I} = wv^{-1} \cdot \mathscr O_{v,I}$. In particular, the following formula holds:
$$
	|\mathscr B(G/H)| = \sum_{I \subset \mathscr D^*} |W/W_I|.
$$
\end{theorem-var}

A very special example of a strongly spherical subgroup was treated by Timashev in \cite{Ti1}, where the corresponding set of $B$-orbits is studied. More precisely, let $U'$ be the derived subgroup of $U$, namely $U' = \prod_{\Phi^+ \senza \grD} U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, and consider the subgroup $TU' \subset G$: this is a spherical subgroup of $G$ contained in $B$, and we have equalities $\mathscr D^* = \Psi = \grD$. In this case the parametrization of the $B$-orbits in terms of reduced pairs can be proved in a simple way by using the commutation relations among root subgroups, and the set of reduced pairs $(w,I)$ is easily seen to be in a $W$-equivariant bijection with the set of faces of the weight polytope associated with any given dominant regular weight of $G$. This elegant description generalizes to the case of any strongly solvable spherical subgroup as follows.

Let $\grl$ be a regular dominant weight and let $P = \conv(W\grl)$ be the associated weight polytope in $\grL_\mQ = \Lambda \otimes_\mathbb Z \mQ$, where $\Lambda$ denotes the weight lattice of $G$. Since $\grl$ is regular, the elements $w\grl$ with $w \in W$ are all distinct and coincide with the vertices of $P$. By a \textit{subpolytope} of $P$ we mean the convex hull of a subset of vertices of $P$. Notice that $P$ is naturally endowed with an action of $W$ which permutes the subpolytopes of $P$, and we denote by $\mathscr S(P)$ the set of subpolytopes of $P$.

Given a spherical subgroup $H \subset G$ contained in $B$, to any reduced pair $(w,I)$ we may associate a subpolytope of $P$ by setting $\mathscr S_{w,I} = \conv(wW_I\rho)$. This enables us to reformulate our combinatorial model of $\mathscr B(G/H)$ as follows.

\begin{theorem-var}[Theorem~\ref{teo:subpolytopes}]\label{teo:injS}
Let $H$ be a spherical subgroup of $G$ contained in $B$, then the map $(w,I) \longmapsto \mathscr S_{w,I}$ defines a $W$-equivariant injective map $\mathscr B(G/H) \longrightarrow \mathscr S(P)$. Moreover, $\mathscr S_{w,I}$ is described as the intersection of $P$ with a cone in $\grL_\mQ$ as follows
$$
	\mathscr S_{w,I} = P \cap \big(w\grl+\mQ_{\geq0}(w(\Phi^-_I)) \big).
$$
\end{theorem-var}

The map $(w,I) \longmapsto \mathscr S_{w,I}$ is not surjective in general, but applying the last part of the previous theorem it is possible in any given example to describe explicitly the image of the map. Moreover, this map has the advantage of being compatible with the Bruhat order, i.e.\ the inclusion relation between $B$-orbit closures: if $(v,J)$ and $(w,I)$ are reduced pairs such that $\mathscr S_{v,J} \subset \mathscr S_{w,I}$, then $\mathscr O_{v,J} \subset \ol{\mathscr O_{w,I}}$ (see Proposition~\ref{prop: bruhat-compatibile}). Unfortunately the converse of the previous statement is false in general, and a complete description of the Bruhat order of $\mathscr B(G/H)$ remains an open problem.

In case $H = TU'$ is easy to see that the image of the map $(w,I) \longmapsto \mathscr S_{w,I}$ equals the set of faces of $P$. It was conjectured by Knop that $G/TU'$ has the largest number of $B$-orbits among all the spherical homogeneous spaces for $G$. Using our formula for the cardinality of $\mathscr B(G/H)$ we prove this in the solvable case.

\begin{theorem-var} [Theorem~\ref{teo:knop-conj}]\label{thm-var:knopconj}
The spherical homogeneous space $G/TU'$ has the largest number of $B$-orbits among the homogeneous spherical varieties $G/H$ with $H$ a solvable subgroup of $G$.
\end{theorem-var}

We explain now the structure of the paper. In Section~\ref{s:preliminaries} we explain our notations and collect some basic facts about spherical varieties and toric varieties. Then we restrict to the case of a spherical subgroup $H \subset G$ contained in $B$, and in Section~\ref{s:toricvar} we study the variety $B/H$: this is an affine toric $T$-variety whose set of $T$-stable prime divisors is naturally parametrized by $\mathscr D^*$, and whose set of $T$-orbits is naturally parametrized by the subsets of $\mathscr D^*$. In Section~\ref{s:active} we introduce the notion of \textit{weakly active root}. These are the roots $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$ whose associated root subgroup $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset B$ acts non-trivially on $B/H$, and we use them to attach a root system $\Phi_I$ to every subset $I \subset \mathscr D^*$, thus to every $T$-orbit in $B/H$. In Section~\ref{s:Borbits} we introduce the notion of reduced pair (and the analogous one of \textit{extended pair}), and we prove Theorem~\ref{thm-var:bij}. Finally, in Section~\ref{s:W} we study the action of $W$ on $\mathscr B(G/H)$ and we prove Theorems~\ref{thm-var:W}, \ref{teo:injS} and \ref{thm-var:knopconj}.\\

\textit{Acknowledgements.} We thank P.~Bravi, M.~Brion, F.~Knop and A.~Maffei for useful conversations on the subject, and especially R.S.~Avdeev for numerous remarks and suggestions on previous versions of the paper which led to significative improvements.

This work originated during a stay of the first named author in Friedrich-Alexander-Universit\"at Erlangen-N\"urnberg during the fall of 2012 partially supported by a DAAD fellowship, he is grateful to F.~Knop and to the Emmy Noether Zentrum for hospitality. Both the authors were partially supported by the DFG Schwerpunktprogramm 1388 -- Darstellungstheorie.

\section{Notations and preliminaries}\label{s:preliminaries}

\subsection{Generalities}

All varieties and algebraic groups that we will consider will be defined over the complex numbers. Let $G$ be a connected reductive algebraic group. Given a subgroup $K \subset G$, we denote by $\mathcal X} \newcommand{\calU}{\mathcal U(K)$ the group of characters of $K$ and by $K^\mru$ the unipotent radical of $K$. The Lie algebra of $K$ will be denoted either by $\Lie K$ or by the corresponding fraktur letter (here $\mathfrak k$). Given $g \in G$ we set $K^g = g^{-1}Kg$ and ${}^g\!K = gKg^{-1}$. If $K$ acts on an algebraic variety $X$, we denote by $\Div^K(X)$ the set of $K$-stable prime divisors of $X$. If $S$ is a group acting on an algebraic variety $X$ and if $x \in X$, we denote by $\Stab_S(x)$ the stabilizer of $x$ in $S$.  If $M$ is a lattice (that is, a free and finitely generated abelian group), the dual lattice is $M^\vee = \Hom_{\mathbb Z}(M,\mathbb Z)$, and the corresponding $\mQ$-vector space $M \otimes_{\mathbb Z} \mQ$ is denoted by $M_\mQ$.

Let $B$ be a connected solvable group and let $T \subset B$ be a maximal torus. The \textit{weight lattice} of $Z$ is the lattice
$$
\mathcal X} \newcommand{\calU}{\mathcal U_B(Z) = \{\text{$B$-weights of non-zero rational $B$-eigenfunctions } f \in \mathbb C} \newcommand{\mN}{\mathbb N(Z) \},
$$
and the \textit{rank} of $Z$ is by definition the rank of its weight lattice. When the acting group is clear from the context, we will drop the subscript and write simply $\mathcal X} \newcommand{\calU}{\mathcal U(Z)$. By $\mathbb C} \newcommand{\mN}{\mathbb N(Z)^{(B)}$ we will denote the set of non-zero rational $B$-eigenfunctions of $Z$.

Given a $B$-homogeneous variety $Z$ and $z \in Z$, we say that $z$ is a \textit{standard base point} (with respect to $T$) if $\Stab_T(z)$ is a maximal diagonalizable subgroup of $\Stab_B(z)$. Notice that standard base points always exist, and that if $z \in Z$ is standard then every $z' \in Tz$ is standard as well.

\begin{lemma}	\label{lemma:B-rank}
Let $Z$ be a $B$-homogeneous variety and let $z_0 \in Z$ be a standard base point, then $T z_0$ is a closed $T$-orbit. If moreover $H = \Stab_B(z_0)$, then $H = (T \cap H) H^\mru$ and the followings equalities hold:$$\mathcal X} \newcommand{\calU}{\mathcal U_B(Z) = \mathcal X} \newcommand{\calU}{\mathcal U(B)^H = \mathcal X} \newcommand{\calU}{\mathcal U_T(Tz_0)$$
(where $\mathcal X} \newcommand{\calU}{\mathcal U(B)^{H}$ denotes the subgroup of $\mathcal X} \newcommand{\calU}{\mathcal U(B)$ of characters that are trivial on $H$).
\end{lemma}

\begin{proof}
For all $z \in Z$ the stabilizer $\Stab_T(z)$ is a diagonalizable subgroup of $\Stab_B(z)$. On the other hand $Z$ is $B$-homogeneous, therefore all the stabilizers $\Stab_B(z)$ are isomorphic and the maximal dimension for $\Stab_T(z)$ is the dimension of a maximal torus of $\Stab_B(z)$. By definition $\Stab_T(z_0)$ contains a maximal torus of $\Stab_B(z)$, therefore $T z_0$ has minimal dimension in $Z$, hence it is closed. The last claim is immediate.
\end{proof}

From now on $B$ will denote a Borel subgroup of $G$, $T\subset B$ a maximal torus, and $U = B^\mru$ the unipotent radical of $B$. Let $\Phi \subset \mathcal X} \newcommand{\calU}{\mathcal U(T)$ be the root system of $G$ associated with $T$, $\Phi^+$ (resp.\ $\Phi^-$) the set of positive (resp.\ negative) roots determined by $B$ and $\grD \subset \Phi^+$ the corresponding set of simple roots. When dealing with an explicit irreducible root system we will enumerate the simple roots following Bourbaki's notation \cite{Bou}. We also set $\grD^- = -\grD$.

If $\grb \in \Phi^+$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$ we denote by $[\grb:\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma]$ the coefficient of $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ in $\grb$ as a sum of simple roots, and we define the \textit{support} of $\grb$ as
$$
	\supp(\grb) = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD \st [\grb:\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma] > 0\}.
$$

Let $W = N_G(T)/T$ be the Weyl group of $G$ with respect to $T$. If $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi$, we denote by $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in W$ the corresponding reflection and by $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset G$ the unipotent root subgroup associated with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$. If $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$, we denote by $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ the minimal parabolic subgroup of $G$ containing $B$ associated with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$. If $w \in W$, we denote by $\Phi^+(w)$ the corresponding \textit{inversion set}, i.e.
$$
	\Phi^+(w) = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \st w(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \Phi^-\},
$$
and by $l(w)$ the length of $w$, that is the cardinality of $\Phi^+(w)$. Denote by $w_0$ be the longest element of $W$. If $\mathscr O$ is a $T$-stable subset of a $G$-variety and $n\in N_G(T)$, then $n\mathscr O$ only depends on the class $w$ of $n$ in $W$, therefore we will denote $n\mathscr O$ simply by $w\mathscr O$.

\subsection{Spherical varieties and toric varieties}	\label{ssec:spherical}

An irreducible normal $G$-variety $X$ is called \textit{spherical} if it contains an open $B$-orbit. See \cite{Kn1} as a general reference for spherical varieties. In particular, $X$ contains an open $G$-orbit, which is a spherical homogeneous variety. Following \cite{Br1}, \cite{Po}, \cite{Vi}, $X$ is spherical if and only if $B$ possesses finitely many orbits on it. Therefore, following the definition we gave in the introduction, a homogeneous variety $G/H$ is spherical if and only if $H$ is a spherical subgroup of $G$.

Let $X$ be a spherical $G$-variety. Every $B$-stable prime divisor $D \in \Div^B(X)$ induces a discrete valuation $\nu_D$ on $\mathbb C} \newcommand{\mN}{\mathbb N(X)$, which is trivial on the constant functions. On the other hand, since $X$ contains an open $B$-orbit, every function $f \in \mathbb C} \newcommand{\mN}{\mathbb N(X)^{(B)}$ is uniquely determined by its weight up to a scalar factor. By restricting valuations to $\mathbb C} \newcommand{\mN}{\mathbb N(G/H)^{(B)}$ we get then a map
$$
	\rho : \Div^B(X) \lra \mathcal X} \newcommand{\calU}{\mathcal U(X)^\vee,
$$
defined by $\langle \rho(D), \chi \rangle = \nu_D(f_\chi)$ where $f_\chi \in \mathbb C} \newcommand{\mN}{\mathbb N(X)^{(B)}$ is any non-zero $B$-semiinvariant function of weight $\chi$.

When $G = B = T$ is a torus, we will also say that $X$ is a \textit{toric variety}. Notice that, differently form the standard literature on toric varieties, we will not assume that the action of $T$ on $X$ is effective. As a general reference on toric varieties see \cite{CLS}.

Let $X$ be an affine toric $T$-variety. The \textit{cone of} $X$ is the rational polyhedral cone in $\mathcal X} \newcommand{\calU}{\mathcal U(X)^\vee_\mQ$ defined as
$$
	\grs_X = \cone(\rho(D) \st D \in \Div^T(X))
$$
This cone encodes important information on the geometry of $X$. In particular, there is an order reversing one-to-one correspondence between the set of $T$-orbits on $X$ (ordered with the inclusion of orbit closures) and the set of faces of $\grs_X$ (ordered with the inclusion). Given a face $\tau$ of $\grs_X$ we denote by $\mathscr U_\tau$ the corresponding $T$-orbit in $X$, in particular $\mathscr U_\grs$ is the unique closed orbit of $X$ and $\mathscr U_0$ is the open orbit of $X$.

\begin{definition}[{see \cite[Definition 2.3]{Ll}}]	\label{def:radici-toriche}
Let $X$ be an affine toric $T$-variety and let $\alpha\in \mathcal X} \newcommand{\calU}{\mathcal U(X)$. Then $\alpha$ is a \textit{root} of $X$ if there exists $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \Div^T(X)$ such that $\langle \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \alpha \rangle = -1$, and $\langle \rho(D), \alpha \rangle \geq 0$ for all $D \in \Div^T(X) \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}$.
\end{definition}

We will denote by $\Root(X)$ the set of roots of an affine toric variety $X$. Notice that by its definition $\Root(X)$ comes with a map
\begin{equation}	\label{eqn:mappa-delta-torica}
\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma : \Root(X) \lra \Div^T(X).
\end{equation}

For the notion of root of a toric variety, which goes back to Demazure, see also \cite[Definition 2.1]{AZK} and \cite[Proposition 3.13]{Oda}. Following \cite{De} and \cite{Ll}, there is a one-to-one correspondence between the roots of $X$ and the one parameter unipotent subgroups of $\Aut(X)$ normalized by $T$, see \cite[\S 2]{Ll} and \cite[\S 2]{AZK}. More precisely, every $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Root(X)$ defines a locally nilpotent derivation $\partial_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ of the graded algebra $\mathbb C} \newcommand{\mN}{\mathbb N[X]$, which acts on $\mathbb C} \newcommand{\mN}{\mathbb N[X]$ by the rule
\begin{equation}		\label{eqn:rooot-action}
\partial_\alpha(f_\chi) = \langle \rho (\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \chi \rangle f_{\chi + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma}
\end{equation}
where $\chi \in \mathcal X} \newcommand{\calU}{\mathcal U(X)$ and where $f_\chi \in \mathbb C} \newcommand{\mN}{\mathbb N(X)^{(T)}$ has weight $\chi$. Notice that the line $\mathbb C} \newcommand{\mN}{\mathbb N \partial_\alpha$ is fixed by the action of $T$, which acts on it via the weight $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$. By exponentiating $\partial_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, we get then a one parameter unipotent subgroup $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset \Aut(Z)$ normalized by $T$, that is $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grl_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma(\mathbb C)$ where $\grl_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ denotes the one parameter subgroup $\xi \longmapsto \exp(\xi \partial_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.

We gather in the following proposition some properties, that we will need later, of the action of the group $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ on the $T$-orbits of $X$.

\begin{proposition} [{\cite[Lemma 2.1, Proposition 2.1 and Lemma 2.2]{AZK}}] \label{prop:sottovarietà toriche stabili}
Let $X$ be an affine toric $T$-variety, let $\tau, \tau'$ be faces of $\grs_X$ and let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Root(X)$.
\begin{itemize}
	\item[i)] $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr U_\tau$ decomposes at most in the the union of two $T$-orbits.
	\item[ii)] $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ is the unique $T$-stable prime divisor of $X$ which is not stable under the action of $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$.
	\item[iii)] Suppose that $\mathscr U_{\tau'} \subset \ol{\mathscr U_\tau}$, then $\mathscr U_{\tau'} \subset V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr U_{\tau}$ if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \ristretto_{\tau'} \leq 0$ and $\tau$ is the codimension one face of $\tau'$ defined by the equation $\tau = \tau' \cap \ker \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$.
\end{itemize}
\end{proposition}

\section{Strongly solvable spherical subgroups and\\ associated toric varieties}\label{s:toricvar}

From now on, if not differently stated, $H$ will be a strongly solvable spherical subgroup of $G$. Up to conjugating $H$ in $G$ we may assume that $H$ is contained in $B$, in which case $H^\mru = U \cap H$. Up to conjugating $H$ in $B$ we may and will also assume that $T_H = T \cap H$ is a maximal diagonalizable subgroup of $H$. By Lemma~\ref{lemma:B-rank} it follows then that $TH/H \simeq T/T_H$ is a closed $T$-orbit in $B/H$, and that $\mathcal X} \newcommand{\calU}{\mathcal U_B(B/H) = \mathcal X} \newcommand{\calU}{\mathcal U_T(T/T_H)$. In particular we get the equality $\rk B/H = \rk G - \rk H$.

In order to classify the strongly solvable spherical subgroups $H$ of $G$, Avdeev introduced the \textit{active roots} of $H$, defined as
$$\Psi_H = \{ \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \st U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \not \subset H^\mru \},$$
and developed a combinatorial theory of such roots. We refer to \cite{Avd1}, \cite{Avd2} for details on this construction. When no confusion is possible, we will simply write $\Psi$ instead of $\Psi_H$.

\subsection{The structure of $B/H$ as a toric variety}\label{s:BmodH}

Consider the projection $G/H \lra G/B$, denote by $\mathscr B^*(G/H)$ the set of the $B$-orbits in $G/H$ which project dominantly on $G/B$ and by $\mathscr D^*_H$ the set of the $B$-stable prime divisors of $G/H$ which project dominantly on $G/B$. When no confusion is possible, we will simply write $\mathscr D^*$ instead of $\mathscr D^*_H$. Since $G/H$ possesses finitely many $B$-orbits, $\mathscr D^*$ equals the set of closures of the codimension one $B$-orbits in $\mathscr B^*(G/H)$.

\begin{proposition}	\label{prop:corrispondenza-divisori}
The map $\mathscr O \longmapsto w_0^{-1} \mathscr O \cap B/H$ induces a bijection between $\mathscr B^*(G/H)$ and the set of $T$-orbits in $B/H$, which preserves codimensions and inclusions of orbit closures. In particular, the map $D \longmapsto w_0^{-1} D \cap B/H$ induces a bijection between $\mathscr D^*$ and $\Div^T(B/H)$.
\end{proposition}

\begin{proof}
Let $\mathscr O \in \mathscr B^*(G/H)$. Then the projection of $\mathscr O$ in $G/B$ is the dense $B$-orbit $Bw_0B/B$, therefore there exists $u \in B$ such that $\mathscr O = Bw_0 uH/H$. Then
$$w_0^{-1} \mathscr O \cap B/H = (B \cap B^{w_0}) u H/H = T uH/H,$$
that is, $w_0^{-1} \mathscr O \cap B/H$ is a single $T$-orbit. Conversely, if $\mathscr U \subset B/H$ is a $T$-orbit, then we get an element $\mathscr O \in \mathscr B^*(G/H)$ by setting $\mathscr O = Bw_0 \mathscr U$. The equalities $\mathscr O = Bw_0(w_0^{-1} \mathscr O \cap B/H)$ and $\mathscr U = w_0^{-1} (Bw_0\mathscr U) \cap B/H$ imply the first claim, and the rest is an obvious consequence.
\end{proof}

\begin{corollary}	\label{cor:torica}
$B/H$ is a smooth affine toric $T$-variety.
\end{corollary}

\begin{proof}
Being homogeneous under the action of a solvable group, $B/H$ is smooth and affine (see e.g. \cite[Lemma 2.12]{Ti1}). Notice also that $B/H$ is irreducible since $B$ is connected. Therefore the fact that $B/H$ is toric under the action of $T$ follows by the sphericity of $H$ thanks to Proposition~\ref{prop:corrispondenza-divisori}, since $\mathscr B^*(G/H)$ is a finite set.
\end{proof}

\begin{proposition}\label{prop:B/H-toric}
\begin{itemize}
	\item[i)] The weight lattice of $B/H$, as a toric $T$-variety, is equal to $w_0\mathcal X} \newcommand{\calU}{\mathcal U(G/H)$.
	\item[ii)] For any $D\in \mathscr D^*$, the $T$-invariant valuation of $B/H$ defined by the $T$-stable prime divisor $w_0 D\cap B/H$ coincides with $w_0 \rho(D)$.
\end{itemize}
\end{proposition}

\begin{proof}
Up to twist the $T$-action by $w_0$, the $T$-varieties $B/H$ and $w_0B/H$ are isomorphic. Since the $B$-orbit of $w_0B/B$ is dense in $G/B$ and since $B\cap B^{w_0}=T$, the $T$-stable prime divisors of $w_0 B/H$ coincide with the intersections $D \cap w_0 B/H$, where $D \in \mathscr D^*$. To prove the proposition we may then replace $B/H$ with $w_0 B/H$, and show instead the equality $\mathcal X} \newcommand{\calU}{\mathcal U_T(w_0 B/H) = \mathcal X} \newcommand{\calU}{\mathcal U_B(G/H)$ and that the $T$-invariant valuation of $w_0B/H$ defined by $D\cap w_0B/H$ is $\rho(D)$.

Consider the open subset $Bw_0B/H \subset G/H$, and notice that we have an isomorphism
$Bw_0B/H \simeq B\times^T w_0 B/H$. Then restriction to $w_0 B/H$ induces a bijection between $B$-semiinvariant rational functions on $G/H$ and $T$-semiinvariant rational functions on $w_0 B/H$, and i) follows.

Identifying $\mathbb C} \newcommand{\mN}{\mathbb N(G/H)^{(B)}$ and $\mathbb C} \newcommand{\mN}{\mathbb N(w_0B/H)^{(T)}$, we get as well an identification between the discrete valuation of $\mathbb C} \newcommand{\mN}{\mathbb N(G/H)$ associated with a $B$-stable prime divisor $D \subset G/H$ which intersects $Bw_0B/H$ (hence $w_0 B/H$) with the discrete valuation of $\mathbb C} \newcommand{\mN}{\mathbb N(w_0 B/H)$ associated with $D\cap w_0 B/H$, and ii) follows.
\end{proof}

The set of active roots of $H$ is closely related to the structure of $B/H$ as a toric variety. Consider indeed the Levi decompositions $B =TU \simeq T \ltimes U$ and $H = T_HH^\mru \simeq T_H \ltimes H^\mru$. These induce a projection $B/H \lra T/T_H$, and a $T$-equivariant isomorphism
\begin{equation}		\label{eqn:fiber-product}
	B/H \simeq T\times^{T_H} U/H^\mru.
\end{equation}
In other words, $B/H$ is a homogeneous vector bundle over $T/T_H$, with fiber the $T_H$-module $U/H^\mru$. Therefore the $T$-orbits in $B/H$ correspond naturally to the $T_H$-orbits in $U/H^\mru$, and it follows that $U/H^\mru$ possesses an open $T_H$-orbit.

Denote $\tau : \mathcal X} \newcommand{\calU}{\mathcal U(T) \lra \mathcal X} \newcommand{\calU}{\mathcal U(T_H)$ the restriction map. Since $T_H$ is a diagonalizable group, the $T_H$-module structure of $U/H^\mru$ is completely determined by its $T_H$-weights, which are the weights $\tau(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$. The fact that $U/H^\mru$ possesses an open $T_H$-orbit is then equivalent to the fact that it is a {\em multiplicity-free} $T_H$-module, that is, the set $\tau(\Psi)$ is linearly independent. Given $\pi \in \mathcal X} \newcommand{\calU}{\mathcal U(T_H)$, we denote by $\mathbb C} \newcommand{\mN}{\mathbb N_\pi$ the one dimensional $T_H$-module defined by $\pi$. By \cite[Lemma 1.4]{Mo} the exponential map induces a $T_H$-equivariant isomorphism $\Lie(U)/\Lie(H^\mru) \lra U/H^\mru$, hence we get isomorphisms of $T_H$-modules
$$
	U/H^\mru \simeq \Lie(U)/\Lie(H^\mru) \simeq \bigoplus_{\pi \in \tau(\Psi)} \mathbb C} \newcommand{\mN}{\mathbb N_\pi.
$$

In particular, the set $\tau(\Psi)$ parametrizes the set $\Div^{T_H}(U/H^\mru)$, and summarizing the previous discussion we get canonical bijections
\begin{equation}		\label{eqn:identificazioni-divisori}
	\mathscr D^* \longleftrightarrow \Div^T(B/H) \longleftrightarrow \Div^{T_H}(U/H^\mru) \longleftrightarrow \tau(\Psi).
\end{equation}

Notice that $\Psi \subset \mathcal X} \newcommand{\calU}{\mathcal U_T(B/H)$, indeed by (\ref{eqn:fiber-product}) for every $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ there is a $T$-eigenfunction $f_{-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma} \in \mathbb C} \newcommand{\mN}{\mathbb N[B/H]$ of weight $-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$.
Therefore $\Psi \subset \mathcal X} \newcommand{\calU}{\mathcal U_T(B/H) = w_0 \mathcal X} \newcommand{\calU}{\mathcal U(G/H)$. Since $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \not \subset H$, if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ it follows that $\Aut(B/H)$ contains a one parameter unipotent subgroup $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \simeq U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ which is normalized by $T$ and such that $\Lie(V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ is a one dimensional $T$-module of weight $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$. Since such one parameter subgroups are parametrized by the roots of $B/H$, it follows the inclusion $\Psi \subset \Root(B/H)$.

The same argument applies to a more general notion of root.

\begin{definition}
A root $\alpha \in \Phi^+$ is called \textit{weakly active} for $H$ if $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ acts non-trivially on $B/H$.
\end{definition}

We denote by $\Psi^\mathrm{\sharp}_H \subset \Phi^+$ the set of weakly active roots of $H$, when no confusion is possible we will simply write $\Psi^\mathrm{\sharp}$ instead of $\Psi^\mathrm{\sharp}_H$.  Clearly $\Psi \subset \Psi^\mathrm{\sharp}$, and reasoning as above we also have an inclusion $\Psi^\mathrm{\sharp} \subset \Root(B/H)$. Following Definition~\ref{def:radici-toriche} we get then by restriction a map $$\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma\ristretto_{\Psi^\mathrm{\sharp}} : \Psi^\mathrm{\sharp} \lra \Div^T(B/H).$$
By regarding $B/H \simeq T \times^{T_H} U/H^\mru$ as a homogeneous vector bundle over $T/T_H$, it follows that the evaluation of a weakly active root $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ on a divisor $D \in \Div^T(B/H)$ coincides with the evaluation of its restriction $\tau(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \mathcal X} \newcommand{\calU}{\mathcal U_{T_H}(U/H^\mru)$ along the divisor $D \cap U/H^\mru \in \Div^{T_H}(U/H^\mru)$. In particular $\tau(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ is a root of the toric variety $U/H^\mru$ for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, and the divisor $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ is uniquely determined by $\tau(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.

We will need the following characterization of the active roots in terms of the corresponding evaluation.

\begin{theorem} [{\cite[Theorem 5.34 (b)]{Avd2}}]  \label{teo:equivalent}
Let $\alpha\in \mN\Phi^+$ and assume that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \mathbb Z \Psi$. Then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ if and only if there exists $D_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \mathscr D^*$ such that
\[
	\langle w_0\rho(D), \alpha \rangle =
	\left\{ \begin{array}{rl}
			-1 & \text{ if } D = D_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma		\\
			0 & \text{ if } D \in \mathscr D^* \senza \{D_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\}
	\end{array}\right..
\]
\end{theorem}

In the notation of the previous theorem, notice that if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\in \Psi$ then the $B$-stable divisor $D_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \mathscr D^*$ is the divisor corresponding to $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \Div^T(B/H)$ under the natural bijection between $\mathscr D^*$ and $\Div^T(B/H)$. To summarize:

\begin{corollary}	\label{cor:equivalent}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$. Then $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = w_0 D_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cap B/H$, and the $T$-eigenfunction $f_{-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma} \in \mathbb C} \newcommand{\mN}{\mathbb N[B/H]$ of weight $-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is a global equation for $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{corollary}

Given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, it follows by the theory of the automorphisms of affine toric varieties that we may define the $T$-stable divisor $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ in terms of the natural action of $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ on $B/H$.

\begin{lemma} \label{lem:divisori-stabili}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, then $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \Div^T(B/H)$ is the unique divisor which is not $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$-stable. If moreover $\langle w_0 \rho (D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \leq 0$ for all $D \in \Div^T(B/H)$, then $u \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \vuoto$ for all non-trivial elements $u \in U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$.
\end{lemma}

\begin{proof}
By definition, the multiplication induces a non-trivial action of $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ on $B/H$. In particular, considering the weight of $T$ acting on $\Lie(U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, it follows that the morphism of algebraic groups $B \lra \Aut(B/H)$ maps $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ onto the one parameter unipotent subgroup $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset \Aut(B/H)$, determined by $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ as a root of $B/H$. Proposition~\ref{prop:sottovarietà toriche stabili} implies then that $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \Div^T(B/H)$ is the unique $T$-stable divisor which is not $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$-stable.

Suppose that $\langle w_0 \rho (D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \leq 0$ for all $D \in \Div^T(B/H)$. Since $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Root(B/H)$, it follows that $\langle w_0 \rho (D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = 0$ for all $D \in \Div^T(B/H) \senza \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, and $\langle w_0 \rho (\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = -1$. If $f_{-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma} \in \mathbb C} \newcommand{\mN}{\mathbb N[B/H]$ is any non-zero $T$-eigenfunction of weight $-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, it follows that $f_{-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma}$ is a global equation for $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ in $B/H$.

Let $\partial_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ be the locally nilpotent derivation of $\mathbb C} \newcommand{\mN}{\mathbb N[B/H]$ associated to $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ as a root of $B/H$. By (\ref{eqn:rooot-action}) it follows that $\partial_\alpha(f_{-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma})$ is a non-zero function, which is constant on $B/H$. Exponentiating $\partial_\alpha$ we obtain $f_{-\alpha}(u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma(\xi)x) = f_{-\alpha}(x)+\xi$ for all $\xi \in \mathbb C} \newcommand{\mN}{\mathbb N$ (where $\mathbb C} \newcommand{\mN}{\mathbb N$ is regarded as the $T$-module of weight $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ and where we fix a $T$-equivariant isomorphism $u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma : \mathbb C} \newcommand{\mN}{\mathbb N \lra U_\alpha$), and the second claim follows.
\end{proof}

\begin{corollary}	\label{cor:weak-vs-active}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ if and only if $\langle w_0 \rho (D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \leq 0$ for all $D \in \Div^T(B/H)$.
\end{corollary}

\begin{proof}
If $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$, then the claim follows by Theorem \ref{teo:equivalent}. Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and suppose that $\langle w_0 \rho (D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \leq 0$ for all $D \in \Div^T(B/H)$. By the previous lemma it follows that $u \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \vuoto$ for all non-trivial element $u \in U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$. In particular, since $B/H$ contains a unique closed orbit, it follows that $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma T H \not \subset T H$, that is $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma H \neq H$. Therefore $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \not \subset H$, and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$.
\end{proof}

\subsection{Orbits of $T$ on $B/H$ via active roots}

Being affine, the toric variety $B/H$ possesses a unique closed orbit, namely $TH/H$, which is contained in every $T$-stable divisor of $B/H$. Since $B/H$ is smooth its associated cone is simplicial, therefore the $T$-orbits in $B/H$ are parametrized the subsets of $\Div^T(B/H)$. By (\ref{eqn:identificazioni-divisori}), these correspond to the subsets of $\mathscr D^*$, and given $I \subset \mathscr D^*$ we denote by $\mathscr U_I$ the corresponding $T$-orbit in $B/H$. In other words, $\mathscr U_I$ is defined by the equality
$$
	I = \{D \in \mathscr D^* \st \mathscr U_I \not \subset w_0 D\}.
$$
In particular, $\mathscr U_\vuoto$ is the closed orbit of $B/H$, and $\mathscr U_{\mathscr D^*}$ is the open orbit. The goal of this subsection is to describe explicitly the $T$-orbits $\mathscr U_I$, by giving canonical base points defined in terms of the active roots of $H$.

\begin{proposition}	\label{prop:Torb_onD}
Fix an enumeration $\Div^T(B/H) = \{D_1, \ldots, D_m\}$, fix some indices $1 \leq i_1 < \ldots < i_p \leq m$ and set $I = \{D_{i_1}, \ldots, D_{i_p}\}$. For every choice of elements $\grb_1, \ldots, \grb_p \in \Psi$ with $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_1) = D_{i_1}, \ldots, \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_p) = D_{i_p}$ and for every choice of non-trivial elements $u_1 \in U_{\grb_1}, \ldots, u_p \in U_{\grb_p}$, it holds
\[
	\mathscr U_I = T u_1 \cdots u_p H.
\]
\end{proposition}

\begin{proof}
Denote $\mathscr U = T u_1 \cdots u_p H$. The point $u_1 \cdots u_p H$ is obtained by successively applying the elements $u_j$ on the point $H \in B/H$, which is contained in every $T$-stable prime divisor of $B/H$. By Lemma~\ref{lem:divisori-stabili} the root group $U_{\beta_j}$ stabilizes every $D \in \Div^T(B/H) \senza \{\delta(\beta_j)\}$. Therefore the definition of $\grb_1, \ldots, \grb_p$ implies that $\mathscr U_I$ is contained in every $D$ belonging to $\Div^T(B/H) \senza \{ D_{i_1}, \ldots, D_{i_p} \}$.

It remains to prove that $\mathscr U_I$ is not contained in $\delta(\beta_j)$ for all $j= 1, \ldots, p$. By Lemma~\ref{lem:divisori-stabili} it follows that $\delta(\beta_j)$ is stable under the action of $U_{\beta_k}$ for all $k \neq j$, whereas $u_j \delta(\beta_j) \cap \delta(\beta_j) = \vuoto$. It follows that $u_{j+1} \cdots u_p H \in \delta(\beta_j)$ and $u_j \cdots u_p H \not \in \delta(\beta_j)$. Therefore $u_1 \cdots u_p H \not \in \delta(\beta_j)$ as well, and the claim follows.
\end{proof}

We deduce immediately from Proposition~\ref{prop:Torb_onD} the following description of the $T$-stable prime divisors of $B/H$ in terms of active roots.

\begin{corollary} \label{cor:divisors}
With the notation of Proposition~\ref{prop:Torb_onD}, it holds
$$
	D_i = \ol{T U_{\grb_1} \cdots \hat{U}_{\grb_i} \cdots U_{\grb_m} H}.
$$
Moreover, $D_i = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_i)$ is the unique element in $\Div^T(B/H)$ which  is not stable under the action of $U_{\beta_i}$.
\end{corollary}

In particular, $\Div^T(B/H)$ is identified with $\tau(\Psi)$, and we get two canonical maps
$$
\Psi \lra \Div^T(B/H),
$$
namely the one defined by $\tau : \Psi \lra \tau(\Psi)$ and the one obtained by restricting the map
$$
\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma : \Root(B/H) \lra \Div^T(B/H).
$$
The following proposition shows that these two maps are actually the same.

\begin{proposition}	\label{prop:delta-vs-tau}
Let $\alpha,\beta\in \Psi$, then we have $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)$ if and only if $\tau(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \tau(\grb)$. In particular, the map $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma\ristretto_\Psi : \Psi \lra \Div^T(B/H)$ is surjective.
\end{proposition}

\begin{proof}
Denote $C_\alpha = TU_\alpha H/H$, and notice that $C_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is both $T$-stable and $U_\alpha$-stable. By Proposition~\ref{prop:Torb_onD}, $C_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ decomposes in two $T$-orbits. Since $TH/H$ is the unique closed $T$-orbit in $B/H$, it follows that $C_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is a closed subvariety of $B/H$, equal to the intersection of all the $T$-stable prime divisors of $B/H$ but $\delta(\alpha)$.

Since $B/H$ is a homogeneous vector bundle over $TH/H \simeq T/T_H$ with fiber equal to $UH/H \simeq U/H^\mru$, it follows that $C_\alpha$ is a line bundle over $T/T_H$ with fiber $L = U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma H/H$, and $T_H$ acts on $L$ via the restriction $\tau(\alpha)$. On the other hand, Corollary~\ref{cor:divisors} shows that $L$ is transversal to $\delta(\alpha)\cap U/H^\mru$, which is a $T_H$-stable hyperplane of $U/H^\mru = \bigoplus_{\tau(\Psi) }\mathbb C} \newcommand{\mN}{\mathbb N_\pi$. Therefore
\[
\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap U/H^\mru = \bigoplus_{\substack{ \pi\in\tau(\Psi), \\ \pi\neq\tau(\alpha) }}\mathbb C} \newcommand{\mN}{\mathbb N_\pi,
\]
and the claim follows by noticing that $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)$ if and only if $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap U/H^\mru = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb) \cap U/H^\mru$.
\end{proof}

Following Corollary~\ref{cor:divisors} we can identify $\Div^T(B/H)$ with $\mathscr D^*$, by abuse of notation from now on we will regard $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma$ as a map
$$
	\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma : \Root(B/H) \lra \mathscr D^*.
$$
Similarly, we will often identify $\mathscr D^*$, $\Div^T(B/H)$ and $\tau(\Psi)$, as well as the maps $\tau : \Psi \lra \tau(\Psi)$ and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma\ristretto_\Psi : \Psi \lra \Div^T(B/H)$.

Recall the \textit{dominance order} on $\mathcal X} \newcommand{\calU}{\mathcal U(T)$, defined by $\grl \leq \mu$ if and only if $\mu - \grl \in \mN\grD$. Given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$, denote $F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ the \textit{family of active roots generated by} $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, defined as
$$
	F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \{ \grb \in \Psi \st \grb \leq \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\}.
$$
We collect in the following proposition some properties of the families of active roots from \cite{Avd1} that will be very useful in the following.

\begin{proposition}[{\cite[Lemma 5, Corollaries 2 and 3]{Avd1}}]	\label{prop:family}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\in \Psi$.
\begin{itemize}
	\item[i)] The family $F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ is a linearly independent set of roots, and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb) \neq \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb')$ for all  for all $\grb, \grb' \in F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
	\item[ii)] If $\grb \in \Psi$, then $\grb \in F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ if and only if $\supp(\grb) \subset \supp(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{itemize}
\end{proposition}

\section{Weakly active roots and root systems}\label{s:active}

In this section we will extend the study of the active roots to the weakly active roots, and we will give two alternative combinatorial definition of them. This will provide the technical tools needed for the description of $\mathscr B(G/H)$ given in the next section. We keep the assumptions of the previous section.

\subsection{Weakly active roots}

We will give two different combinatorial characterizations of the weakly active roots. To do this, we will need a couple of preliminary lemmas.

\begin{lemma}	\label{lemma:somma-vive1}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and $\grb \in \Psi$ be such that $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)),\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle > 0$. Then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Psi^\mathrm{\sharp}$, and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{lemma}

\begin{proof}
Notice that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb \in \Root(B/H)$: indeed $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \mathcal X} \newcommand{\calU}{\mathcal U(B/H)$ and by Theorem~\ref{teo:equivalent} it satisfies the condition of Definition \ref{def:radici-toriche}. Moreover, being $\langle w_0\rho(\delta(\grb)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma +\grb \rangle \geq 0$, it follows that $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.

To conclude the proof, we need to show that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma +\grb \in \Phi^+$ and that $U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb}$ acts non-triviallt on $B/H$. As in Section~\ref{ssec:spherical}, given $\grg \in \Root(B/H)$, we denote by $V_\grg$ the associated one parameter unipotent subgroup of $\Aut(B/H)$, so that $\Lie(V_\gamma)$ acts as a derivation on $\mathbb C} \newcommand{\mN}{\mathbb N(B/H)$ according to (\ref{eqn:rooot-action}). In particular, given $\grg_1, \grg_2 \in \Root(B/H)$, we get that $\partial_{\grg_1}$ and $\partial_{\grg_2}$ commute if and only if
$$
\langle w_0 \rho (\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg_1)), \grg_2 \rangle = \langle w_0 \rho (\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg_2)), \grg_1 \rangle.
$$

Being $\langle w_0\rho(\delta(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \grb \rangle > 0$ and $\langle w_0 \rho (\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\beta)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = 0$, it follows by the previous discussion that $\partial_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ and $\partial_\grb$ do not commute, hence $\Lie(V_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb}) = [\Lie(V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma),\Lie(V_\grb)]$. Consider now the homomorphism of algebraic groups $\psi : B \lra \Aut(B/H)$ induced by the action of $B$ on $B/H$. By definition, $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ and $U_\grb$ act non-trivially on $B/H$, hence $\psi(U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ and $\psi(U_\grb) = V_\grb$. Therefore
$$
	\Lie(V_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb}) = [\Lie(V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma),\Lie(V_\grb)] = d\psi [\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma,\mathfrak u_\grb].
$$
It follows that $[\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma,\mathfrak u_\grb]$ is non-zero, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Phi^+$ and $\psi(U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb}) = V_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb}$.
\end{proof}

Given a root $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$, we define the following sets:
$$\Phi^+(\alpha) = (\alpha + \mN \Phi^+) \cap \Phi^+, \qquad \qquad
\Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \mN\Psi) \cap \Psi.$$

\begin{lemma}	\label{lemma:somma-radici}
Let $\alpha\in\Phi^+$ and $\grb \in \Phi^+(\alpha) \senza \{\alpha\}$. There exist $\grb_0, \ldots, \grb_n \in \Phi^+$ with $\grb_0 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, $\grb_n = \grb$ and $\grb_i - \grb_{i-1} \in \Phi^+$ for all $i \leq n$.
\end{lemma}

\begin{proof}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1, \ldots, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m \in \Phi^+$ with $\grb = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m$. Assume that $m$ is minimal with this property, we prove the claim by induction on $m$.
Set $\grg = \beta - \alpha$. If $m=1$ the claim is true. Assume $m>1$, then by the minimality of $m$ it follows that $\grg \not \in \Phi^+$. Since $\grg \in \mN \Phi^+$, it follows then $\grg \not \in \Phi$, hence $(\grb,\alpha) \leq 0$, where $(-,-)$ denotes an ad-invariant scalar product on $\mathfrak t^*$. Hence
\[
(\alpha,\alpha) + (\grg,\alpha) = (\alpha + \grg, \alpha) = (\grb, \alpha) \leq 0
\]
an we get $(\grg, \alpha) \leq -(\alpha,\alpha) < 0$. Up to reorder the indices, we may assume that $(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1,\alpha) < 0$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 \in \Phi^+$. On the other hand $\grb \in \Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma')$ and $\grb = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2 + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m$, therefore the claim follows by induction.
\end{proof}

\begin{lemma}	\label{lemma:ideali}
Given $\alpha \in \Phi^+$, the ideal $(\mathfrak u_\alpha)$ generated by $\mathfrak u_\alpha$ in $\mathfrak u$ is the direct sum of the root spaces $\mathfrak u_\grb$ with $\grb \in \Phi^+(\alpha)$.
\end{lemma}

\begin{proof}
Denote $\mathfrak r = \bigoplus_{\grb \in \Phi^+(\alpha)} \mathfrak u_\grb$. We show that $\mathfrak r \subset \mathfrak u$ is an ideal and that $\mathfrak r \subset (\mathfrak u_\alpha)$, whence the lemma. To show that $\mathfrak r$ is an ideal, it is enough to notice that for all $\grb \in \Phi^+(\alpha)$ and for all $\gamma \in\Phi^+$ it holds either $[\mathfrak u_\grb, \mathfrak u_\grg] = 0$ or $[\mathfrak u_\grb, \mathfrak u_\grg] = \mathfrak u_{\grb + \grg}$, in which case $\grb + \grg \in \Phi^+(\alpha)$.

Let $\grb \in \Phi^+(\alpha)$. By Lemma~\ref{lemma:somma-radici}, there exist $\grb_0, \ldots, \grb_n \in \Phi^+$ with $\grb_0 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, $\grb_n = \grb$ and $\grb_i - \grb_{i-1} \in \Phi^+$ for all $i \leq n$. For every $i=1, \ldots, n-1$ define $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i = \grb_i - \grb_{i-1}$, then we have
\[
[\mathfrak u_{\beta_i},\mathfrak u_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_{i+1}}] = \mathfrak u_{\beta_{i+1}} \neq 0.
\]
It follows that $\mathfrak u_\grb \subset (\mathfrak u_\alpha)$ for all $\grb \in \Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, hence $\mathfrak r \subset (\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{proof}

\begin{theorem}\label{teo:deadker}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$. The following statements are equivalent:
\begin{itemize}
	\item[i)] $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$;
	\item[ii)] $\Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \neq \vuoto$;
	\item[iii)] $\Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap \Psi \neq \vuoto$.
\end{itemize}
If moreover $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, then we have $\delta(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = \delta(\grb)$ for all $\beta \in \Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{theorem}

\begin{proof}

i)  $\Rightarrow$ ii) For all $D \in \mathscr D^*$, fix a root $\beta_D \in \Psi$ such that $\delta(\beta_D) = D$. This is possible thanks to Proposition~\ref{prop:delta-vs-tau}, which implies that the map $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma\ristretto_\Psi : \Psi \lra \mathscr D^*$ is surjective. Let now $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and consider the element
\begin{equation}\label{eqn:activate}
\grb = \alpha + \sum_{D\in\mathscr D^* \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}} \langle w_0 \rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \grb_D.
\end{equation}
By Lemma \ref{lemma:somma-vive1} it follows that $\grb \in \Psi^\mathrm{\sharp}$. On the other hand, if $D \in \mathscr D^*$, then Theorem~\ref{teo:equivalent} implies
$$
	\langle w_0\rho(D), \grb \rangle =
	\left\{ \begin{array}{ll}
			0 & \text{ if } D \neq \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \\
			-1 & \text{ if } D = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)
	\end{array}\right.
$$
Thereofore $\grb \in \Psi$ by Corollary \ref{cor:weak-vs-active}, and we get ii).

ii) $\Rightarrow$ iii) Obvious.

iii) $\Rightarrow$ i) Suppose that $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ acts trivially on $B/H$, we show that $\Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap \Psi = \vuoto$. Denote $N$ the kernel of the action of $U$ on $B/H$. Given $u \in U$, we have $u \in N$ if and only if $ubH = bH$ for all $b\in B$. On the other hand, for all $b\in B$, the equality $ubH = bH$ holds if and only if $u\in bHb^{-1}$. Therefore $N$ is the intersection of $U$ with the biggest normal subgroup of $B$ contained in $H$. Equivalently, $N$ is the biggest $T$-stable normal subgroup of $U$ contained in $H$. It follows that $N$ is stable under conjugation by $T$, hence it is the product of the root subgroups $U_\alpha$ which are contained in it.

By assumption we have $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset N$, hence $\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset \mathfrak n$. By Lemma~\ref{lemma:ideali} the ideal $(\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ generated by $\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ in $\mathfrak u$ is the direct sum of the root spaces $\mathfrak u_\grb$ with $\grb \in \Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. On the other hand $\mathfrak n$ is an ideal of $\mathfrak u$, and by definition no root space $\mathfrak u_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ is contained in $\mathfrak h$. Therefore $\Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap \Psi = \vuoto$.

For the last statement of the theorem, let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and $\beta \in \Psi(\alpha)$. By Theorem~\ref{teo:equivalent} it follows that $\langle w_0\rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \grb \rangle \leq -1$. On the other hand $\langle w_0\rho(D), \grb \rangle = 0$ for all $D \in \mathscr D^* \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$, therefore $\delta(\beta) = \delta(\alpha)$.
\end{proof}

As a first consequence of the previous theorem, we generalize Lemma \ref{lemma:somma-vive1} to arbitrary pairs of weakly active roots.

\begin{lemma}	\label{lemma:somma-vive}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma, \grb  \in \Psi^\mathrm{\sharp}$ be such that $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)),\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle > 0$. Then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Psi^\mathrm{\sharp}$, and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{lemma}

\begin{proof}
Let $\grb' \in \Psi(\grb)$, then $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb') = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)$ and Lemma \ref{lemma:somma-vive1} implies that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb' \in \Psi^\mathrm{\sharp}$, and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb') = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. Let $\grg \in \Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb')$, then $\grg = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb + \grg_1+ \ldots+\grg_n$ for some $\grg_1, \ldots, \grg_n \in \Psi$ and Theorem \ref{teo:deadker} implies $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. Therefore, by Theorem \ref{teo:equivalent}, $\langle w_0 \rho(D), \grg \rangle = 0$ for all $D \in \mathscr D^*\senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}$, and $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \grg \rangle = -1$.
Being $\{\grg_1, \ldots, \grg_n\} \subset F(\grg)$, by Proposition~\ref{prop:family} i) it follows $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg_i) \neq \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg)$ for all $i$, hence $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \grg_i \rangle = 0$ for all $i$ by Theorem \ref{teo:equivalent}, and it follows
$$\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb \rangle = \langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)), \grg \rangle = -1.$$
On the other hand the hypothesis of the lemma implies $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \rangle \geq 0$, hence $\langle w_0 \rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \rangle \geq 0$ for all $D \in \mathscr D^* \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}$ and following Definition \ref{def:radici-toriche} we get $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Root(B/H)$ and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.

To conclude the proof, we need to show that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma +\grb \in \Phi^+$ and that $U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb}$ acts non-trivially on $B/H$. This is shown with the same argument used in Lemma \ref{lemma:somma-vive1}, which applies without any change in this more general case.
\end{proof}

Given $I\subset \mathscr D^*$ we denote
$$
	\Psi_I = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi \st \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \subset w_0 D \quad  \exists D \in I\}
$$
Under the identifications $\mathscr D^* \longleftrightarrow \tau(\Psi) \longleftrightarrow \Div^T(B/H)$, if we regard $I \subset \tau(\Psi)$ then $\Psi_I = \{ \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi \st \tau(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in I\}$, whereas if we regard $I \subset \Div^T(B/H)$ then $\Psi_I = \delta^{-1}(I) \cap \Psi$. In case $I = \{D\}$ is a single element, then we will also denote $\Psi_I$ simply by $\Psi_D$.

\begin{definition}	\label{def:activated-roots}
Let $I \subset \mathscr D^*$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, then we say that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is {\em activated by} $I$ if there is $\grb \in \mN\Psi_I$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Psi$, and we say that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ \textit{stabilizes} $I$ if there is $\grb \in \mN\Psi_I$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Psi_I$.
\end{definition}

We denote by $\Psi^\mathrm{\sharp}(I)$ the set of weakly active roots activated by $I$, and by $\Psi^\mathrm{\sharp}_I$ the subset of $\Psi^\mathrm{\sharp}(I)$ of the roots which stabilize $I$. Notice that, for all $I \subset \mathscr D^*$, we have $\Psi \subset \Psi^\mathrm{\sharp}(I)$ and $\Psi_I \subset \Psi^\mathrm{\sharp}_I$. In addition, a weakly active root $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$ stabilizes $I$ if and only if $\delta(\alpha) \in I$. Given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$ and $I \subset \mathscr D^*$ we denote
$$
	\Psi_I(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) = (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \mN \Psi_I) \cap \Psi.
$$
In the language just introduced, we have the following consequence to Theorem~\ref{teo:equivalent}.

\begin{proposition} \label{prop:radici-stabili}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and $I \subset \mathscr D^*$.  Then
\begin{itemize}
	\item[i)] $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$ if and only if $\langle w_0\rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \leq 0$ for all $D \in \mathscr D^* \senza I$,
	\item[ii)] $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$ if and only if $\langle w_0\rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = 0$ for all $D \in \mathscr D^* \senza I$.
\end{itemize}
In particular, we have the equality $\Psi^\mathrm{\sharp}_I = \mN \Psi^\mathrm{\sharp}_I \cap \Psi^\mathrm{\sharp}$.
\end{proposition}

\begin{proof}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$, applying Theorem~\ref{teo:equivalent} to any element in $\Psi_I(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ it follows $\langle w_0\rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \leq 0$ for all $D \in \mathscr D^* \senza I$. Conversely, if such inequalities are all satisfied, then we may consider the element $\grb \in \Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ defined as in (\ref{eqn:activate}), and since $\langle w_0\rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle \geq 0$ for all $D \in \mathscr D^* \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}$ it follows $\grb \in \Psi_I(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. This shows i), and ii) follows by noticing that, if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$, then by definition $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ stabilizes $I$ if and only if $\delta(\alpha) \in I$, if and only if $\langle w_0\rho(D), \grb \rangle \geq 0$ for all $D \in \mathscr D^* \senza I$.
\end{proof}

Given $I \subset \mathscr D^*$, let $\grs_I \subset \mathcal X} \newcommand{\calU}{\mathcal U(B/H)^\vee_\mQ = w_0\mathcal X} \newcommand{\calU}{\mathcal U(G/H)^\vee_\mQ$ be the cone of the affine open subset of $B/H$ with closed $T$-orbit $\mathscr U_I$. By definition $\grs_I$ is generated by the one dimensional rays in $w_0\mathcal X} \newcommand{\calU}{\mathcal U(G/H)^\vee_\mQ$ which correspond to the $T$-stable prime divisors of $B/H$ which contain $\mathscr U_I$, namely the elements of $\mathscr D^* \senza I$. Hence
$$
	\grs_I = \cone(w_0 \rho(D) \st D \in \mathscr D^* \senza I),
$$
and we may rephrase Proposition~\ref{prop:radici-stabili} as follows.

\begin{proposition} \label{prop:radici-stabili2}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and $I \subset \mathscr D^*$. Then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$ if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \ristretto_{\grs_I} \leq 0$, and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$ if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \ristretto_{\grs_I} = 0$.
\end{proposition}

Given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, we are now ready to describe combinatorially the action of the root subgroup $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ on the set of $T$-orbits on $B/H$ in terms of weakly active roots.

\begin{proposition}\label{prop:Ualphaactionfinal}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and $I \subset \mathscr D^*$. Then $\mathscr U_I$ is not stable under the action of $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$. If this is the case, then we have
$$
	U_\alpha \mathscr U_I  = \left\{ \begin{array}{ll}
	\mathscr U_I \cup \mathscr U_{I \cup \{\delta(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}} & \text{if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_I$}	\\
	\mathscr U_I \cup \mathscr U_{I \senza \{\delta(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\}} & \text{if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$}
	\end{array} \right.
$$
In particular, $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr U_I \subset \ol \mathscr U_I$ if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$.
\end{proposition}

\begin{proof}
Consider the homomorphism of algebraic groups $B \lra \Aut(B/H)$, then in the notation of Section~\ref{ssec:spherical} the root group $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ acts on $B/H$ as the one parameter unipotent subgroup $V_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset \Aut(B/H)$. Suppose that $\mathscr U_I$ is not stable under the action of $U_\alpha$. Then by Proposition~\ref{prop:sottovarietà toriche stabili} i) the set $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr U_I$ decomposes in the union of two $T$-orbits $\mathscr U_I \cup \mathscr U_J$ for some $J$.

Suppose $|I| < |J|$, then $\mathscr U_I \subset \ol{\mathscr U_J}$ and Proposition~\ref{prop:sottovarietà toriche stabili} iii) implies $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\ristretto_{\grs_I} \leq 0$ and $\grs_J = \grs_I \cap \ker \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is a facet of $\grs_I$. By Proposition~\ref{prop:radici-stabili2} we get then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\sharp(I) \senza \Psi^\mathrm{\sharp}_I$. If instead $|I| > |J|$, then $\mathscr U_J \subset \ol{\mathscr U_I}$ and the same proposition implies $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\ristretto_{\grs_J} \leq 0$ and $\grs_I = \grs_J \cap \ker \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$ by Proposition~\ref{prop:radici-stabili2}.

Conversely, suppose that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I)$. Then Proposition~\ref{prop:radici-stabili2} implies $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\ristretto_{\grs_I} \leq 0$, and by the description of $\grs_I$ it follows $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\ristretto_{\grs_I} = 0$ if and only if $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in I$. Then the claim follows by Proposition~\ref{prop:sottovarietà toriche stabili} iii) applied to the pair $\mathscr U_{I \senza \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)} \subset \ol{\mathscr U_I}$ or to the pair $\mathscr U_I \subset \ol{\mathscr U_{I \cup \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)}}$, depending on whether $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in I$ or $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \not \in I$.
\end{proof}

\begin{remark}
In particular, given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and $I \subset \mathscr D^*$, we have the following properties:
\begin{itemize}
	\item[i)] If $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr U_I \neq \mathscr U_I$, then the closure $\ol{\mathscr U_I}$ is $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ stable if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$.
	\item[ii)] If $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(I \senza \{\delta(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\})$.
\end{itemize}
\end{remark}

\subsection{Combinatorics related to the weakly active roots.}

Recall the following fundamental property of the active roots.

\begin{lemma}[{\cite[Lemma 7]{Avd1}}]	\label{lemma:differenza-attive-avdeev}
For all $\grb \in F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)\senza\{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\}$ it holds $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma - \grb \in \Phi^+$.
\end{lemma}

We will need the following generalization of this property, which holds for any combination of roots in $F(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.

\begin{lemma} \label{lemma:differenza-attive}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ and let $\grb \in \mN \Psi$ be such that $\grb < \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma - \grb \in \Phi^+$.
\end{lemma}

\begin{proof}
Write $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma-\grb = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i \in \Phi^+$, we show the claim by induction on $n$. We have $\alpha\in\Phi^+(\alpha_i)\cap\Psi$ for all $i\in\{1,\ldots,n\}$, so Theorem~\ref{teo:deadker} yields $\alpha_i\in\Psi^\mathrm{\sharp}$.

Notice that by Theorem \ref{teo:equivalent} we have $\langle w_0 \rho(D),\beta \rangle \leq 0$ for all $D \in \mathscr D^*$, hence $\langle w_0 \rho(D),\alpha-\beta \rangle < 0$ at most for one $D \in \mathscr D^*$, and  if such $D$ exists then it is $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. On the other hand, write $\grb = \grb_1 + \ldots + \grb_m$ for $\grb_1,\ldots,\grb_m \in \Psi$: then $\grb_j\leq\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ for all $j$ and from Proposition~\ref{prop:family} i) we deduce $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_j) \neq \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. Therefore $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)),\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma-\grb \rangle =-1$, and following Definition \ref{def:radici-toriche} we get $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1+\ldots+\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n \in \Root(B/H)$.

Suppose that $i > 1$, then there is at least one index $i$ with $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i) \neq \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, say $i=1$: this implies $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = 0$ by Theorem \ref{teo:equivalent}. Being $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 \rangle = -1$ and $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)), \grb \rangle \leq 0$ it follows that there is at least one index $i$ such that $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i \rangle > 0$, say $i=2$. It follows by Lemma \ref{lemma:somma-vive} that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1+\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2 \in \Psi^\mathrm{\sharp}$, therefore we can apply the inductive hypothesis and we get $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$.
\end{proof}

Thanks to Theorem~\ref{teo:deadker}, we deduce the following descriptions.

\begin{corollary}	\label{cor:weak-as-differences}
Let $I \subset \mathscr D^*$, then we have the equalities
\[
	\Psi^\mathrm{\sharp}(I) = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma - \grb \st \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi, \; \grb \in \mN \Psi_I, \; \grb < \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\},
\]
\[
	\Psi^\mathrm{\sharp}_I = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma - \grb \st \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi_I, \; \grb \in \mN \Psi_I, \; \grb <  \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\}.
\]
\end{corollary}

Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ and let $\grb \in \Psi$ be such that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Psi$, then \cite[Proposition 1]{Avd1} shows that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb' \in \Psi$ for all $\grb' \in \Psi$ with $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb') = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)$. We will need the following generalization of this property.

\begin{proposition} 	\label{prop:attivazioni-equivalenti}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \mN \Phi^+$ and let $\grb_1, \ldots, \grb_n \in \Psi$ be such that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum_i a_i \grb_i \in \Psi$ for some $a_1, \ldots, a_n \in \mN$. Then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum_i a_i \grb_i' \in \Psi$ for all $\grb'_1, \ldots, \grb'_n \in \Psi$ such that $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_1) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_1'), \ldots, \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_n) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_n')$. Moreover we have $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum_i a_i \grb_i) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum_i a_i \grb'_i)$.
\end{proposition}

\begin{proof}
Denote $\grg = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum_i a_i \grb_i$. We show the claim by induction on the sum $a = \sum_i a_i$. Suppose that $a = 1$, then $n=1$ and Lemma~\ref{lemma:differenza-attive-avdeev} shows $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grg - \grb_1 \in \Phi^+$. Therefore the claim follows by \cite[Proposition 1]{Avd1}.

Suppose now that $n >1$, then $\grb_n \in F(\grg)$ and Lemma~\ref{lemma:differenza-attive-avdeev} implies
$$
	\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + a_1 \grb_1 + \ldots + a_{n-1} \grb_{n-1} + (a_n-1)\grb_n \in \Phi^+.
$$
Setting $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb_n'$, by \cite[Proposition 1]{Avd1} it follows
$$
	\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' + a_1 \grb_1 + \ldots + a_{n-1} \grb_{n-1} + (a_n-1)\grb_n \in \Psi,
$$
and we conclude that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum_i a_i \grb_i' \in \Psi$ by the inductive hypothesis.

To show the last claim, notice that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ by Lemma~\ref{lemma:differenza-attive}. Denote $\grg' = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \sum a_i \grb'_i$, then we have $\grg, \grg' \in \Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, therefore $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg') = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ by Theorem~\ref{teo:deadker}.
\end{proof}

We now show that the semigroup generated by the set of weakly active roots $\Psi^\mathrm{\sharp}_I$ which stabilize a given subset $I \subset \mathscr D^*$ is \textit{saturated} in the root lattice $\mathbb Z \grD$, that is
\[
	\mN \Psi^\mathrm{\sharp}_I = \cone(\Psi^\mathrm{\sharp}_I) \cap \mathbb Z \grD.
\]
More precisely, we will show that $\mN\Psi^\mathrm{\sharp}_I$ is the intersection of the semigroup generated by the positive roots with the rational vector space generated by $\Psi_I$. In order to do this, we will need a couple of preliminary lemmas.

Define
$$
	\grS = \bigcup_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi} \supp(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)
$$

\begin{lemma}	\label{lem:spherical-roots}
We have the equality $\grS = \Psi^\mathrm{\sharp} \cap \grD$. In particular $\mathbb Z \Psi = \mathbb Z \Psi^\mathrm{\sharp} = \mathbb Z \grS$ and $\mathbb Z\Psi \cap \mN\grD = \mN \Psi^\mathrm{\sharp} = \mN \grS$.
\end{lemma}

\begin{proof}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp} \cap \grD$, then by Theorem~\ref{teo:deadker} it follows $\Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \neq \vuoto$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grS$. Suppose conversely that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grS$, then by definition $\Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cap \Psi \neq \vuoto$, therefore $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$ by Theorem~\ref{teo:deadker} again. The last equalities follow immediatley by the inclusions $\mathbb Z \grS \subset \mathbb Z \Psi^\mathrm{\sharp} \subset \mathbb Z \Psi \subset \mathbb Z \grS$ and $\mN \grS \subset \mN\Psi^\mathrm{\sharp} \subset \mathbb Z \Psi \cap \mN \grD \subset \mN \grS$.
\end{proof}

\begin{remark}\label{rem:spherical roots}
The set $-w_0(\grS)$ is the set of \textit{spherical roots} that is attached to $H$ as a spherical subgroup of $G$ following the theory of spherical varieties (see e.g. \cite[Theorem 1.3]{Kn3} for the general case, and \cite[Theorem 5.28]{Avd2} for strongly solvable spherical subgroups of $G$). The ``twist'' appearing here is due to the fact that in \cite{Avd2} the subgroup $H$ is assumed to be contained in the opposite Borel subgroup $B^{w_0}$.
\end{remark}

\begin{proposition}  \label{prop:semigruppi-attive}
Given $I \subset \mathscr D^*$, we have
$$\mN\Psi^\mathrm{\sharp}_I = \mathbb Z \Psi_I \cap \mN \grD = \mQ \Psi_I \cap \mN \grD.$$
\end{proposition}

\begin{proof}
By the definition of $\grS$ together with Lemma~\ref{lem:spherical-roots} we have
$$
\mQ \Psi \cap \mN \grD = \mQ \Psi \cap \mN \grS = \mN \grS = \mN \Psi^\mathrm{\sharp}.
$$
Therefore the claim is equivalent to the equalities
$$\mN\Psi^\mathrm{\sharp}_I = \mathbb Z \Psi_I \cap \mN \Psi^\mathrm{\sharp} = \mQ \Psi_I \cap \mN \Psi^\mathrm{\sharp}.$$

By the definition of $\Psi^\mathrm{\sharp}_I$ we have the inclusions $\mN\Psi^\mathrm{\sharp}_I \subset \mathbb Z \Psi_I \cap \mN \Psi^\mathrm{\sharp} \subset \mQ \Psi_I \cap \mN \Psi^\mathrm{\sharp}$. Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \mQ \Psi_I \cap \mN\Psi^\mathrm{\sharp}$ and write $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb_1 + \ldots + \grb_n$ with $\grb_i \in \Psi^\mathrm{\sharp}$. By Proposition~\ref{prop:radici-stabili}, we have $\langle w_0 \rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = 0$ for every $D \in \mathscr D^* \senza I$. Proceeding by induction on $n$, we show that we may choose $\grb_i$ in $\Psi^\mathrm{\sharp}_I$ for all $i$. If $n=1$, then we have $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$ by Proposition~\ref{prop:radici-stabili}. Suppose $n >1$ and assume that $\grb_i \not \in \Psi^\mathrm{\sharp}_I$ for some $i$. Then the same corollary implies $\langle w_0\rho(D), \grb_i \rangle \neq 0$ for some $D \in \mathscr D^* \senza I$.

Suppose that $\langle w_0\rho(D), \grb_i \rangle > 0$. Being $\langle w_0\rho(D), \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \rangle = 0$, it follows $\langle w_0\rho(D), \grb_j \rangle < 0$ for some $j \neq i$, hence
$D = \delta(\grb_j)$ and $\grb_i + \grb_j \in \Psi^\mathrm{\sharp}$ by Lemma~\ref{lemma:somma-vive}. Similarly, if $\langle w_0\rho(D), \grb_i \rangle < 0$, then $D = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_i)$ and there is some $j \neq i$ with $\langle w_0\rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_i)), \grb_j \rangle > 0$, hence $\grb_i + \grb_j \in \Psi^\mathrm{\sharp}$ by Lemma~\ref{lemma:somma-vive} again. Therefore $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ can be written as a sum of $n-1$ weakly active roots, and we conclude by the inductive assumption.
\end{proof}

\subsection{The root system associated to a $T$-orbit in $B/H$.}

Recall that a root subsystem $\Phi' \subset \Phi$ is \textit{closed} if, for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma, \grb \in \Phi'$ such that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Phi$, it holds $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb \in \Phi'$ as well. Given $I \subset \mathscr D^*$ we denote
$$
	\Phi_I = \mathbb Z \Psi_I \cap \Phi, \qquad \qquad \Phi_I^{\pm} = \mathbb Z \Psi_I \cap \Phi^\pm.
$$
It follows easily by its definition that $\Phi_I$ is a closed root subsystem of $\Phi$, and that $\Phi_I^+ \subset \Phi_I$ is a system of positive roots. We denote by $\grD_I \subset \Phi^+_I$ the corresponding basis and by $W_I$ the Weyl group of $\Phi_I$ (notice that in general $\grD_I \not \subset \grD$).

Proposition~\ref{prop:semigruppi-attive} readily implies the following property of the root system $\Phi_I$.

\begin{proposition}  \label{prop: I-rootsyst}
Given $I \subset \mathscr D^*$, we have
$$\Phi^+_I = \mQ \Psi_I \cap \Phi^+ = \mN\Psi^\mathrm{\sharp}_I \cap \Phi^+
\qquad \text{ and } \qquad \Psi^\mathrm{\sharp}_I = \Phi^+_I \cap \Psi^\mathrm{\sharp}.$$
In particular $\grD_I \subset \Psi^\mathrm{\sharp}_I$, and $I$ is recovered by $\Phi_I$.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:semigruppi-attive} we get the equalities $\Phi^+_I = \mQ \Psi_I \cap \Phi^+ = \mN\Psi^\mathrm{\sharp}_I \cap \Phi^+$, and the inclusion $\grD_I \subset \Psi^\mathrm{\sharp}_I$ follows as well. Combining with Proposition~\ref{prop:radici-stabili} we get then $\Psi^\mathrm{\sharp}_I = \Phi^+_I \cap \Psi^\mathrm{\sharp}$. Finally, being $\Psi_I \subset \Psi^\mathrm{\sharp}_I$, we have $I = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Psi^\mathrm{\sharp}_I) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Phi^+_I \cap \Psi^\mathrm{\sharp})$, and the last claim follows.
\end{proof}

We say that a root subsystem $\Phi' \subset \Phi$ is \textit{parabolic} if there exists $w \in W$ such that $w(\Phi)$ is generated by a subset of simple roots of $\Phi$. This is equivalent to the property that $\Phi' = \mQ \Phi' \cap \Phi$ (see \cite[Ch.~VI, \S~1, Proposition~24]{Bou}), therefore we get the following corollary.

\begin{corollary} \label{cor: sistemi-parabolici}
Let $I \subset \mathscr D^*$, then $\Phi_I$ is a parabolic root subsystem of $\Phi$.
\end{corollary}

The following proposition provides a first link between the root system $\Phi_I$ and the geometry of the corresponding $T$-orbit $\mathscr U_I$.

\begin{proposition}	\label{prop:radici-reticoli}
Let $I \subset \mathscr D^*$, then $\Phi_I \subset \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_I)$. Moreover we have the equality $\mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_I) = \mathbb Z \Phi_I + \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_\vuoto)$.
\end{proposition}

\begin{proof}
Since $\mathbb Z\Phi_I = \mathbb Z \Psi_I$, it is enough to prove that $\Psi_I \subset \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_I)$. On the other hand if $J \subset I$ we have $\mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_J) \subset \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_I)$, therefore it is enough to consider the case where $I$ contains a single element.

Suppose $I = \{D\}$, let $\grb \in \Psi_I$ and let $B_0 = TU_\grb$. Then $\ol{\mathscr U_I} = B_0 H/H$ is homogeneous under $B_0$, and we have an isomorphism $\ol{\mathscr U_I} \simeq B_0/B_0 \cap H$. The unipotent radical $(B_0 \cap H)^\mru$ is a connected subgroup of $U_\grb$, hence is trivial since $\mathscr U_I \neq \mathscr U_\vuoto$. On the other hand by construction $T \cap H \subset H$ is a maximal diagonalizable subgroup, hence $B_0 \cap H = T \cap H$. Therefore we get an isomorphism of $T$-varieties $\ol{\mathscr U_I} \simeq T/T \cap H  \ltimes U_\grb$, and it follows that $\ol{\mathscr U_I}$ possesses a $T$-eigenfunction of weight $\grb$. This shows the first claim, and the second claim is easily proved by induction on $|I|$.
\end{proof}

\begin{corollary}
Let $I \subset \mathscr D^*$, then the following inequalities hold
\[
	|I| \leq \rk \Phi_I \leq \dim(T/T_H) + |I|
\]
\end{corollary}

\begin{proof}
This follows from the equality $\rk \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr U_I) = \dim(\mathscr U_I) = \dim(T/T_H) + |I|$ together with the inclusion $\Psi_I \subset \Phi_I$.
\end{proof}

Since $\Phi_I \subset \Phi$ is a closed root subsystem, to every $I \subset \mathscr D^*$ we may also attach a reductive subgroup $G_I$ of $G$, namely the subgroup generated by $T$ together with the root subgroups $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi_I$. We denote by $B_I$ the Borel subgroup of $G_I$ associated with $\Phi^+_I$, that is $B_I = G_I \cap B$.

\begin{corollary} \label{cor: B_I action}
We have the equality $\ol{\mathscr U_I} = \ol{B_I H/H}$.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:Ualphaactionfinal} it follows that $\ol{\mathscr U_I}$ is $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$-stable for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}_I$. On the other hand by Proposition~\ref{prop: I-rootsyst} every $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+_I \senza \Psi^\mathrm{\sharp}_I$ acts trivially on $B/H$, and the claim follows.
\end{proof}

\begin{remark}	\label{oss:ranks}
By the definition of $\Phi_I$ we have $\rk (\Phi_I) = \dim(\mQ \Psi_I)$. In some cases this rank is easy computable:
\begin{itemize}
	\item[-] $0 \leq \rk(\Phi_I) \leq |\grS|$.
	\item[-] $\rk(\Phi_I) = 0$ if and only if $I = \vuoto$.
	\item[-] $\rk(\Phi_I) = |\grS|$ if and only if $I = \mathscr D^*$.
	\item[-] If $I = \{D\}$ then $\rk(\Phi_I) = |\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma^{-1}(D)|$ (this follows by \cite[Corollary 1]{Avd1}).
	\item[-] If $T \subset H$ then $\rk(\Phi_I) = |I|$ for all $I \subset \mathscr D^*$ (this follows by \cite[Theorem 1]{Avd1}).
\end{itemize}
\end{remark}

\subsection{Explicit description of the basis $\grD_I$}

By making use of the classification of the active roots, we will give in this subsection some more explicit description of the root systems $\Phi_I$ and of their bases $\grD_I$. We will not use these results until the last section of the paper, where we will prove a bound for the number of $B$-orbits in $G/H$.

Suppose that $H \subset B$ is a spherical subgroup of $G$ (possibly not containing $T$). In the following theorem we recall Avdeev's classification of the active roots, as well as some of their properties (see \cite{Avd1}, Proposition 3, Corollary 6, Theorem 3, and Lemma 10), and deduce some corollaries.

\begin{theorem} [{\cite{Avd1}}]	\label{teo:avdeev}
Let $\grb \in \Psi$, then the following properties hold.
\begin{itemize}
	\item[i)] There exists a unique simple root $\pi(\grb) \in \supp(\grb)$ with the following property: if $\grb = \grb_1 + \grb_2$ for some $\grb_1, \grb_2 \in \Phi^+$, then $\grb_1 \in \Psi$ if and only if $\pi(\grb) \not \in \supp(\grb_1)$.
	\item[ii)] The map $\grb' \longmapsto \pi(\grb')$ induces a bijection between $F(\grb)$ and $\supp(\grb)$.
	\item[iii)] If $\grb_1, \grb_2 \in \Psi$ and $\pi(\grb_1) = \pi(\grb_2)$, then $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_1) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_2)$.
	\item[iv)] The active root $\grb$ appears in Table~\ref{tab: active roots}\footnote{Notice that when $\supp(\grb)$ is of type $\sfF_4$ our enumeration of the set $\supp(\grb)$ in Table~\ref{tab: active roots} differs from the one in \cite{Avd1}: following \cite{Bou} for us $\alpha_1$ and $\alpha_2$ are the short simple roots.}, and $[\grb : \pi(\grb)] = 1$.
\end{itemize}
\end{theorem}

\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
Type & $\supp(\grb)$ & $\grb$ \\
\hline
\hline
1. & any of rank $n$ & $\alpha_1+\cdots+\alpha_n$ \\
\hline
2. & $\mathsf{B}_n$ & $\alpha_1+\cdots+\alpha_{n-1} + 2\alpha_n$  \\
\hline
3. & $\mathsf{C}_n$ & $2\alpha_1+\cdots+2\alpha_{n-1} + \alpha_n$  \\
\hline
4. & $\mathsf{F}_4$ & $\alpha_1 + \alpha_2 + 2\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3 + 2\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4$ \\
\hline
5. & $\mathsf{G}_2$ & $2\alpha_1+\alpha_2$ \\
\hline
6. & $\mathsf{G}_2$ & $3\alpha_1+\alpha_2$ \\
\hline
\end{tabular}
\end{center}
\caption{Active roots} \label{tab: active roots}
\end{table}

We will call the integer appearing in the first column of Table~\ref{tab: active roots} the \textit{type} of an active root.

Let $\grb \in \Psi$, we say that a subset $A \subset \supp(\grb)$ is \textit{connected} (resp. \textit{co-connected}) if $A$ (resp. $\supp(\grb) \senza A$) is connected as a set of vertices in the Dynkin diagram of $\Phi$. If $\grb' \in F(\grb)$, it follows by Lemma~\ref{lemma:differenza-attive-avdeev} that $\supp(\grb') \subset \supp(\grb)$ is connected and co-connected, whereas Theorem~\ref{teo:avdeev} i) shows that, if $\grb' \neq \grb$, then $\pi(\grb) \not \in \supp(\grb')$.

Notice that Theorem~\ref{teo:avdeev} allows to construct the whole family $F(\grb)$ from $\grb$ and $\pi(\grb)$. In particular we have the following property.

\begin{corollary} \label{cor:support-family}
Let $\grb \in \Psi$ and let $A \subset \supp(\grb)$. Then $A = \supp(\grb')$ for some $\grb' \in F(\grb) \senza \{\grb\}$ if and only if $A$ is connected and co-connected, and $\pi(\grb) \not \in A$.
\end{corollary}

\begin{proof}
As already noticed, every active root $\grb' \in F(\grb) \senza \{\grb\}$ satisfies the properties of the claim. The corollary follows then by Theorem~\ref{teo:avdeev} ii) by noticing that the number of subsets $A\subset \supp(\grb)$ which are connected and co-connected such that $\pi(\grb) \not \in A$ is the number of edges of the Dynkin diagram of $\supp(\grb)$. Indeed, associating to $A$ the unique edge connecting a simple root of $A$ to a simple root of $\supp(\grb)\senza A$ is a bijection. On the other hand, the number of such edges is precisely $|\supp(\grb)|-1$.
\end{proof}

In particular we get the following characterization of the pairs $(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma, \grb) \in \Psi^\mathrm{\sharp} \times \Psi$ with $\grb \in \Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.

\begin{corollary} \label{cor:weak-family}
Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$ and $\grb \in \Psi$, and suppose that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma < \grb$. Then $\grb \in \Psi(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ if and only if $\pi(\grb) \in \supp(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$.
\end{corollary}

\begin{proof}
Being $\grb \in \Phi^+(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, by Theorem~\ref{teo:deadker} it follows that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}$. If $\supp(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \neq \supp(\grb)$, let $A \subset \supp(\grb) \senza \supp(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ be a connected component. Notice that $A$ is both connected and co-connected in $\supp(\grb)$, therefore by Corollary~\ref{cor:support-family} there is $\grb' \in \Psi$ with $\supp(\grb') = A$. On the other hand $\langle \grb', \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma^\vee \rangle < 0$, therefore $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb' \in \Phi^+$ and being $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb' \leq \grb$ we still have $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \grb' \in \Psi^\mathrm{\sharp}$. It follows that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma+\grb'$ still satisfies the assumptions, and we may assume that $\supp(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) =\supp(\grb)$. In this case the claim can be easily checked by making use of Theorem~\ref{teo:avdeev} and Table~\ref{tab: active roots}:
\begin{itemize}
	\item[i)] If $\grb$ is of type 1, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb$, which is absurd.
	\item[ii)] If $\grb$ is of type 2, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$, and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n \in \Psi$.
	\item[iii)] If $\grb$ is of type 3, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i)$ for some $i < n$, and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i \in \Psi$ for all $i < n$.
	\item[iv)] If $\grb$ is of type 4, then either $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3 + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4)$ or $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4$, and 	both $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3 + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4$ are active roots.
	\item[v)] If $\grb$ is of type 5 or 6, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1$, and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 \in \Psi$ \qedhere.
\end{itemize}
\end{proof}

In particular, the previous corollary shows that $\grb \in \Psi(\pi(\grb))$ for all $\grb \in \Psi$. In this case we can be even more precise.

\begin{corollary}	\label{cor:massimali}
Let $\grb \in \Psi$ and $\grb' \in F(\grb)$, then $\grb'$ is maximal in $F(\grb) \senza \{\grb\}$ if and only if $\supp(\grb')$ is a connected component of $\supp(\grb) \senza \{\pi(\grb)\}$ if and only if $\pi(\grb)$ and $\pi(\grb')$ are non-orthogonal. Moreover
$$\pi(\grb) = \grb - \sum_{\grg \in \max (F(\grb) \senza \{\grb\}) } a_{\grb,\grg} \grg$$
where $\max (F(\grb) \senza \{\grb\})$ denotes the set of maximal elements in $F(\grb) \senza \{\grb\}$ and where $a_{\grb,\grg} \in \mN$ is the maximum such that $a_{\grb,\grg} \grg < \grb$.
\end{corollary}

\begin{proof}
Notice that every connected component of $\supp(\grb) \senza \{\pi(\grb)\}$ is co-connected. The first claim follows then by Corollary~\ref{cor:support-family}, together with Proposition~\ref{prop:family} ii).

Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' \in \supp(\grb')$ non-orthogonal to $\pi(\grb)$, notice that such root is unique since $\supp(\grb)$ contains no loops. Then $\supp(\grb') \senza \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'\}$ is connected and co-connected in $\supp(\grb)$, and since it does not contain $\pi(\grb)$ by Corollary~\ref{cor:support-family} there is $\grb'' \in F(\grb)$ such that $\supp(\grb'') = \supp(\grb') \senza \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'\}$. Being $\supp(\grb'') \subset \supp(\grb')$, Proposition~\ref{prop:family} implies that $\grb'' \in F(\grb')$ and $\grb'-\grb'' \in \Phi^+$. On the other hand by Theorem~\ref{teo:avdeev} i) $\pi(\grb') \in \supp(\grb') \senza \supp(\grb'')$, therefore it must be $\pi(\grb') = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'$.

The last claim can be easily deduced by the previous discussion together with a direct inspection based on Theorem~\ref{teo:avdeev} and Table~\ref{tab: active roots}.
\end{proof}

Another technical property that we will need and that we can deduce from Theorem~\ref{teo:avdeev} is the following.

\begin{lemma}	\label{lemma:radici-adiacenti}
Let $\grb \in \Psi$ and let $\grb_1, \grb_2 \in F(\grb)$. If $\supp(\grb_1) \cup \supp(\grb_2)$ is connected, then $\grb_1$ and $\grb_2$ are comparable.
\end{lemma}

\begin{proof}
By Corollary~\ref{cor:support-family} the complementary
$$
\supp(\grb) \senza (\supp(\grb_1)\cup\supp(\grb_2)) = (\supp(\grb)\senza\supp(\grb_1))\cap (\supp(\grb)\senza\supp(\grb_2))
$$
is the intersection of two connected subsets of $\supp(\grb)$, so it is connected since the Dynkin diagram of $\supp(\grb)$ has no loops. Therefore $\supp(\grb_1)\cup\supp(\grb_2)$ is connected and co-connected, and by Corollary~\ref{cor:support-family} there exists $\gamma\in F(\grb)$ such that $\supp(\gamma)=\supp(\grb_1)\cup\supp(\grb_2)$.

Suppose that $\gamma$ is different both from $\grb_1$ and $\grb_2$. For $i = 1,2$, we have then $\grb_i\in F(\gamma)$ by Proposition~\ref{prop:family} ii), and $\gamma-\grb_i\in\Phi^+$ by Lemma~\ref{lemma:differenza-attive-avdeev}. Therefore by Theorem~\ref{teo:avdeev} i) $\pi(\gamma)\notin\supp(\grb_i)$, which is absurd by the definition of $\grg$. Therefore $\gamma$ is either $\grb_1$ or $\grb_2$, hence $\grb_1$ and $\grb_2$ are comparable by Proposition~\ref{prop:family} ii).
\end{proof}

Let $I\subset \mathscr D^*$ and $\grb \in \Psi_I$. Denote $F_I(\grb) = (F(\grb) \cap \Psi_I) \senza \{\grb\}$ and define
$$\grb^\sharp_I = \grb - \sum_{\grg \in \max (F_I(\grb)) } a_{\grb,\grg} \grg$$
where $\max (F_I(\grb))$ is the set of maximal elements in $F_I(\grb)$ and where $a_{\grb,\grg} \in \mN$ is the maximum such that $a_{\grb,\grg} \grg < \grb$. By Theorem~\ref{teo:deadker} it holds $\grb_I^\sharp \in \Psi^\mathrm{\sharp}_I$, and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_I^\sharp) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)$: indeed, being pairwise uncomparable, the elements in $\max (F_I(\grb))$ have pairwise disjoint support by Lemma~\ref{lemma:radici-adiacenti}.

\begin{theorem}	\label{teo:DeltaI-rango-massimo}
Let $I \subset \Psi$, then $\grD_I = \{\grb^\sharp_I \st \grb \in \Psi_I\}$.
\end{theorem}

\begin{proof}
Suppose that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD_I$, then by Corollary~\ref{cor:weak-as-differences} we may write $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - (\grb_1 + \ldots + \grb_p)$ for some $\grb \in \Psi_I$ and some $\grb_1, \ldots, \grb_p \in F_I(\grb)$. We claim that
$$\grb_1 + \ldots + \grb_p \leq \sum_{\grg \in \max (F_I(\grb)) } a_{\grb,\grg}\grg.$$
This is clear if the supports of $\grb_1, \ldots, \grb_p$ are pairwise disconnected, e.g. if the active root $\grb$ is of type 1. Suppose that this is not the case and that (up to reordering the indices) $\supp(\grb_1) \cap \supp(\grb_2) \neq \vuoto$. Then $\grb_1$ and $\grb_2$ are comparable by Lemma~\ref{lemma:radici-adiacenti}, and we may assume that $\grb_1 \leq \grb_2$. Since $\grb_1 + \grb_2 < \grb$, arguing with Theorem~\ref{teo:avdeev} and Table~\ref{tab: active roots} we have the following possibilities:
\begin{itemize}
	\item[i)] $\grb$ is of type 2, and $\grb_1 = \grb_2 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$;
	\item[ii)] $\grb$ is of type 3, and there are indices $i,j$ with $1 \leq i \leq j < n$ such that $\grb_1 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i$ and $\grb_2 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1+ \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_j$;
	\item[iii)] $\grb$ is of type 4, and there are indices $i,j$ with $3 \leq i \leq j \leq 4$ such that $\grb_1 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_j + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4$ and $\grb_2 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i+ \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_4$;
	\item[iv)] $\grb$ is of type 5 or 6, and $\grb_1 = \grb_2 = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1$.
\end{itemize}
If $\grg \in \max(F_I(\grb))$ is such that $\grb_2 \leq \grg$, in all these cases we get that $\grb_1 + \grb_2 \leq a_{\grb,\grg} \grg$. Moreover, no root $\grb_i$ with $i > 2$ is in $F(\grg)$, since otherwise $\grb - (\grb_1 + \grb_2 + \grb_i) \not \in \mN \Phi^+$, and the inequality follows. Therefore $\grb^\sharp_I \leq \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, and since $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is minimal in $\Phi^+_I$ we get $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb^\sharp_I$. This proves the inclusion $\grD_I \subset \{\grb^\sharp_I \st \grb \in I\}$.

We now show that for all $\grb \in \Psi$ the element $\grb_I^\sharp$ is indecomposable in $\Phi_I^+$. Suppose indeed that $\grb \in \Psi_I$ and let $\grg_1, \ldots, \grg_n \in \Psi_I$ be such that
$$
	\grb_I^\sharp = (\grg_1)_I^\sharp + \ldots + (\grg_n)_I^\sharp.
$$
Being $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_I^\sharp) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)$, since for all $i$ the evaluation of $(\grg_i)_I^\sharp$ takes exactly one negative value on $\mathscr D^*$ by Definition~\ref{def:radici-toriche}, there is $\grg \in \{\grg_1, \ldots, \grg_n\}$ with $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grg)$. Therefore $\grg_I^\sharp \leq \grb_I^\sharp < \grb$, and $\pi(\grg) = \pi(\grg_I^\sharp) \in \supp(\grb)$. By \cite[Corollary 11]{Avd1} we get then $\pi(\grb) = \pi(\grg_I^\sharp)$, and Corollary~\ref{cor:weak-family} shows that $\grb \in \Psi(\grg_I^\sharp)$. Therefore there are $\grb_1, \ldots, \grb_m \in F(\grb)$ such that $\grb = \grg_I^\sharp + \grb_1+ \ldots + \grb_m$. Then $\langle w_0 \rho(\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_i)),\grg_I^\sharp \rangle > 0$ for all $i=1, \ldots, m$, and by Proposition~\ref{prop:radici-stabili} it follows $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb_i) \in I$, that is $\grb_i \in \Psi_I$. Therefore the definition of $\grb_I^\sharp$ implies $\grb_I^\sharp \leq \grg_I^\sharp$, that is $n=1$ and $\grb_I^\sharp = \grg_I^\sharp$.
\end{proof}

We assume for the rest of the section that $H \subset G$ is a spherical subgroup such that $T \subset H \subset B$. Then $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma\ristretto_\Psi : \Psi \lra \mathscr D^*$ is bijective by Proposition~\ref{prop:delta-vs-tau}, and $\pi : \Psi \lra \grD$ is injective by Theorem~\ref{teo:avdeev} iii). We will identify the set of divisors $\mathscr D^*$ with the set of active roots $\Psi$.

Define a graph $\mathcal G(\Psi)$ with set of vertices $\Psi$ as follows: two active roots $\grb, \grb'$ are connected by an edge if and only if they are not strongly orthogonal (namely, neither their sum nor their difference is in $\Phi$). Given $I \subset \Psi$, we denote by $\mathcal G(I)$ the associated subgraph of $\mathcal G(\Psi)$. Notice that, if $\grb, \grb'$ belong to the same connected component of $\mathcal G} \newcommand{\calL}{\mathcal L(I)$, then they belong to the same irreducible component of $\Phi_I$: if indeed $\{\grb, \grb'\} \subset I$ is an edge of $\mathcal G(I)$, then either $\grb + \grb' \in \Phi_I$ or $\grb - \grb' \in \Phi_I$.

If $I \subset \Psi$ we denote by $\Phi'_I$ the root system generated by the simple roots $\pi(\grb)$ with $\grb \in I$, and by $W'_I$ the Weyl group of $\Phi'_I$.

\begin{proposition}	\label{prop:intervalli}
Suppose that $T \subset H$, let $I \subset \Psi$ and suppose that $\pi(I)$ is connected. Then the followings hold:
\begin{itemize}
	\item[i)] $\Phi_I$ is an irreducible parabolic subsystems of $\Phi$ of rank $|I|$.
	\item[ii)] If $\Phi$ is simply laced, then $\Phi_I$ and $\Phi'_I$ are isomorphic root systems.
	\item[iii)] If $\Phi$ is not simply laced and $\pi(I)$ contains both long and short roots, then $\Phi_I$ and $\Phi'_I$ are isomorphic root systems.
\end{itemize}
\end{proposition}

\begin{proof}
i) Let $\grb, \grb' \in I$, denote $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma =\pi(\grb)$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' =\pi(\grb')$ and suppose $\langle \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma', \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma^\vee\rangle < 0$. Suppose that there is some active root whose family contains both $\grb$ and $\grb'$: then $\grb$ and $\grb'$ are comparable by Lemma ~\ref{lemma:radici-adiacenti}, hence $\{\grb,\grb'\}$ is an edge of $\mathcal G} \newcommand{\calL}{\mathcal L(I)$. Therefore $\grb, \grb'$ belong to the same component of $\Phi_I$.

Suppose that there is no active root whose family contains both $\grb$ and $\grb'$. We claim that in this case $\supp(\grb) \cap \supp(\grb') = \vuoto$. Indeed, if this intersection is not empty then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma\in\supp(\grb')$ or $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'\in\supp(\grb)$ (because the Dynkin diagram of $G$ has no loops). Assume that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'\in\supp(\grb)$, then by Theorem~\ref{teo:avdeev} ii) there is $\grb''\in F(\grb)$ with $\pi(\grb'')=\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'$, which implies $\grb''=\grb'$ by the injectivity of $\pi$. Therefore $\grb'\in F(\grb)$, contradicting the assumption.

Therefore $\supp(\grb) \cap \supp(\grb') = \vuoto$, and since $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \supp(\grb)$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' \in \supp(\grb')$ are non-orthogonal Theorem~\ref{teo:avdeev} iv) implies that $\langle \grb', \grb^\vee \rangle = \langle \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma', \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma^\vee\rangle < 0$. Therefore $\grb + \grb' \in \Phi^+_I$, hence $\grb, \grb'$ belong to the same component of $\Phi_I$. It follows that $\Phi_I$ is an irreducible subsystem of $\Phi$ (whose rank is $|I|$ by Remark~\ref{oss:ranks}), and it is parabolic by Corollary~\ref{cor: sistemi-parabolici}.

ii) By Theorem~\ref{teo:DeltaI-rango-massimo} $\grD_I = \{\grb^\sharp_I \st \grb \in I\}$ is a basis of $\Phi_I$. Let $\grb \in I$.
By Corollary~\ref{cor:massimali}, if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'\in\pi(I)$ adjacent to $\pi(\grb)$ then the root $\beta'\in I$ such that $\pi(\beta')=\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ is maximal in $F(\grb)\senza\{\grb\}$, hence $\beta'\in\max(F_I(\grb))$, which implies that $\grb^\sharp_I$ doesn't have $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'$ in its support. In other words $\supp(\grb_I^\sharp) \cap \pi(I)$ contains $\pi(\grb)$ but none of the roots of $\pi(I)$ adjacent to it: thanks to the fact that $\supp(\grb_I^\sharp) \cap \pi(I)$ is connected, we deduce that $\supp(\grb_I^\sharp) \cap \pi(I) = \{\pi(\grb)\}$. Notice also that all elements of $\supp(\grb_I^\sharp)\senza\{\pi(\grb)\}$ are orthogonal to the elements of $\pi(I)\senza\{\pi(\grb)\}$, otherwise the Dynkin diagram of $\Phi$ would have a loop. Since all the roots in $\Phi$ have the same length, it follows that the root systems generated by $\grD_I$ and by $\pi(I)$ are isomorphic.

iii) Let $\grD_0 \subset \grD$ be the connected component containing $\pi(I)$. Denote $\Phi_0$ the corresponding irreducible subsystem of $\Phi$ and enumerate $\grD_0 = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1, \ldots, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n\}$ as in \cite{Bou}. We also enumerate $I$ by setting $\grb_i = \pi^{-1}(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i)$ for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i \in \pi(I)$. If $\pi(I) = \grD_0$, then the claim is clear, since i) implies then $\Phi_I = \Phi'_I = \Phi_0$. In particular we may assume that $\Phi'_I$ is not of type $\sfF_4$ nor of type $\mathsf G_2$.

Suppose first that $\Phi_0$ is of type $\sfB_n$ or $\mathsf C} \newcommand{\sfD}{\mathsf D_n$. Then $\pi(I) = \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_p, \ldots, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n\}$ for some $p$ with $1 < p < n$. Notice that the claim follows once we prove that $\Phi_I$ contains roots of different length: indeed by i) $\Phi_I$ and $\Phi'_I$ are both irreducible parabolic subsystems of $\Phi_0$ of rank $|I|$. By Corollary~\ref{cor:massimali} it follows that $I$ contains all the maximal elements in $F(\grb_n) \senza \{\grb_n\}$, therefore $(\grb_n)_I^\sharp = \pi(\grb_n) = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$ and we get $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n \in \Phi^+_I$.

Suppose that $\Phi_0$ is of type $\sfB_n$, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$ is short. On the other hand a positive root $\grg$ with $[\grg : \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_{n-1}] = 1$ is short if and only if $\grg = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i + \ldots + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$ for some $i \leq n$, therefore either $\grb_{n-1}$ is long or $\grb_{n-1}-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$ is long, and the claim follows since $\Phi_I$ is a parabolic subsystem of $\Phi_0$.

Suppose that $\Phi_0$ is of type $\mathsf C} \newcommand{\sfD}{\mathsf D_n$, then $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n$ is long. On the other hand a positive root $\grg$ with $[\grg:\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_{n-1}] = 1$ is necessarily short, and the claim follows.

Suppose now that $\Phi_0$ is of type $\sfF_4$, then by the assumption $\pi(I) \supset \{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3\}$. If $\bigcup_{\grb \in I} \supp(\grb)$ is of type $\sfB$ or $\mathsf C} \newcommand{\sfD}{\mathsf D$ the same arguments as before apply, suppose that this is not the case. A direct inspection based on Theorem~\ref{teo:avdeev} shows that $\grb_3$ is necessarily a short root, and that either $\grb_3$ is a long root or $\grb_3 - \grb_2$ is a long root. Therefore $\Phi_I$ contains roots of different lengths, hence it is of type $\sfB_{|I|}$ or $\mathsf C} \newcommand{\sfD}{\mathsf D_{|I|}$. If $|I| = 2$ then the claim follows.

Suppose that $|I| = 3$. If $\grb_1 \in I$, then it follows by Corollary \ref{cor:massimali} that $I$ contains all the maximal elements in $F(\grb_1) \senza \{\grb_1\}$ and in $F(\grb_2) \senza \{\grb_2\}$. Therefore $(\grb_i)^\sharp_I = \pi(\grb_i) = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i$ for $i=1,2$, and it follows that $\grD_I$ contains two long roots, hence $\Phi_I$ is of type $\sfB_3$. Similarly, if $\grb_4 \in I$ we conclude that $\grD_I$ contains two short roots, hence $\Phi_I$ is of type $\mathsf C} \newcommand{\sfD}{\mathsf D_3$.
\end{proof}

\section{Combinatorial parameters for the orbits of $B$ in $G/H$}\label{s:Borbits}

We are now ready to give a combinatorial parametrization of the $B$-orbits in $G/H$, where $H$ is a spherical subgroup of $G$ contained in $B$, in terms of the root systems introduced in the previous sections.

\subsection{Reduced and extended pairs}

Consider the projection $G/H \lra G/B$. By the Bruhat decomposition, every $B$-orbit $\mathscr O$ in $G/H$ uniquely determines a Weyl group element, namely that one corresponding to the image of $\mathscr O$ in $G/B$, which is a Schubert cell $BwB/B$. Given $w \in W$ and $I \subset \mathscr D^*$, notice that $w\mathscr U_I$ is a well defined $T$-orbit in $G/H$, mapping on the $T$-fixed point $wB/B \in G/B$. Therefore to every pair $(w,I)$ with $w \in W$ and $I \subset \mathscr D^*$, we may associate a $B$-orbit in $G/H$ by setting $\mathscr O_{w,I} = Bw \mathscr U_I$.

Notice that every $B$-orbit in $G/H$ is of the shape $\mathscr O_{w,I}$ for some $w \in W$ and some $I \subset \mathscr D^*$. Suppose indeed that $\mathscr O$ is a $B$-orbit in $G/H$ which projects on the Schubert cell $BwB/B$. Then, reasoning as in Proposition~\ref{prop:corrispondenza-divisori}, the intersection $w^{-1} \mathscr O \cap B/H$ is a $(B \cap B^w)$-orbit in $B/H$, hence it is $T$-stable. If $\mathscr U_I$ is any $T$-orbit in $w^{-1} \mathscr O \cap B/H$, we have then the equality $\mathscr O = \mathscr O_{w,I}$. While $w$ is uniquely determined in $W$, in general there are several choices for $I \subset \mathscr D^*$.

We will say that $I \subset \mathscr D^*$ is a \textit{representative} for a $B$-orbit $\mathscr O$ if $\mathscr O = \mathscr O_{w,I}$ for some $w \in W$. The goal of the following theorem is to show that there are canonical minimal and maximal representatives for the $B$-orbits in $G/H$, and to give combinatorial characterizations of such pairs.

\begin{theorem}	\label{teo:parametrizzazione}
Let $\mathscr O$ be a $B$-orbit in $G/H$. There exist a unique minimal representative $\mathrm m$ and a unique maximal representative $\mathrm M$, and $I$ is a representative for $\mathscr O$ if and only if $\mathrm m \subset I \subset \mathrm M$. Suppose moreover that $\mathscr O = \mathscr O_{w,I}$ and denote
$$I(w) = \delta\big(\Psi^\mathrm{\sharp}(I) \senza \Phi^+(w) \big).$$
\begin{itemize}
\item[i)]  We have the equalities $\mathrm m = I \senza I(w)$ and $\mathrm M = I \cup I(w)$.
\item[ii)] The following formulas hold:
\begin{align*}
	\dim(\mathscr O) &= \dim(B/H) + l(w) - |\mathscr D^* \senza \mathrm M|,\\
	&\rk(\mathscr O) = \rk(B/H) + |\mathrm m|.
\end{align*}
More precisely, $\mathcal X} \newcommand{\calU}{\mathcal U_B(\mathscr O) = w \mathcal X} \newcommand{\calU}{\mathcal U_T(\mathscr U_\mathrm m)$, and in particular $w(\Phi_\mathrm m) \subset \mathcal X} \newcommand{\calU}{\mathcal U_B(\mathscr O)$.
\end{itemize}
\end{theorem}

\begin{proof}
By the discussion at the beginning of this section, the first claim is equivalent to the fact that $w^{-1}\mathscr O \cap B/H$ contains a unique minimal $T$-orbit and a unique maximal $T$-orbit (ordered via inclusion of closures). Let $I \subset \mathscr D^*$ be a representative for $\mathscr O$, then $w^{-1}\mathscr O \cap B/H = (B \cap B^w) \mathscr U_I$. Therefore $w^{-1}\mathscr O \cap B/H$ is homogeneous under the action of a connected solvable group, and reasoning as in Corollary~\ref{cor:torica} it follows that it is an irreducible $T$-stable affine subvariety of $B/H$. In particular, $w^{-1}\mathscr O \cap B/H$ is itself a toric variety under the action of $T$. Being irreducibile, it contains a unique maximal $T$-orbit, and being affine it contains a unique closed $T$-orbit, hence a unique minimal orbit. It also follows that $w^{-1}\mathscr O \cap B/H$ is open in its closure, whence it contains exactly all $T$-orbits whose closures contain the minimal $T$-orbit and which are contained in the closure of the maximal one.

i) Since $\mathrm m$ is contained in every representative of $\mathscr O$, it is enough to consider the case $I = \mathrm m$. By the description of the action of the root groups on the $T$-orbits in $B/H$, we claim that
\begin{equation}	\label{eqn:formulone}
	\forall D \in \mathrm M \senza \mathrm m \qquad \exists \grb \in \Psi_D \quad \exists \grg \in \mN \Psi_\mathrm m \quad \st \quad \grb - \grg \in \mN (\Psi^\mathrm{\sharp} \senza \Phi^+(w)).
\end{equation}
Indeed, given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$, we have $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \subset B \cap B^w$ if and only if $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \not \in \Phi^+(w)$. Being $\mathscr U_\mathrm M \subset (B \cap B^w) \mathscr U_\mathrm m$, it follows then by Proposition~\ref{prop:Ualphaactionfinal} that there is a sequence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1, \ldots, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n \in \Psi^\mathrm{\sharp} \senza \Phi^+(w)$ such that
$\mathrm M \senza \mathrm m = \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1), \ldots, \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n)\}$ and $U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \cdots U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} \mathscr U_\mathrm m \supset \mathscr U_M$. Moreover, every root $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i$ has to be activated by $\mathrm m \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1), \ldots, \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_{i-1})\}$, that is: for all $i = 1, \ldots, n$ there are $\grg_i \in \mN \Psi_\mathrm m$, $\grb_{i,1} \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)}$, $\ldots$, $\grb_{i,i-1} \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_{i-1})}$ and non-negative integers $a_{i,1}$, $\ldots$, $a_{i,i-1}$ such that
\begin{align*}
	\grb_1 := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \grg_1 \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)}	\\
	\grb_2 := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2 + \grg_2 + a_{1,1} \grb_{1,1} \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2)} \\
	\grb_3 := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3 + \grg_3 + a_{2,1} \grb_{2,1} + a_{2,2} \grb_{2,2} \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3)} \\
	\qquad \qquad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \qquad \qquad \\
	\grb_n := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n + \grg_n + a_{n,1} \grb_{n,1} + \ldots + a_{n,n-1} \grb_{n,n-1} \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n)}
\end{align*}
By Proposition~\ref{prop:attivazioni-equivalenti} it follows then
\begin{align*}
	\grb_1' := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \grg_1 \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1)}	\\
	\grb_2' := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2 + \grg_2 + a_{1,1} \grb_1' \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2)} \\
	\grb_3' := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3 + \grg_3 + a_{2,1} \grb_1' + a_{2,2} \grb_2' \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_3)} \\
	\qquad \qquad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \qquad \qquad \\
	\grb_n' := \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n + \grg_n + a_{n,1} \grb_1' + \ldots + a_{n,n-1} \grb_{n-1}' \in \Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n)}
\end{align*}
In particular, for every $i = 1, \ldots, n$, it follows that $\grb'_i = \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'_i + \grg'_i$ with $\grg'_i \in \mN \Psi_\mathrm m$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma'_i \in \mN \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1 + \ldots + \mN \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i$, and (\ref{eqn:formulone}) follows.

Let now $D \in \mathrm M \senza \mathrm m$, $\grb \in \Psi_D$ and $\grg \in \mN \Psi_\mathrm m$ as in (\ref{eqn:formulone}), and set $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = \grb - \grg$. Then by Lemma~\ref{lemma:differenza-attive} we have $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(\mathrm m)$. On the other hand we have $w(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \in \mN \Phi^+$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \senza \Phi^+(w)$. Therefore $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi^\mathrm{\sharp}(\mathrm m) \senza \Phi^+(w)$, and by Theorem~\ref{teo:deadker} we get $D = \delta(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$, which shows $D \in \mathrm m(w)$.

ii) For the dimension formula, notice that
$$
\dim(B wB/H) - \dim(\mathscr O) = \dim(B/H) - \dim(w^{-1}\mathscr O \cap B/H).
$$
On the other hand by the definition of $\mathrm M$ we have $\dim(B/H) - \dim(w^{-1}\mathscr O \cap B/H) = |\mathscr D^* \senza \mathrm M|$, and the claim follows since $\dim(B wB/H) = l(w) + \dim(B/H)$.

For the rank formula, fix a point $x_\mathrm m \in \mathscr U_\mathrm m$. Notice that $x_\mathrm m$ is a standard base point in $(B \cap B^w)\mathscr U_\mathrm m$, as a homogeneous space under the solvable connected group $B \cap B^w$. Indeed, if $x \in (B \cap B^w)\mathscr U_\mathrm m$ is a standard base point, then $Tx$ is a closed $T$-orbit by Lemma~\ref{lemma:B-rank}. On the other hand by ii) the set $(B \cap B^w)\mathscr U_\mathrm m$ contains a unique closed $T$-orbit, namely $\mathscr U_\mathrm m$, therefore $x_\mathrm m$ is a standard base point. It follows that $wx_\mathrm m \in \mathscr O$ is a standard base point as well, namely $\Stab_T(wx_\mathrm m)$ is a maximal diagonalizable subgroup of $\Stab_B(wx_\mathrm m)$. Indeed $\Stab_T(wx_\mathrm m) = w\Stab_T(x_\mathrm m) w^{-1}$, and being $\Stab_G(x_\mathrm m) \subset B$ we get
$$
\Stab_B(wx_\mathrm m) = B \cap w \Stab_B(x_\mathrm m) w^{-1} = w \Stab_{B \cap B^w} (x_\mathrm m)w^{-1}.
$$
Therefore $wx_\mathrm m \in \mathscr O$ is a standard base point, since $x_\mathrm m \in (B \cap B^w)\mathscr U_\mathrm m$ is a standard base point.

By Lemma~\ref{lemma:B-rank} we have then
$$
	\mathcal X} \newcommand{\calU}{\mathcal U_B(\mathscr O) = \mathcal X} \newcommand{\calU}{\mathcal U_T(w\mathscr U_\mathrm m) = w\mathcal X} \newcommand{\calU}{\mathcal U_T(\mathscr U_\mathrm m),
$$
and by Proposition~\ref{prop:radici-reticoli} we get $w(\Phi_\mathrm m) \subset \mathcal X} \newcommand{\calU}{\mathcal U_B(\mathscr O)$.
\end{proof}

Given $w \in W$ and $I\subset \mathscr D^*$ we will denote by $\mathrm m_{w,I}$ the minimal representative of $\mathscr O_{w,I}$ and by $\mathrm M_{w,I}$ the maximal representative of $\mathscr O_{w,I}$.

\begin{definition}
Let $w \in W$ and $I\subset \mathscr D^*$. We say that $(w,I)$ is a \textit{reduced pair} if $I = \mathrm m_{w,I}$ (i.e., if $I \cap I(w) = \vuoto$), and we say that it is an \textit{extended pair} if $I = \mathrm M_{w,I}$ (i.e., if $I(w) \subset I$). Given $w \in W$ and $I \subset \mathscr D^*$, we will call $(w,\mathrm m_{w,I})$ and $(w,\mathrm M_{w,I})$ respectively the \textit{reduction} and the \textit{extension} of $(w,I)$.
\end{definition}

\begin{remark}\label{rem:pairs}
Given $I \subset \mathscr D^*$, we have by definition $\Psi^\mathrm{\sharp}_I = \Psi^\mathrm{\sharp}(I) \cap \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma^{-1}(I)$, whereas by Theorem~\ref{teo:parametrizzazione} we have the inclusion $\Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_{\mathrm M_{w,I}} \subset \Phi^+(w)$. The following characterizations of reduced pairs and of extended pairs follow immediately:
\begin{itemize}
	\item[i)] The pair $(w,I)$ is reduced if and only if $\Psi^\mathrm{\sharp}_I \subset \Phi^+(w)$, if and only if $\Phi^+_I \subset \Phi^+(w)$.
	\item[ii)] The pair $(w,I)$ is extended if and only if $\Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_I \subset \Phi^+(w)$.
\end{itemize}
\end{remark}

To summarize:

\begin{corollary} \label{cor:parametrizzazione}
The map $(w,I) \longmapsto \mathscr O_{w,I}$ gives a parametrization by reduced (resp. extended) pairs
$$
	\{(w,I) \st \Phi^+_I \subset \Phi^+(w) \} \longleftrightarrow \mathscr B(G/H) \longleftrightarrow \{(w,I) \st \Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_I \subset \Phi^+(w) \} .
$$
Moreover:
\begin{itemize}
	\item[i)] If $(w,I)$ is reduced, then $\rk (\mathscr O_{w,I}) = \rk(B/H) + |I|$.
	\item[ii)] If $(w,I)$ is extended, then $\dim (\mathscr O_{w,I}) = \dim(B/H) + l(w) - |\mathscr D^* \senza I|$.
\end{itemize}
\end{corollary}

\begin{remark}	\label{oss:compatibilità}
The following properties easily follow from Remark~\ref{rem:pairs}:
\begin{itemize}
	\item[i)] All the pairs of the shape $(w,\vuoto)$ are reduced, and all the pairs of the shape $(w,\mathscr D^*)$ are extended. All the pairs of the shape $(w_0, I)$ are both reduced and extended.
	\item[ii)] If $(w,I)$ is reduced (resp. extended) and if $J \subset I$, then $(w,J)$ is also reduced (resp. extended).
	\item[iii)] If $(w,I)$ is reduced (resp. extended) and if $w \preceq v$ (i.e. $w$ is a right subexpression of $v$), then $(v,I)$ is also reduced (resp. extended).
\end{itemize}
\end{remark}

\begin{corollary}	\label{cor: orbite chiuse}
Suppose that $(w,I)$ is reduced. Then $\mathscr O_{w,I}$ is closed in $G/H$ if and only if the following conditions hold:
\begin{itemize}
	\item[i)] $I = \vuoto$;
	\item[ii)] $\Phi^+(w) \subset \Psi$;
	\item[iii)] $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma \ristretto_{\Phi^+(w)} : \Phi^+(w) \lra \mathscr D^*$ is injective and $\Psi_{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Phi^+(w))} = \Phi^+(w)$.
\end{itemize}
\end{corollary}

\begin{proof}
As a consequence of the main theorem in \cite{Re}, every closed $B$-orbit has minimal rank. Therefore every closed $B$-orbit in $G/H$ is of the shape $\mathscr O_{w,\vuoto}$ for some $w \in W$. On the other hand $B/H$ is a closed $B$-orbit, and by \cite[Proposition 2.2]{Br2} all closed $B$-orbits have the same dimension, therefore $\mathscr O_{w,\vuoto}$ is closed if and only if $\dim(\mathscr O_{w,\vuoto}) = \dim(B/H)$.

By Theorem~\ref{teo:parametrizzazione} the maximal representative of $\mathscr O_{w,\vuoto}$ is $\mathrm M = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Psi \senza \Phi^+(w))$, therefore $\dim(\mathscr O_{w,\vuoto}) = \dim(B/H)$ if and only if $|\mathscr D^* \senza \mathrm M| = l(w)$. On the other hand
$$
\mathscr D^* \senza \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Psi \senza \Phi^+(w)) \subset \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Psi \cap \Phi^+(w))
$$
and the latter has cardinality at most $l(w)$, therefore $|\mathscr D^* \senza \mathrm M| = l(w)$ if and only if $\Phi^+(w) \subset \Psi$ and $\mathscr D^* \senza \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Psi \senza \Phi^+(w)) = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\Phi^+(w))$ has cardinality $l(w)$.
\end{proof}

\begin{example}
Assume that $T\subset H$. By Proposition~\ref{prop:delta-vs-tau} it follows that the restriction $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma\ristretto_\Psi : \Psi \lra \mathscr D^*$ is bijective, and we may identify $\Psi = \mathscr D^*$. Moreover Corollary~\ref{cor: orbite chiuse} implies
$$
	BwH \text{ is closed} \iff \Phi^+(w) \subset \Psi,
$$
and the formula for the dimension of an orbit of minimal rank simplifies as follows
$$
	\dim(BwH/H) = \dim(B/H) + l(w) - |\Psi \cap \Phi^+(w)| = \dim (B/H) + |\Phi^+(w) \senza \Psi|.
$$
Given a reduced pair $(w,I)$, the description of the maximal representative of $\mathscr O_{w,I}$ reduces to
$$
	\mathrm M_{w,I} = I \cup \{ \grb \in \Psi \st \grb - \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \mN \Psi_I \quad \exists \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \senza \Phi^+(w)\big\}.
$$
Notice that $\Psi \senza \mathrm M_{w,I} \subset \Phi^+(w)$. Indeed if $I \subset J \subset \mathscr D^*$ then it also holds $\mathrm M_{w,I} \subset \mathrm M_{w,J}$, therefore $\Psi \senza \Phi^+(w) = \mathrm M_{w,\vuoto} \subset \mathrm M_{w,I}$. In particular, if we denote
$$
	\Phi^+(w,I) = \Phi^+(w) \senza (\Psi \senza \mathrm M_{w,I}),
$$
then we may simplify the dimension formula as follows:
$$
	\dim(\mathscr O_{w,I}) = \dim (B/H) + |\Phi^+(w,I)|.
$$
Notice that $\Phi^+_I \subset \Phi^+(w,I)$. Also, by the above inclusions $\mathrm M_{w,I} = \Psi \senza \big(\Phi^+(w) \senza \Phi^+(w,I)\big)$, so that the orbit $\mathscr O_{w,I}$ is uniquely determined by $w$ together with the set of inversions $\Phi^+(w,I)$.
\end{example}

\begin{example}[see \cite{Ti1}] \label{ex:timashev}
Let $U' = \prod_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \senza \grD} U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ be the derived subgroup of $U$, then $H = T U'$ is a spherical subgroup of $G$ which is contained in $B$. Then we can identify $\mathscr D^* = \Psi = \grD$, and given $I \subset \grD$ we have $I(w) = \grD \cap w^{-1}(\Phi^+)$. Therefore, given $w \in W$ and $I \subset \grD$, the pair $(w,I)$ is reduced if and only if $w(I) \subset \Phi^-$, whereas it is extended if and only if $\grD \cap w^{-1}(\Phi^+) \subset I$. In particular if $(w,I)$ is reduced then we have $\rk (\mathscr O_{w,I}) = |I|$ and
$$
	\dim (\mathscr O_{w,I}) = l(w) + |I| + |\grD \cap w^{-1}(\Phi^+)|.
$$
\end{example}

\subsection{Comparison with the wonderful case}\label{s:reduction}

An important class of spherical subgroups of a reductive group $G$ is that of \textit{wonderful subgroups} (see e.g. \cite{BL}). This class plays a prominent role in the theory of spherical varieties, both in their classification and in the study of their geometry. In particular, to any spherical subgroup $H$ one may canonically associate a wonderful subgroup $\ol H$ containing $H$, called the \textit{spherical closure} of $H$. Given a spherical subgroup $H$ contained in $B$, the aim of this subsection is to compare the set of $B$-orbits in $G/H$ with that of $G/\ol H$, and to show that these two sets are canonically identified. We keep the notations introduced in Section~\ref{ssec:spherical}.

Restricting to the strongly solvable case, a spherical subgroup $H \subset G$ contained in $B$ is wonderful if and only if $\mathcal X} \newcommand{\calU}{\mathcal U(B/H) = \mathbb Z\Psi$. This characterization follows from Proposition~\ref{prop:B/H-toric} and Remark~\ref{rem:spherical roots}.

If $H$ is a spherical subgroup of $G$, then the normalizer $N_G(H)$ acts naturally on $G/H$ by $G$-equivariant automorphisms, thus it acts on the finite set $\Div^B(G/H)$. The spherical closure $\overline H$ of $H$ is by definition the kernel of the this action, and we say that $H$ is \textit{spherically closed} if $H = \ol H$. Notice that we have $Z(G) \subset \overline H \subset N_G(H)$, where $Z(G)$ denotes the center of $G$. By \cite[Corollary 5.25]{Avd2}, it follows that if $H$ is contained in $B$, then $\ol H$ is also contained in $B$, hence it is strongly solvable.

As a consequence of a general theorem of Knop \cite[Corollary 7.6]{Kn3}, the spherical closure $\ol H$ is a wonderful subgroup of $G$. On the other hand $\ol{\ol H} = \ol H$ (see e.g.\ \cite[Section 2.4.2]{BL}), therefore every spherically closed spherical subgroup of $G$ is wonderful. In the case of a strongly solvable spherical subgroup $H \subset G$, the converse also is true: $H$ is wonderful if and only if it is spherically closed (see \cite[Corollary 3.42]{Avd2}).

Let $H \subset G$ be wonderful spherical subgroup contained in $B$. Then the weight lattice $\mathcal X} \newcommand{\calU}{\mathcal U(B/H) = \mathbb Z \Psi$ admits a basis of simple roots, namely the set $\grS$ introduced in Lemma~\ref{lem:spherical-roots}. By the equality $\mathcal X} \newcommand{\calU}{\mathcal U(B/H) = w_0\mathcal X} \newcommand{\calU}{\mathcal U(G/H)$ it follows that the subset of simple roots $\grS^* = -w_0(\grS)$ is a basis of $\mathcal X} \newcommand{\calU}{\mathcal U(G/H)$.

\begin{lemma}\label{lemma:connected}
Suppose that $G$ is adjoint and let $H \subset G$ be a strongly solvable wonderful subgroup. Then $H$ is connected.
\end{lemma}

\begin{proof}
The identity component $H^\circ \subset H$ is a spherical subgroup of $G$, and the pull-back of rational functions along the projection $G/H^\circ \lra G/H$ identifies the weight lattice $\mathcal X} \newcommand{\calU}{\mathcal U(G/H)$ with a sublattice of finite index of $\mathcal X} \newcommand{\calU}{\mathcal U(G/H^\circ)$. Since $\grS^* \subset \grD$, $\mathcal X} \newcommand{\calU}{\mathcal U(G/H)$ is a saturated sublattice of the root lattice. Since $G$ is adjoint this implies $\mathcal X} \newcommand{\calU}{\mathcal U(G/H) = \mathcal X} \newcommand{\calU}{\mathcal U(G/H^\circ)$, and the equality $H = H^\circ$ follows by the isomorphism $H/H^\circ \simeq \mathcal X} \newcommand{\calU}{\mathcal U(G/H^\circ)/\mathcal X} \newcommand{\calU}{\mathcal U(G/H)$ (see e.g. \cite[Lemma 2.4]{Ga}).
\end{proof}

\begin{corollary}\label{cor:wonderful}
Suppose that $H \subset G$ is spherical and strongly solvable. Then the projection $G/H\to G/\ol H$ induces a bijection between $\mathscr B(G/H)$ and $\mathscr B(G/\ol H)$.
\end{corollary}

\begin{proof}
Denote $H' = H Z(G)$, so that $H \subset H' \subset \ol H$. The projection $G/H\to G/H'$ induces a bijection between $\mathscr{B}(G/H)$ and $\mathscr{B}(G/H')$. Therefore we may replace $H$ with $H'$, and since $Z(G) \subset H'$ we may also assume that $G$ is adjoint. On the other hand by Lemma~\ref{lemma:connected} the quotient $\ol H/H$ is connected, hence the claim follows by \cite[Lemma 3]{Br3}.
\end{proof}

\newpage
\section{The Weyl group action on $\mathscr B(G/H)$}\label{s:W}

\subsection{Preliminaries}	\label{ssec:prel-W-action}

We denote by $M(W)$ the \textit{Richardson-Springer monoid}, namely the monoid generated by elements $m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ ($\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$) with defining relations $m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)^2 = m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$ and the braid relations
$$m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) m(s_\grb) m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \ldots = m(s_\grb) m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) m(s_\grb) \ldots$$
for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma, \grb \in \grD$, the number of factors on both sides being the order of $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma s_\grb$ in $W$ (see \cite[3.10]{RS1}).

As a set, $M(W)$ is identified with $W$, and given $w \in W$ we will denote by $m(w) \in M(W)$ the corresponding element. Hence we may consider the Richardson-Springer monoid as the Weyl group with a different multiplication, defined by the following rule: if $w \in W$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$, then
$$
	m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) m(w) = \left\{ \begin{array}{ll}
			m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w) & \text{ if } l(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w) > l(w)\\
			m(w) & \text{ if } l(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w) < l(w)
	\end{array}\right.
$$
From a geometrical point of view, the multiplication on $M(W)$ coincides with the one defined on the Weyl group by the multiplication of Bruhat cells, namely we have $\ol{B wB w' B} = \ol{Bw''B}$, where $w'' \in W$ is the element defined by the equality $m(w'') = m(w) m(w')$. Sometime we will identify $M(W)$ and $W$ as sets, in that case we will denote the product in $M(W)$ of two elements $w, w' \in W$ by $w * w'$.

Given a spherical homogeneous variety $G/H$, both $W$ and $M(W)$ act on the set of $B$-orbits $\mathscr B(G/H)$. The action of $M(W)$ was defined by Richardson and Springer in the case of a symmetric homogeneous variety (see \cite[4.7]{RS1}), and the definition carries over without modifications to the spherical case. To define the action of $M(W)$ in the case  of a simple reflection, given $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$ and $\mathscr O \in \mathscr B(G/H)$, consider $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O$. This is an irreducible $B$-variety, and since $G/H$ is spherical it decomposes into finitely many $B$-orbits. The element $m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cdot \mathscr O$ is defined then as the open $B$-orbit in $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma  \mathscr O$. This definition extends to an action of the monoid $M(W)$ on $\mathscr B(G/H)$, and it allows to define a partial order  $\preceq$ on $\mathscr B(G/H)$ (called the \textit{weak order}) as follows:
$$
	\mathscr O \preceq \mathscr O' \text{ if and only if } \quad \exists w \in W \; : \; \mathscr O' = m(w) \cdot \mathscr O
$$

The action of $W$ on $\mathscr B(G/H)$ is much more subtle than that of $M(W)$ and was defined by Knop \cite{Kn}. We recall the definition of this action in the case of a spherical subgroup $H \subset G$ contained in $B$, where the involved considerations turn out to be easier. By a case-by-case consideration (see \cite[Lemma 3.2]{Kn} and \cite[Lemma 5 iv)]{Br3}), the $B$-stable variety $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O$ decomposes in the union of two $B$-orbits or in the union of three $B$-orbits. More precisely we have the following possibilities:
\begin{itemize}
	\item[(U)] Suppose that $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O = \mathscr O \cup \mathscr O'$ decomposes in the union of two orbits, and assume for simplicity that $\mathscr O$ is the open one. Then $\dim \mathscr O' = \dim \mathscr O -1$ and $\mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O') = s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O)$, and we define $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cdot \mathscr O = \mathscr O'$.
	\item[(T)] Suppose that $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O = \mathscr O \cup \mathscr O' \cup \mathscr O''$ decomposes in the union of three orbits, and assume for simplicity that $\mathscr O$ is the open one. Then $\dim \mathscr O' = \dim \mathscr O'' = \dim \mathscr O-1$ and $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O') = \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O'') \subset \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O) = s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O)$, where $\mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O)/\mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O') \simeq \mathbb Z$, and we define $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cdot \mathscr O = \mathscr O$ and $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cdot \mathscr O' = \mathscr O''$.
\end{itemize}

By \cite[Theorem 5.9]{Kn}, the $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$-actions defined above glue together into an action of $W$ on $\mathscr B(G/H)$. As a consequence of the previous analysis, notice that the rank of a $B$-orbit is invariant for this action, which agrees with the action of $W$ on the weight lattices $\mathcal X} \newcommand{\calU}{\mathcal U(\mathscr O)$, where $\mathscr O \in \mathscr B(G/H)$.

\begin{example}
If $H = B$ is a Borel subgroup, then the unique possible rank for a $B$-orbit is zero. Indeed every Schubert cell is an orbit for a suitable unipotent subgroup of $B$, hence every $B$-semiinvariant function is constant. Therefore $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O$ is of type (U) for every $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$, and $\mathscr B(G/B)$ is a homogeneous space under $W$.
\end{example}

In some special cases the rank uniquely identifies the $W$-orbit, in particular this happens when the rank of a $B$-orbit is maximal and minimal. This is summarized in the following theorem, which holds for any spherical subgroup $H$: the case of maximal rank being due to Knop (see \cite[Theorem 6.2]{Kn}) and that of minimal rank to Ressayre (see \cite[Corollary 3.1 and Theorem 4.2]{Re}).

\begin{theorem}	\label{teo: ranghi speciali}
Let $\mathscr O, \mathscr O' \in \mathscr{B}(G/H)$ and suppose that $\rk \mathscr O = \rk \mathscr O' = \rk G/H$ (resp. $\rk \mathscr O = \rk \mathscr O' = \rk G - \rk H$). Then there exists $w \in W$ such that $\mathscr O' = w \cdot \mathscr O$.
\end{theorem}

\subsection{Stabilizers for the $W$-action on $\mathscr B(G/H)$}

We now describe the actions of $W$ and of $M(W)$ on $\mathscr B(G/H)$ in the case of a strongly solvable spherical subgroup $H \subset G$ contained in $B$, in terms of the combinatorial parametrization of Corollary~\ref{cor:parametrizzazione}.

Notice that, if $(w,I)$ is a reduced pair, the inclusion $\Phi^+_I \subset \Phi^+(w)$ implies the equalities
$$
w^{-1}(\grD^-) \cap \Phi^+_I = w^{-1}(\grD^-) \cap \Psi^\mathrm{\sharp}_I = w^{-1}(\grD^-) \cap \grD_I.
$$

In order to study the actions of $W$ and of $M(W)$ in terms of reduced and extended pairs, we turn to a closer look to the possible cases arising in the decomposition of the $B$-stable subsets $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I}$, where $\mathscr O_{w,I} \in \mathscr B(G/H)$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$.

\begin{lemma} \label{lemma:triple-T}
\begin{itemize}
	\item[i)] Let $(w,I)$ be a reduced pair, let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$ and denote $\grb = -w^{-1}(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. Then we have
$$
	s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cdot \mathscr O_{w,I} = \left\{ \begin{array}{ll}
			\mathscr O_{w,I} & \text{ if } \grb \in \Psi^\mathrm{\sharp}_I \\
			\mathscr O_{s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w,I} & \text{ otherwise }
	\end{array}\right.
$$
where all the orbits above are expressed in terms of reduced pairs.
	\item[ii)] Let $(w,I)$ be an extended pair, let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$ and denote $\grb = -w^{-1}(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$. Then we have
$$
	m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cdot \mathscr O_{w,I} = \left\{ \begin{array}{ll}
			\mathscr O_{s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \!* w, I\cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}} & \text{ if } \grb \in \Psi^\mathrm{\sharp}(I) \\
			\mathscr O_{s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \!* w,I} & \text{ if } \grb \not \in \Psi^\mathrm{\sharp}(I)
	\end{array}\right.
$$
where all the orbits above are expressed in terms of extended pairs. In particular, we have $m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cdot \mathscr O_{w,I} = \mathscr O_{w,I}$ if and only if $\grb \in \Phi^+(w) \senza (\Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_I)$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $w \in W$ and $I \subset \mathscr D^*$, and suppose that $(w,I)$ is reduced. By the results in Section~\ref{ssec:prel-W-action} the $B$-stable subset $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w\mathscr U_I$ decomposes either in the union of two $B$-orbits which are permuted by the action of $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, or in the union of three $B$-orbits, an open one fixed by $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ and two of codimension one which are permuted by $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$.

Suppose $\grb \in \Phi^+$ and consider the decomposition
$$
	P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma  = Bs_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cup Bs_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma B s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = B s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \cup B U_{-\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma}.
$$
Denote $v = s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w$, it follows then the decomposition $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = Bv \mathscr U_I \cup Bw U_{\grb} \mathscr U_I$. By Proposition~\ref{prop:Ualphaactionfinal} we get
$$
	U_\grb \mathscr U_I  = \left\{ \begin{array}{ll}
	\mathscr U_I & \text{if $\grb \in \Phi^+ \senza \Psi^\mathrm{\sharp}(I)$}\\
	\mathscr U_I \cup \mathscr U_{I \cup \{\delta(\grb)\}} & \text{if $\grb \in \Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_I$}\\
	\mathscr U_I \cup \mathscr U_{I \senza \{\delta(\grb)\}} & \text{if $\grb \in \Psi^\mathrm{\sharp}_I$}
	\end{array} \right.
$$
Correspondingly we have the following possibilities, where we denote $\mathrm m = \mathrm m_{w,I}$ and $\mathrm M = \mathrm M_{w,I}$:
\begin{itemize}
	\item[U)] Suppose $\grb \in \Phi^+ \senza \Psi^\mathrm{\sharp}(\mathrm m)$ or $\grb \in \Psi^\mathrm{\sharp}(\mathrm m) \cap \Psi^\mathrm{\sharp}_{\mathrm M} \senza \Psi^\mathrm{\sharp}_\mathrm m$.\\
We have in this case $U_\grb \mathscr U_I \subset (B \cap B^w)\mathscr U_I$, hence $BwU_\grb \mathscr U_I = \mathscr O_{w,I}$. Therefore $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = \mathscr O_{v,I} \cup \mathscr O_{w,I}$ decomposes in the union of two orbits as represented in the following diagram, and we have
\begin{itemize}
	\item $\dim \mathscr O_{v,I} = \dim \mathscr O_{w,I} -1$;
	\item $\rk \mathscr O_{v,I} = \rk \mathscr O_{w,I}$;
	\item $\mathrm M_{v,I} = \mathrm M$;
	\item $\mathrm m_{v,I} =\mathrm m$.
\end{itemize}
\[	\xy
0;/r5pc/:*\dir{*};
p+(0,0.2)="a",*+!UL{\mathscr O_{w,\mathrm m} = \mathscr O_{w,\mathrm M}},"a",
p+(0,-.5)*\dir{*}="c",
**\dir{-},*+!UL{\mathscr O_{v,\mathrm m}=\mathscr O_{v,\mathrm M}},"c",
\endxy
\]
	\item[T1)] Suppose $\grb \in \Psi^\mathrm{\sharp}_\mathrm m$.\\
We have in this case $U_\grb \mathscr U_I = \mathscr U_I \cup \mathscr U_{I'}$, where we set $I' = I \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$, and by Theorem~\ref{teo:parametrizzazione} it follows $\mathscr U_{I'} \not \subset (B \cap B^w)\mathscr U_I$. Therefore $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = \mathscr O_{v,I} \cup \mathscr O_{w,I'} \cup \mathscr O_{w,I}$ decomposes in the union of three orbits as represented by the following diagram, and we have
\begin{itemize}
	\item $\dim \mathscr O_{v,I} = \dim \mathscr O_{w,I'}  = \dim \mathscr O_{w,I} - 1$;
	\item $\rk \mathscr O_{v,I} = \rk \mathscr O_{w,I'} = \rk \mathscr O_{w,I} - 1$;
	\item $\mathrm M_{v,I} = \mathrm M$, $\mathrm M_{w,I'} = \mathrm M \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$;
	\item $\mathrm m_{v,I} = \mathrm m_{w,I'} = \mathrm m \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$.
\end{itemize}
\[	\xy
0;/r5pc/:*\dir{*};
p+(0,0.2)="a",*+!UL{\mathscr O_{w,\mathrm m} = \mathscr O_{w,\mathrm M}},"a",
p+(-.5,-.5)*\dir{*},**\dir{-},
p+(-1.3,-.5)="b",*+!UL{\mathscr O_{v,\mathrm m \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}}=\mathscr O_{v,\mathrm M}},"b",
p+(.5,-.5)*\dir{*}="c",
**\dir{-},*+!UL{\mathscr O_{w,\mathrm m \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}}=\mathscr O_{w,\mathrm M \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}}},"c",
\endxy
\]

	\item[T2)] Suppose $\grb \in \Psi^\mathrm{\sharp}(\mathrm m) \senza \Psi^\mathrm{\sharp}_\mathrm M$.\\
Then $U_\grb \mathscr U_I = \mathscr U_I \cup \mathscr U_{I'}$, where we set $I' = I \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$, and by Theorem~\ref{teo:parametrizzazione} we have $\mathscr U_{I'} \not \subset (B \cap B^w)\mathscr U_I$. Therefore $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = \mathscr O_{v,I} \cup \mathscr O_{w,I} \cup \mathscr O_{w,I'}$ decomposes in the union of three orbits as represented by the following diagram, and we have
\begin{itemize}
	\item $\dim \mathscr O_{v,I} = \dim \mathscr O_{w,I}  = \dim \mathscr O_{w,I'} - 1$;
	\item $\rk \mathscr O_{v,I} = \rk \mathscr O_{w,I} = \rk \mathscr O_{w,I'} - 1$;
	\item $\mathrm M_{v,I} = \mathrm M_{w,I'} = \mathrm M \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$;
	\item $\mathrm m_{v,I} = \mathrm m$, $\mathrm m_{w,I'} = \mathrm m \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$.
\end{itemize}
\[	\xy
0;/r5pc/:*\dir{*};
p+(0,0.2)="a",*+!UL{\mathscr O_{w,\mathrm m \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}} = \mathscr O_{w,\mathrm M \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}}},"a",
p+(-.5,-.5)*\dir{*},**\dir{-},
p+(-1.3,-.5)="b",*+!UL{\mathscr O_{v,\mathrm m}=\mathscr O_{v,\mathrm M \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}}},"b",
p+(.5,-.5)*\dir{*}="c",
**\dir{-},*+!UL{\mathscr O_{w,\mathrm m}=\mathscr O_{w,M}},"c",
\endxy
\]
\end{itemize}

Suppose now $\grb \in \Phi^-$ and consider the decomposition
$$
	P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma  = B \cup B s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma B = B \cup B s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma.
$$
Denote $v=s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma w$, then we get $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = Bw\mathscr U_I \cup B v U_{\grb} \mathscr U_I$. Since $(w,I)$ is reduced we have $\grb \not \in \Psi^\mathrm{\sharp}_I$ and $\grb \not \in \Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_M$. Reasoning as before, we get the following possibilities, where we denote $\mathrm m = \mathrm m_{w,I}$, $\mathrm M = \mathrm M_{w,I}$, $\mathrm M' = \mathrm M_{v,I}$:
\begin{itemize}
	\item[U)] Suppose $\grb \in \Phi^+ \senza \Psi^\mathrm{\sharp}(\mathrm m)$ or $\grb \in \Psi^\mathrm{\sharp}(\mathrm m) \cap \Psi^\mathrm{\sharp}_{\mathrm M'} \senza \Psi^\mathrm{\sharp}_\mathrm m$.\\
Then $U_\grb \mathscr U_I \subset (B \cap B^v)\mathscr U_I$, hence $BvU_\grb \mathscr U_I = \mathscr O_{v,I}$. Therefore $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = \mathscr O_{w,I} \cup \mathscr O_{v,I}$ decomposes in the union of two orbits as represented in the following diagram, where we have:
\begin{itemize}
	\item $\dim \mathscr O_{w,I} = \dim \mathscr O_{v,I} +1$;
	\item $\rk \mathscr O_{w,I} = \rk \mathscr O_{v,I}$;
	\item $\mathrm M_{v,I} = \mathrm M$;
	\item $\mathrm m_{v,I} = \mathrm m$.
\end{itemize}
\[	\xy
0;/r5pc/:*\dir{*};
p+(0,0.2)="a",*+!UL{\mathscr O_{v,\mathrm m} = \mathscr O_{v,\mathrm M}},"a",
p+(0,-.5)*\dir{*}="c",
**\dir{-},*+!UL{\mathscr O_{w,\mathrm m}=\mathscr O_{w,\mathrm M}},"c",
\endxy
\]
	\item[T)] Suppose $\grb \in \Psi^\mathrm{\sharp}(\mathrm m) \senza \Psi^\mathrm{\sharp}_{\mathrm M'}$.\\
Then $U_\grb \mathscr U_I = \mathscr U_I \cup \mathscr U_{I'}$, where we set $I' = I \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$, and by Theorem~\ref{teo:parametrizzazione} it follows $\mathscr U_{I'} \not \subset (B \cap B^v)\mathscr U_I$. Therefore $P_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \mathscr O_{w,I} = \mathscr O_{w,I} \cup \mathscr O_{v,I} \cup \mathscr O_{v,I'}$ decomposes in the union of three orbits as represented by the following diagram, where we have
\begin{itemize}
	\item $\dim \mathscr O_{v,I} = \dim \mathscr O_{w,I}  = \dim \mathscr O_{v,I'} - 1$;
	\item $\rk \mathscr O_{v,I} = \rk \mathscr O_{w,I} = \rk \mathscr O_{v,I'} - 1$;
	\item $\mathrm M_{v,I} = \mathrm M \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$, $\mathrm M_{v,I'} = \mathrm M$;
	\item $\mathrm m_{v,I} = \mathrm m$, $\mathrm m_{v,I'} = \mathrm m \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}$.
\end{itemize}
\[	\xy
0;/r5pc/:*\dir{*};
p+(0,0.2)="a",*+!UL{\mathscr O_{v,\mathrm m \cup \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}} = \mathscr O_{v,\mathrm M}},"a",
p+(-.5,-.5)*\dir{*},**\dir{-},
p+(-1.3,-.5)="b",*+!UL{\mathscr O_{w,\mathrm m}=\mathscr O_{w,\mathrm M}},"b",
p+(.5,-.5)*\dir{*}="c",
**\dir{-},*+!UL{\mathscr O_{v,\mathrm m}=\mathscr O_{v,\mathrm M \senza \{\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma(\grb)\}}},"c",
\endxy
\]
\end{itemize}

The claims follow now by applying the definitions of the actions of $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ and of $m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma)$ in all the possibilities presented above.
\end{proof}

\begin{corollary}
Let $(w,I)$ be an extended pair. The orbit $\mathscr O_{w,I}$ is minimal w.r.t. the weak order on $\mathscr B(G/H)$ if and only if
$$
	w^{-1}(\grD^-) \cap \Phi^+ \subset \Psi^\mathrm{\sharp}(I) \senza \Psi^\mathrm{\sharp}_I.
$$
\end{corollary}

\begin{proof}
The claim follows by the previous lemma, indeed $\mathscr O_{w,I}$ is minimal w.r.t. the weak order if and only if $m(s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma) \cdot \mathscr O_{w,I} \neq \mathscr O_{w,I}$ for all $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD$.
\end{proof}

We now focus on the action of $W$ on $\mathscr B(G/H)$. First we show that the minimal representative of an orbit is a complete invariant for the action of $W$, namely it is invariant and it distinguishes the $W$-orbits. Then we will describe the stabilizers of the action in terms of reduced pairs.

\begin{theorem}	\label{teo:stabilizzatori1}
Let $(w,I)$, $(v,J)$ be reduced pairs. Then $\mathscr O_{w,I}$ and $\mathscr O_{v,J}$ are in the same $W$-orbit if and only if $I = J$, in which case we have $\mathscr O_{v,I} = vw^{-1} \cdot \mathscr O_{w,I}$.
\end{theorem}

\begin{proof}
By Lemma~\ref{lemma:triple-T} i) it follows that the minimal representative $I$ is an invariant for the action of $W$. To show that it uniquely determines the $W$-orbit, we show that if $(w,I)$ is reduced then $\mathscr O_{w_0,I} = w_0 w^{-1} \cdot \mathscr O_{w,I}$. In particular, this will imply that every $W$-orbit in $\mathscr B(G/H)$ contains a unique element which projects dominantly on $G/B$.

Suppose that $(w,I)$ is reduced and let $w_0 w^{-1} = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n}$ be a reduced expression. Then $w_{i+1}^{-1}(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i) \in \Phi^+$ for every $i=1, \ldots n$, where we denote
$$
	w_j = \left\{ \begin{array}{ll}
			w & \text{ if } j = n+1 \\
			s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_j} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} w & \text{ if } 0 < j \leq n
	\end{array}\right.
$$
Therefore, for every $i \leq n$, we have $\Phi^+(w_i) = \Phi^+(w_{i+1}) \cup \{w_{i+1}^{-1}(\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i)\}$. Since $(w,I)$ is reduced, it follows that $\Phi^+_I \subset \Phi^+(w)$, thus $\Phi^+_I \subset \Phi^+(w_i)$ for all $i \leq n$. It follows that $(w_i,I)$ is reduced as well for all $i \leq n$, and by Lemma~\ref{lemma:triple-T} we get $\mathscr O_{w_{i-1},I} = s_{i-1} \cdot \mathscr O_{w_i,I}$. Combining all the steps we get $\mathscr O_{w_0,I} = w_0 w^{-1} \cdot \mathscr O_{w,I}$, and the last claim also follows.
\end{proof}

\begin{corollary}	\label{cor: Worbits}
The Weyl group $W$ has $2^{|\mathscr D^*|}$ orbits in $\mathscr B(G/H)$, and a complete set of representatives is given by the orbits which project dominantly on $G/B$, namely by the subsets of $\mathscr D^*$.
\end{corollary}

It follows by Theorem~\ref{teo:stabilizzatori1} that every $W$-orbit contains a distinguished element.

\begin{corollary}
Let $(w,I)$ be a reduced pair, then the element $w^{-1} \cdot\mathscr O_{w,I}$ depends only on $I$ and not on $w$.
\end{corollary}

Given a reduced pair $(w,I)$, we set $\mathscr O_I^\sharp = w^{-1} \cdot\mathscr O_{w,I}$. We now turn to the description of the stabilizers for the action of $W$ on $\mathscr B(G/H)$, and we prove the following theorem.

\begin{theorem}	\label{teo:stabilizzatori2}
Let $(w,I)$ be a reduced pair. Then $\Stab_W(\mathscr O_{w,I}) = wW_Iw^{-1}$.
\end{theorem}

Together with Theorem \ref{teo:stabilizzatori1}, the previous theorem gives a formula to compute the number of $B$-orbits in $G/H$ in terms of the root systems $\Phi_I$.

\begin{corollary}	\label{cor:numero-Borbite}
The number of $B$-orbits in $G/H$ is given by the formula
\[
	\sum_{I \subset \mathscr D^*} |W/W_I|.
\]
\end{corollary}

To prove Theorem \ref{teo:stabilizzatori2} we proceed by steps. First we prove one of the two inclusions.

\begin{lemma}	\label{lemma:stabilizzatori2}
Let $(w,I)$ be a reduced pair. Then $\Stab_W(\mathscr O_{w,I}) \subset wW_Iw^{-1}$.
\end{lemma}

\begin{proof}
Let $v \in \Stab_W(\mathscr O_{w,I})$, we proceed by induction on the length of $v$. If $l(v) = 1$, then by Lemma~\ref{lemma:triple-T} we have $v = s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ for some $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in w(\Psi^\mathrm{\sharp}_I)$, hence we assume $l(v) > 1$. Let $v = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1}$ be a reduced expression, for $i \leq m$ we set $v_i = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1}$ and $\mathscr O_i = v_i \cdot \mathscr O_{w,I}$. Suppose $\mathscr O_i \neq \mathscr O_{i-1}$ for all $i \leq m$, then Lemma~\ref{lemma:triple-T} implies $v \cdot \mathscr O_{w,I} = \mathscr O_{vw,I}$, hence $vw = w$ and $v=e$. Otherwise, let $n \leq m$ be such that $\mathscr O_n = \mathscr O_{n-1}$, and assume that $n$ is minimal with this property.

If $n = 1$ the claim follows by the inductive hypothesis, suppose $n > 1$. Denote $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = v_{n-1}^{-1} (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n)$, then $v = v' s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, where $v' = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m} \cdots \hat{s}_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1}$. By construction
$$
\mathscr O_{n-1} = v_{n-1} \cdot \mathscr O_{w,I} = \mathscr O_{v_{n-1} w,I}
$$
and $s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \in \Stab_W(\mathscr O_{n-1})$, and it follows $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Stab_W(\mathscr O_{w,I})$. On the other hand we have $w^{-1} v_{n-1}^{-1} (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n) \in  \Psi^\mathrm{\sharp}_I$ by Lemma~\ref{lemma:triple-T}, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in w(\Psi^\mathrm{\sharp}_I)$ is of the desired shape and $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in w W_I w^{-1}$. Finally $v' \in \Stab_W(\mathscr O_{w,I})$ and $l(v') < l(v)$, therefore we may apply the inductive hypothesis and it follows $v \in w W_I w^{-1}$.
\end{proof}

\begin{corollary}
Let $w_I \in W_I$ be the longest element, then we have $\mathscr O_I^\sharp = \mathscr O_{w_I,I}$.
\end{corollary}

\begin{proof}
Let $w \in W$ be such that $\mathscr O^\sharp_I = \mathscr O_{w,I}$. By the definition of $\mathscr O^\sharp_I$ we have the equality $\mathscr O_{w,I} = w^{-1} \cdot \mathscr O_{w,I}$, hence $w \in \Stab_W(\mathscr O_{w,I})$, and by Lemma~\ref{lemma:stabilizzatori2} we get $w \in W_I$. On the other hand $W_I$ contains a unique element such that $\Phi^+_I \subset \Phi^+(w)$, namely $w_I$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{teo:stabilizzatori2}]
Since $\mathscr O_{w,I} = w \cdot \mathscr O^\sharp_I$, by the previous corollary it is enough to show the equality $\Stab_W(\mathscr O^\sharp_I) = W_I$. The inclusion $\Stab_W(\mathscr O^\sharp_I) \subset W_I$ follows from Lemma~\ref{lemma:stabilizzatori2}, we show the other inclusion.

Let $v \in W_I$, we show $v \in \Stab_W(\mathscr O^\sharp_I)$ proceeding by induction on $l(v)$, where we regard $v$ as an element of $W$. If $l(v) = 1$, then we have $v = s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ for some $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \grD \cap \Phi^+_I \subset \grD_I$, hence $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = -w_I(\grb)$ for some other root $\grb \in \grD_I$. On the other hand $\grD_I \subset \Psi^\mathrm{\sharp}_I$, and Lemma~\ref{lemma:triple-T} implies $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Stab_W(\mathscr O_I^\sharp)$.

Suppose now that $l(v) > 1$, and let $v = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1}$ be a reduced expression of $v$ as an element of $W$. If $i \leq n$, we denote $v_i = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1}$ and $\mathscr O_i = v_i \cdot \mathscr O^\sharp_I$. Suppose $\mathscr O_i \neq \mathscr O_{i-1}$ for all $i \leq n$, then Lemma~\ref{lemma:triple-T} implies that $v \cdot \mathscr O_{w_I,I} = \mathscr O_{vw_I,I}$ and that $(vw_I,I)$ is a reduced pair. On the other hand $vw_I \in W_I$, and $w_I$ is the unique element in $W_I$ such that $\Phi^+_I \subset \Phi^+(w_I)$. Therefore $vw_I = w_I$, which is absurd since $v \neq e$. Therefore $s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_k} \in \Stab_W(\mathscr O_{k-1})$ for some $k \leq n$. Assume that $k$ is minimal with this property, then Lemma~\ref{lemma:triple-T} implies
$$
	\mathscr O_{k-1} = v_{k-1} \cdot\mathscr O_I^\sharp = \mathscr O_{v_{k-1} w_I,I}.
$$

By construction $s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_k} \in \Stab_W(\mathscr O_{k-1})$, hence $-w_I v^{-1}_{k-1} (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_k) \in \Psi^\mathrm{\sharp}_I$  by Lemma~\ref{lemma:triple-T}. Denote $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma = v^{-1}_{k-1} (\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_k)$, then $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Stab_W(\mathscr O_I^\sharp)$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+_I$, hence $s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in W_I$. On the other hand $v = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \cdots \hat{s}_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_k} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$, therefore $s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \cdots \hat{s}_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_k} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} \in W_I$ and applying the inductive hypothesis it follows $v \in \Stab_W(\mathscr O_I^\sharp)$.
\end{proof}

\subsection{Reduced pairs and weight polytopes}

Building upon Theorem~\ref{teo:stabilizzatori2}, we now produce a combinatorial model for the action of $W$ on $\mathscr B(G/H)$ in terms of weight polytopes.

Consider the Weyl group $W_I$. The system of positive roots $\Phi^+_I$ induces a Bruhat order $\leq_I$ on $W_I$, which is compatible with the restriction of the Bruhat order $\leq$ on $W$ in the following sense.

\begin{lemma} \label{lem: I-bruhat}
Let $(w,I)$ be a reduced pair. Let $v_1, v_2 \in W_I$ be such that $v_1 \leq_I v_2$, then $wv_2 \leq wv_1$. In particular, the left coset $wW_I$ possesses a unique minimal element and a unique maximal element with respect to $\leq$, namely $ww_I$ and $w$.
\end{lemma}

\begin{proof}
Recall that $G_I \subset G$ is the reductive subgroup generated by $T$ together with the root spaces $U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$ with $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi_I$, and $B_I \subset G_I$ is the Borel subgroup $B \cap G_I$.
Being $v_1 \leq_I v_2$, it follows $w_I v_2 \leq_I w_I v_1$, hence $w_I v_2 \in \ol{B_I w_I v_1 B_I}$.

Being $\Phi_I^+ \subset \Phi^+(w) \cap \Phi^+(w_I)$, it follows $\Phi^+(ww_I) \cap \Phi^+_I = \vuoto$, hence we get the equality $Bww_IT = Bww_I B_I$. Therefore
$$
w v_2 \in Bw w_I B_I w_I v_2 B_I \subset \ol{Bw w_I B_I w_I v_1 B_I} \subset \ol{Bwv_1 B_I} \subset \ol{Bwv_1 B}
$$
and the claim follows.
\end{proof}

Denote by $\grL$ the weight lattice of $T$ and let $\grl \in \grL$ be a regular dominant weight. Denote $P = \conv(W\grl)$ the weight polytope of $\grl$ in $\grL_\mQ$, then the vertices of $P$ correspond bijectively to the elements of $W$. By a \textit{subpolytope} of $P$ we mean the convex hull of a subset of vertices of $P$. Denote $\mathscr S(P)$ the set of subpolytopes of $P$, then the Weyl group acts naturally on $\mathscr S(P)$. Given $I \subset \mathscr D^*$ we set $\mathscr S_I = \conv(W_I \grl)$, and for a reduced pair $(w,I)$ we set
$$
	\mathscr S_{w,I} = w \mathscr S_I = \conv (wW_I \grl) = \conv\big(vw (\grl) \st v \in \Stab_W(\mathscr O_{w,I})\big)
$$
(where the last equality follows from Theorem~\ref{teo:stabilizzatori2}). Denote moreover
$$
	\mathscr C_{w,I} = \cone(-w(\Psi^\mathrm{\sharp}_I)) = \cone(-w(\Phi^+_I)) = -w\big(\mQ \Psi_I \cap \mQ_{\geq 0} \grD\big)
$$

\begin{theorem}	\label{teo:subpolytopes}
The map $\mathscr O_{w,I} \longmapsto \mathscr S_{w,I}$ is a $W$-equivariant embedding of $\mathscr B(G/H)$ into $\mathscr S(P)$. Moreover we have the equality
$$\mathscr S_{w,I} = P \cap (\mathscr C_{w,I}+w\grl),$$
and in particular $\dim (\mathscr S_{w,I}) = \rk(\Phi_I)$.
\end{theorem}

\begin{proof}
Let $I \subset \mathscr D^*$ and denote by $\mathscr B_I(G/H)$ the corresponding $W$-orbit in $\mathscr B(G/H)$. We claim that the map $\mathscr O_{w,I} \longmapsto \mathscr S_{w,I}$, regarded as a map $\mathscr B_I(G/H) \lra \mathscr S(P)$, is injective and $W$-equivariant. Indeed, since $(w,I)$ is reduced, by construction we have
$$
\Stab_W(\mathscr O_{w,I}) = \Stab_W(\mathscr S_{w,I}).
$$
Since $\mathscr B_I(G/H)$ is a single $W$-orbit, this shows that the map is injective, and since by Lemma~\ref{lemma:triple-T} the actions of the simple reflections on $\mathscr O_{w,I}$ and on $\mathscr S_{w,I}$ coincide, it shows that it is $W$-equivariant as well.

In particular the map $\mathscr B(G/H) \lra \mathscr S(P)$ is $W$-equivariant and we need only to show the injectivity, namely that $I$ is determined by $\mathscr S_{w,I}$. First of all, notice that $w$ is determined by $\mathscr S_{w,I}$. Indeed, by Lemma~\ref{lem: I-bruhat}, $w$ is maximal in $wW_I$ w.r.t. the Bruhat order, and is uniquely determined by this property. It follows that $w\grl$ is the unique minimal vertex of $\mathscr S_{w,I}$ w.r.t. the dominance order, hence $w$ is uniquely determined by $\mathscr S_{w,I}$. Then the equality $\mathscr S_{w,I} = \conv (wW_I \grl)$ implies that the set of differences $\{v\grl - w\grl \st v\grl \in \mathscr S_{w,I} \}$ generates the semigroup $w(\mN\Phi^+_I)$, hence we recover $I$ from $w$ and $\mathscr S_{w,I}$ thanks to Proposition~\ref{prop: I-rootsyst}. The last claim also follows.
\end{proof}

We conclude this subsection by showing that the parametrization of orbits via subpolytopes of $P$ of Theorem~\ref{teo:subpolytopes} is compatible with the Bruhat order on $\mathscr B(G/H)$ in the following sense.

\begin{proposition}	\label{prop: bruhat-compatibile}
Let $(w,I)$ and $(v,J)$ be reduced pairs and suppose that $\mathscr S_{v,J} \subset \mathscr S_{w,I}$. Then we have $\mathscr O_{v,J} \subset \ol{\mathscr O_{w,I}}$.
\end{proposition}

The proposition is an easy consequence of the following lemma.

\begin{lemma}\label{lemma:closure}
Let $(w,I)$ be a reduced pair and let $J \subset \mathrm M_{w,I}$. If $v \in wW_I$, then $\mathscr O_{v,J} \subset \ol{\mathscr O_{w,I}}$.
\end{lemma}

\begin{proof}
We will make use of the following fact, which follows by the description of the $T$-stable curves in $G/B$ (see \cite[Proposition 3.9]{Ca2}): if $x \in W$ and $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+(x)$, then $x s_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \ol{B x U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma}$.

Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1, \ldots, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m \in \grD_I$ be such that $w^{-1} v = s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_m}$ and, if $i \leq m$, set $w_i = w s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} \cdots s_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i}$. Then by Lemma~\ref{lem: I-bruhat} we have the following chain in the Bruhat order
$$
	v = w_n < \ldots < w_i < \ldots < w_1 < w.
$$
Denote $\mathrm M = \mathrm M_{w,I}$. Being $J \subset \mathrm M$, it follows $\mathscr O_{v,J} \subset \ol{\mathscr O_{v,\mathrm M}}$, therefore to prove the theorem it is enough to consider the case $J = \mathrm M$. In particular, we have the inclusion $I \subset J$, hence $\grD_I \subset \Psi^\mathrm{\sharp}_I \subset \Psi^\mathrm{\sharp}_J$. It follows then by Corollary~\ref{cor: B_I action} that $\ol{\mathscr U_J} = \ol{B_J \mathscr U_J}$ is $U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_i}$-stable for all $i$. In particular we have that $U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2} \cdots U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \mathscr U_J \subset \ol{\mathscr U_J}$, and by the remark at the beginning of the proof we get the inclusions
\[
	\ol{\mathscr O_{w,I}} = \ol{Bw \mathscr U_J} = \ol{Bw U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_1} \cdots U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \mathscr U_J} \supset \ol{Bw_1 U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_2} \cdots U_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma_n} \mathscr U_J} \supset \cdots \supset \ol{Bw_n \mathscr U_J} \supset Bv \mathscr U_J.
\qedhere
\]
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop: bruhat-compatibile}]
Notice that $vW_J \subset wW_I$, hence $w^{-1}v \in W_I$ implies $W_J \subset W_I$. In particular it follows $\Phi^+_J \subset \Phi^+_I$, therefore
$$
	J = \delta(\Phi^+_J \cap \Psi^\mathrm{\sharp}) \subset \delta(\Phi^+_I \cap \Psi^\mathrm{\sharp}) = I,
$$
and the claim follows by Lemma~\ref{lemma:closure}.
\end{proof}

\section{A bound for the number of $B$-orbits in $G/H$.}

A conjecture of Knop states that the homogeneous variety $G/TU'$ of Example~\ref{ex:timashev} has the largest number of $B$-orbits among all the homogeneous spherical $G$-varieties. In this section we prove such bound in the setting of solvable spherical subgroups. That is, we prove the following theorem.

\begin{theorem} \label{teo:knop-conj}
The spherical variety $G/TU'$ has the largest number of $B$-orbits among the homogeneous spherical varieties $G/H$ with $H$ a solvable subgroup of $G$.
\end{theorem}

The set $\mathscr B(G/TU')$ is nicely described in terms of faces $\mathscr F(P)$ of the weight polytope $P$ of a regular dominant weight. Indeed, in this case the embedding of Theorem~\ref{teo:subpolytopes} induces a $W$-equivariant bijection between $\mathscr B(G/TU')$ and $\mathscr F(P)$ (see also \cite[Proposition 3.5]{Ti1} and \cite[Section 3]{Vi1} for a description of $\mathscr F(P)$ and of $\mathscr B(G/TU')$ in terms of the $W$-action).

We first prove Theorem~\ref{teo:knop-conj} in the basic case of a maximal rank strongly solvable spherical subgroup of $G$.

\begin{lemma} \label{lemma:Knop-maximal-rank}
The spherical variety $G/TU'$ has the largest number of $B$-orbits among the homogeneous spherical varieties $G/H$ with $H$ a strongly solvable subgroup of $G$ of maximal rank.
\end{lemma}

\begin{proof}
Suppose that $\pi(I) \subset \Psi$ is connected. By Proposition~\ref{prop:intervalli} $\Phi_I$ and $\Phi_I'$ are both irreducible of rank $|I|$. In particular, if $\Phi_I'$ is of type $\mathsf A} \newcommand{\sfB}{\mathsf B$, then it follows the inequality $|W_I'| \leq |W_I|$. Otherwise, if $\Phi_I'$ is not of type $\mathsf A} \newcommand{\sfB}{\mathsf B$, then Proposition~\ref{prop:intervalli} shows that $W'_I = W_I$.

Consider now an arbitrary subset $I \subset \Psi$ and let $I = I_1 \cup \ldots \cup I_m$ correspond to the decomposition of $\pi(I)$ into connected components. Then $\Phi'_I = \Phi'_{I_1} \times \ldots \times \Phi'_{I_m}$, and for all $j=1, \ldots, m$ it holds $|W'_{I_j}| \leq |W_{I_j}|$. Since $\Psi$ is linearly independent, $\Phi_{I_h} \cap \Phi_{I_k} = \vuoto$ for all $h \neq k$, hence $W_{I_h} \cap W_{I_k} = \{e\}$ for all $h \neq k$, and it follows that
$$
|W'_I| = |W'_{I_1} \times \ldots \times W'_{I_m}| \leq |W_{I_1} \times \ldots \times W_{I_m}| \leq |W_I|.$$
Therefore by Corollary \ref{cor:numero-Borbite} we get
\[
	|\mathscr B(G/H)| = \sum_{I \subset \Psi} |W/W_I| \leq \sum_{I \subset \Psi} |W/W'_I| \leq  \sum_{I' \subset \grD} |W/W_{I'}| = |\mathscr B(G/H')|.	\qedhere
\]

\end{proof}

\begin{proof}[Proof of Theorem \ref{teo:knop-conj}]
We reduce the proof of the theorem to the case of a strongly solvable spherical subgroup of $G$, which is treated in the previous lemma.

Suppose that $H$ is a solvable spherical subgroup and denote by $H^\circ \subset H$ the identity component. Clearly $G/H^\circ$ has a larger number of $B$-orbits than $G/H$, and being solvable and connected $H^\circ$ is contained inside a Borel subgroup of $G$. Therefore we may assume that $H$ is strongly solvable.

Suppose now that $H$ is a strongly solvable spherical subgroup, assume that $H \subset B$ and that $T \cap H$ contains a maximal torus $T_H \subset H$. Recall the restriction $\tau : \Psi \lra \mathcal X} \newcommand{\calU}{\mathcal U(T_H)$, following the discussion at the end of Section \ref{s:toricvar} we identify the maps $\tau : \Psi \lra \tau(\Psi)$ and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma : \Psi \lra \mathscr D^*$. Denote $\mathscr D^* = \{D_1, \ldots, D_m\}$ and set $\Psi_i = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma^{-1}(D_i)$. As in \cite[\S 2]{Avd1}, define a partial order on $\{\Psi_1, \ldots, \Psi_m\}$ as follows: $\Psi_i \prec \Psi_j$ if there are $\grb \in \Psi_i$ and $\grb' \in \Psi_j$ with $\grb \leq \grb'$. Let $M_{\Psi} = \{\Psi_{i_1}, \ldots, \Psi_{i_p}\}$ be the set of the maximal elements and pick a representative $\grb_{i_k} \in \Psi_{i_k}$ for all $k = 1, \ldots, p$. Denote
$$\Psi' = \bigcup_{k=1}^p F(\grb_{i_k})$$
and $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma' = \delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma \ristretto_{\Psi'}$. Notice that $\delta} \newcommand{\grl}{\lambda}  \newcommand{\grs}{\sigma' : \Psi' \lra \mathscr D^*$ is surjective by construction, and is injective by \cite[Lemma~5 and Lemma~8]{Avd1}. On the other hand the set of restrictions $\tau(\Psi') = \tau(\Psi)$ is linearly independent in $\mathcal X} \newcommand{\calU}{\mathcal U(T_H)$ by \cite[Theorem 1]{Avd1}, therefore $\Psi'$ is a linearly independent set of roots.

Let $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma, \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' \in \Phi^+$ and suppose that $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma + \alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' \in \Psi'$: then by Lemma \ref{lemma:differenza-attive-avdeev} either $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi$ or $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' \in \Psi$, therefore by the definition of $\Psi'$ either $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Psi'$ or $\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma' \in \Psi'$. Therefore
$$H' = T \prod_{\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma \in \Phi^+ \senza \Psi'} U_\alpha} \newcommand{\grb}{\beta}    \newcommand{\grg}{\gamma$$
is a strongly solvable subgroup of $G$, and it is spherical by \cite[Theorem 1]{Avd1}.

We claim that $|\mathscr B(G/H)| \leq |\mathscr B(G/H')|$. Indeed, by construction we have a bijection between $\mathscr D^*_H$ and $\mathscr D^*_{H'}$, that we identify under the map $\tau : \Psi' \lra \tau(\Psi)$. Therefore by Corollary \ref{cor: Worbits} $\mathscr B(G/H)$ and $\mathscr B(G/H')$ possess the same number of $W$-orbits, indexed by the subsets of $\mathscr D^* = \mathscr D^*_{H} = \mathscr D^*_{H'}$.

Let $I \subset \mathscr D^*$, let $\Phi'_I = \mathbb Z \Psi'_I \cap \Phi$ be the root system associated to the corresponding $T$-orbit in $B/H'$ and let $W'_I$ its Weyl group. By construction $\Psi'_I \subset \Psi_I$, therefore $\Phi'_I \subset \Phi_I$ and it follows that $W'_I \subset W_I$. Therefore by Corollary \ref{cor:numero-Borbite} it follows
$$
	|\mathscr B(G/H)| = \sum_{I \subset \mathscr D^*} |W/W_I| \leq \sum_{I \subset \mathscr D^*} |W/W'_I| = |\mathscr B(G/H')|.
$$

On the other hand $H'$ is a strongly solvable spherical subgroup of $G$ of maximal rank, therefore $|\mathscr B(G/H')| \leq |\mathscr B(G/TU')|$ by Lemma~\ref{lemma:Knop-maximal-rank} and the proof is complete.
\end{proof}

redpajama_set_name: RedPajamaArXiv