text:
\section{Introduction}

Connected algebraic groups constitute a remarkable class of irreducible quasiprojective algebraic varieties.\;It contains the subclasses of abeli\-an varieties and affine algebraic groups.\;These
subclasses are basic:\;by
Chevalley's theorem, every connected algebraic group $G$  has a unique
connected normal affine algebraic subgroup $L$ such that $G/L$ is an abelian variety;
whence
the variety $G$ is an $L$-torsor over the abelian variety $G/L$.\;The varieties from these subclasses can be embedded in many ways as closed subvarieties in, respectively, projective and affine spaces.\;A natural question then arises as to whether there are distinguished embeddings and equations of their images, which are
canonically determined by the group structure.\;For abelian varieties, this is the existence problem for canonically defined bases in linear systems and that of
presentating homogeneous coordinate rings of ample  invertible sheafs by generators and relations.\;These problems were explored and solved by D.\;Mumford  \cite{Ma66}.\;For
affine algebraic groups, it is the existence problem of the canonically defined presentations of the coordinate algebras of such groups by generators and relations.\;We explore this problem in the present paper.

We fix as the base field an algebraically closed field $k$ of arbitrary characteristic.\;In this paper,
as in \cite{Bor91}, ``variety'' means ``algebraic variety'' in the sense of
Serre
\cite[Subsect.\;34]{Se55}; every variety is taken over $k$.

Let $G$ be a connected affine algebraic group and let $R_u(G)$ be its unipotent radical.\;In view of \cite[Props.\,1,\,2]{Gr58}, \cite[Thm.\,10]{Ro56}, the underlying variety of $G$
is isomorphic to the product of that of $G/R_u(G)$ and $R_u(G)$, and the latter is isomorphic to an affine space.\;Therefore,
the problem under consideration is reduced to the case of reductive groups.\;Given this, henceforth $G$ stands for a connected {\it reductive} algebraic group.

The simplest
case of ${\rm SL}_{2}$ is the guiding example.\;Take  the polynomial $k$-algebra $k[x_1,x_2,x_3,x_4]$ in four variables $x_i$.\;The usual presentation of $k[{\rm SL}_{2}]$ is given by the surjective homomorphism
\begin{equation}\label{SL2}
\mu\colon k[x_1, x_2, x_3, x_4]\to  k[{\rm SL}_{2}],\quad\mbox
$\mu(x_i)\biggl(\!\renewcommand{\arraycolsep}{2pt}\begin{bmatrix}
a_1 & a_2\\
a_3 & a_4
\end{bmatrix}\!\biggr)\!=\!a_i$},
\end{equation}
whose kernel is the ideal $(x_1x_4-x_2x_3-1)$.\;After rewriting, this presentation can be interpreted in
terms of the group structure of ${\rm SL}_2$ as follows.

We have $k[x_1, x_2, x_3, x_4]=
k[x_1, x_3]\otimes_k k[x_2, x_4]$ and the restriction of $\mu$ to the subalgebra
$k[x_1, x_3]$ (respectively, $k[x_2, x_4]$) is an isomorphism with the subalgebra
${\mathcal S}^+$ (respectively, ${\mathcal S}^-$) of $k[{\rm SL}_{2}]$
consisting of all regular functions
invariant with respect to
the subgroup $U^+$ (respectively, $U^-$) of all unipotent upper (respectively, lower) triangular matrices acting by right translations.\;Hence
\eqref{SL2} yields the following presentation of $k[{\rm SL_2}]$ by generators and relations:
\begin{equation}\label{SL2P}
\begin{split}
k[{\rm SL_2}]&\cong ({\mathcal S}^+\otimes _{k}{\mathcal S}^{-})/{\mathcal I},\\
{\mathcal S}^+&=k[\mu (x_1),\mu (x_3)]\cong k[x_1, x_3],\\
{\mathcal S}^{-}&=
k[\mu(x_2),\mu (x_4)]\cong k[x_2, x_4],\\
{\mathcal I}&=\big(\mu (x_1)\otimes \mu(x_4)-\mu(x_2)\otimes \mu(x_3)-1\big).
\end{split}
\end{equation}\vskip 1mm

The subgroups $U^+$, $U^-$ are opposite maximal unipotent subgroups of ${\rm SL}_2$. The subalgebras ${\mathcal S}^+$, ${\mathcal S}^-$ are stable with respect to ${\rm SL}_2$ acting by left translations, and $f\!:=\!\mu (x_1)\otimes \mu(x_4)\!-\!\mu(x_2)\otimes \mu(x_3)\!-\!1$ is the unique element of $({\mathcal S}^+\otimes_k{\mathcal S}^-)^{{\rm SL}_2}$
determined by the conditions
$f(e,e)\!=\!1$, $k[f]\!=\!({\mathcal S}^+\otimes_k{\mathcal S}^-)^{{\rm SL}_2}$.

We show that there is an analogue of \eqref{SL2P} for every connected reductive
algebraic group $G$.\;Namely, we endow $k[G]$ with the $G$-module structure determined by left translations and fix in $G$ a pair of opposite Borel subgroups $B^+$ and $B^-$.\;Let $U^\pm$ be the unipotent radical of $B^\pm$.\;Consider the $G$-stable subalgebras
\begin{equation}\label{S+}
\begin{split}
{\mathcal S}^+&:=\{ f\in k[G]\mid  f(gu)=f(g)\; \mbox{for all $g\in G, u\in U^+$} \},\\
{\mathcal S}^-&:=\{ f\in k[G]\mid  f(gu)=f(g)\; \mbox{for all $g\in G, u\in U^-$} \}
\end{split}
\end{equation}
of $k[G]$ and the natural multiplication homomorphism of $k$-algebras
\begin{equation}\label{mu}
\mu \colon  {\mathcal S}^+\otimes _{k}{\mathcal S}^{-}\hskip -1mm\to k[G],\quad f_1\otimes f_2\mapsto f_1f_2.
\end{equation}

 For $k=\mathbb C$, the following conjectures were put forward  in \cite{FT92}:

 \begin{conjsnonumber}[D.\,E.\,Flath and J. Towber, 1992]\

 \begin{enumerate}[\hskip 2.2mm\rm(i)]
 \item[\rm (S)] The homomorphism $\mu$ is surjective.
 \item[\rm (K)] The ideal ${\rm ker}\,\mu$ in ${\mathcal S}^+\otimes_k {\mathcal S}^-$
is generated by $({\rm ker}\,\mu)^G$.
 \end{enumerate}
 \end{conjsnonumber}

 If these conjectures are true, then the problem under consideration is reduced to
 the following:

\begin{enumerate}[\hskip 4.2mm \rm(a)]
 \item find the canonically defined generators of  the $k$-algebra $({\rm ker}\,\mu)^G$,
 \item find the canonically defined presentations of ${\mathcal S}^\pm$ by generators and relations.
 \end{enumerate}

 In \cite{FT92}, Conjectures (S) and (K) were proved  for $k=\mathbb C$ and $G={\rm SL}_n$,
 ${\rm GL}_n$, ${\rm SO}_n$, ${\rm Sp}_n$ by means of lengthy
 direct
 computations of
 some Laplace decompositions, minors, and algebraic identities between them.\;In Theorems \ref{sur} and \ref{ker}
 below we prove Conjectures (S) and (K) in full generality, with no restrictions on $k$ and $G$.

In Theorems  \ref{vect}, \ref{iG} below we describe ${\rm ker}\,\mu$ as a vector space over $k$.\;In Theorem \ref{gi}, we
solve the above part (a) of the problem, finding the cano\-ni\-cally
defined generators of the $k$-algebra $({\rm ker}\,\mu)^G$.\,We call them {\it ${\rm SL}_2$-type relations} of the sought-for canonical presentation of $k[G]$ because
for $G={\rm SL}_2$ the element
$\mu (x_1)\otimes \mu(x_4)-\mu(x_2)\otimes \mu(x_3)-1$
is just such a generator of
$\mathcal I$ (see \eqref{SL2P}).\;All
of them are inhomogeneous of degree $2$.\;If $G$ is
semisimple,
  they are indexed by the elements of the Hilbert basis ${\mathscr H}$ of the monoid of dominant weights of $G$.\;Note that the cardinality $|{\mathscr H}|$ of ${\mathscr H}$ is at least ${\rm rank}\,G$ with equality for simply connected $G$, but in the general case it may be much bigger.\;For instance, if $G={\rm PGL}_{r}$, then $|{\mathscr H}|\geqslant p(r)+\varphi(r)-1$, where $p$ and $\varphi$ are, respectively, the classical partition function and the Euler function (see \cite[Example\;3.15]{Po11}).\;Note that
the problem of determining a full set of generators of the ideal ${\rm ker}\,\mu$ was formulated in
\cite[Sect.\,4]{Fl94} and, for $k=\mathbb C$,  $G={\rm SL}_n$, ${\rm GL}_n$, ${\rm SO}_n$, ${\rm Sp}_n$, solved in
\cite{FT92} by lengthy direct computations.

For a semisimple group $G$
whose monoid of dominant weights is freely generared (i.e., with $|{\mathscr H}|={\rm rank}\, G$),
 a solution to the above part (b) of the problem  in characteristic $0$ was
 obtained (but not published) by B.\;Kostant; his proof appeared in \cite[Thm.\;1.1]{LT79}.\;In arbitrary characteristic, such a solution is given by Theorems
 \ref{RR}, \ref{fg},
 \ref{grS} below, which are
heavily based on
the main results of \cite{RR85} and \cite{KR87}.\;All relations in this case are homogeneous
of degree $2$.\;We call them {\it Pl\"ucker-type relations} of the sought-for canonical presentation of $k[G]$ because  the $k$-algebra ${\mathcal S}^\pm$ for $G={\rm SL}_n$ is the coordinate algebra of the affine multicone over the flag variety, and if ${\rm char}\,k=0$, these relations are generated by the classical
Pl\"ucker-type relations, obtained by Hodge \cite{Ho42}, \cite{Ho43}, that determine this multicone
(see below Section \ref{example}).\;The
set of these relations is a union of
finite-dimensional vector spaces canonically determined by the group structure of $G$; these spaces are
indexed by the elements of ${\mathscr H}\times {\mathscr H}$ and
different spaces have zero intersection (see Theorem \ref{grS}).\;Thus
in this case, we obtain a canonical presentation of $k[G]$, in which all relations are quadratic and divided into two families: homogeneous relations
of Pl\"ucker type and inhomogeneous relations
 of ${\rm SL}_2$-type.\;As a parallel, we
recall that any abelian variety is canonically presented as an intersection of quadrics in a projective space given by the Riemann equations; see \cite{Ke98}, \cite{LB92}.

For an arbitrary reductive group $G$, let $\tau\colon \widehat G\to G$ be the universal covering.\;Then $\widehat G=Z\times C$, where $Z$ is a torus, $C$ is a simply connected semisimple group, $G=\widetilde G/{\rm ker}\,\tau$, and ${\rm ker}\,\tau$ is a finite central subgroup.\;The algebra ${\mathcal S}^\pm$ for $\widehat G$ is then
the tensor product of $k[Z]$ and the algebra ${\mathcal S}^\pm$ for $C$.\;Since the presentation of
$k[Z]$ is clear, and that of ${\mathcal S}^\pm$ for $C$ are given by Theorems \ref{RR}, \ref{fg}, \ref{grS}, the above part (b) of the problem is reduced to finding a presentation for the invariant algebra of the finite abelian group ${\rm ker}\,\tau$.

As an illustration, in the last Section \ref{example} we consider the example of $G={\rm SL}_n$, ${\rm char}\,k=0$ and
describe explicitly how the ingredients of our construction
and the canonical presentation of $k[G]$ look like in this case.

\vskip 3mm

The preprints \cite{Po95}, \cite{Po00} of these results in characteristic $0$ have been disseminated long ago.\;The
validity of the results
in arbitrary characteristic was announced in \cite{Po00}.\;The author is pleased by the arisen occasion to finally present
the complete proofs.

\vskip 4mm

\noindent{\it Notation and conventions.}

\vskip 1mm

Below we use freely the standard notation and conventions of
\cite{Bor91}, \cite{Ja87}, \cite{PV94}, and \cite{Sh13}.\;In
parti\-cu\-lar, the algebra of functions regular on a variety $X$ is denoted by $k[X]$, the field of rational functions on an irreducible $X$ is denoted by $k(X)$, and the local ring of $X$ at a point $x$ is denoted by ${\mathcal O}_{x, X}$.\;For a morphism $\varphi\colon X\to Y$ of varieties, $\varphi^*\colon k[Y]\to k[X]$ denotes its co\-mor\-phism.

All topological terms refer to the Zariski topology; the closure of $Z$ in $X$
is denoted by $\overline Z$ (each time it is clear from the context  what is $X$).

  The fixed point set of an action of a group $P$ on a set $S$ is denoted by\;$S^P$.\;Every action $\alpha\colon H\times X\to X$ of an algebraic group $H$ on a variety $X$ is always assumed to be regular (the latter means that $\alpha$ is  a morphism).\;For every $h\in H$, $x\in X$, we write $g\cdot x$ in place of $\alpha(g, x)$.\;The $H$-orbit and the $H$-stabilizer of $x$ are denoted respectively by $H\cdot x$ and $H_x$.\;Every homomorphism of algebraic groups is assumed to be algebraic.

  The additively written group of characters (i.e., homomorphisms to the multiplicative group of $k$) of an algebraic group $H$ is denoted by ${\rm X}(H)$.\;The value of a character $\lambda\in {\rm X}(H)$ at an element $h\in H$ is denoted by $h^\lambda$.\;Given
  a $kH$-module $M$, its weight space with weight $\lambda\in X(H)$ is denoted by $M_\lambda$.

  We fix in $G$ the maximal torus
  \begin{equation*}
  T:=B^+\cap B^-
  \end{equation*}
 and identify ${\rm X}(B^\pm)$ with ${\rm X}(T)$ by means of the restriction isomorphisms ${\rm X}(B^\pm)\to {\rm X}(T)$, $\lambda\mapsto \lambda|_{T}^{\ }$.

 By $\XT+$ we denote
the monoid of dominant weights of $T$ determined by $B^+$.\;Below the highest weight of every simple $G$-module is assumed to be
the highest weight with respect to $T$ and $B^+$.

 We denote by $w_0$ be the longest element of the Weyl group of $T$ and fix in the normalizer of $T$ a representative ${\overset{.}{w}}_0$ of $w_0$.\;We then have ${\overset{.}{w}}_0B^\pm{\overset{.}{w}}_0^{-1}=B^\mp$ and
${\overset{.}{w}}_0U^\pm{\overset{.}{w}}_0^{-1}=U^\mp$.\;For every $\lambda \in \XT+$, we put $\lambda^*:=-w_0(\lambda)\in \XT+$.

The set of all nonnegative rational numbers is denoted by ${\mathbb Q}_{\geqslant 0}$ and we put
${\mathbb N}:={\mathbb Z}\cap {\mathbb Q}_{\geqslant 0}$.

If $m\in \mathbb Z$, $m>0$, we put $[m]:=\{a\in \mathbb Z\mid 1\leqslant a\leqslant m\}$.

 For $d\in \mathbb N$, we
 denote by $[m]_d$ the set of all increasing sequences
 of $d$ elements of $[m]$ (if $d\notin [m]$, then $[m]_d=\varnothing$).

\section{Proof of Conjecture (S)}

For every $\lambda \in \Xt$, the spaces
\begin{equation}\label{S+la}
\begin{split}
{\mathcal S}^+(\lambda)&:=\{ f\in {\mathcal S}^+\mid f(gt)=t^{\lambda }f(g)\; \mbox{for all $g\in G, t\in T$} \},\\
{\mathcal S}^-(\lambda)&:=\{ f\in {\mathcal S}^-\mid f(gt)=t^{w_0(\lambda) }f(g)\; \mbox{for all $g\in G, t\in T$} \}.
\end{split}
\end{equation}
are the finite-dimensional (see, e.g.,\;\cite[I.5.12.c)]{Ja87}) $G$-submodules of the $G$-modules ${\mathcal S}^+$ and ${\mathcal S}^-$ respectively.\;Since ${\mathcal S}^-(\lambda)$ is the right translation of ${\mathcal S}^+(\lambda)$ by ${\overset{.}{w}}_0,$ these $G$-submodules are isomorphic.\;In the notation of \cite[II.2.2]{Ja87}, we have
\begin{equation}\label{H}
{\mathcal S}^-(\lambda)=H^0(\lambda^*),
\end{equation}
so by \eqref{H} and \cite[II.2.6, 2.2, 2.3]{Ja87}, the following properties hold:
\begin{equation}\label{Sm}
\left.
\begin{split}
&{\hskip 1mm\rm (i)}\;{\mathcal S}^\pm(\lambda)\neq 0 \iff \lambda\in \XT+;\\[-1.2mm]
&{\rm (ii)}\;\mbox{${\rm soc}_G^{\ }{\mathcal S}^\pm(\lambda)$
is a simple $G$-module with the highest weight $\lambda^*$.}
\end{split}
\right\}
\end{equation}

If ${\rm char}\, k=0$, then the $G$-module ${\mathcal S}^+(\lambda)$ is semisimple and hence ${\mathcal S}^+(\lambda)={\rm soc}_G^{\ }{\mathcal S}^+(\lambda)$ by \eqref{Sm}(ii).\;If ${\rm char}\, k>0$, then, in general,  this equality does not hold.
From \eqref{S+}, \eqref{S+la}, and \eqref{Sm}(i) we infer that
\begin{equation}\label{grS+-}
\begin{split}
{\mathcal S}^+&=\displaystyle \bigoplus_{\lambda \in \XT+}
{\mathcal S}^+(\lambda), \quad {\mathcal S}^+(\lambda){\mathcal S}^+(\mu)\subseteq {\mathcal S}^+(\lambda+\mu),\\
{\mathcal S}^-&=\displaystyle \bigoplus_{\lambda \in \XT+}
{\mathcal S}^-(\lambda), \quad {\mathcal S}^-(\lambda){\mathcal S}^-(\mu)\subseteq {\mathcal S}^-(\lambda+\mu),
\end{split}
\end{equation}
i.e., \eqref{grS+-} are the $\XT+$-gradings of the algebras ${\mathcal S}^+$ and ${\mathcal S}^-$.\;They are obtained from each other by the right translation by ${\overset{.}{w}}_0$.

\begin{theorem}\label{RR} The
linear span of ${\mathcal S}^\pm(\lambda){\mathcal S}^\pm(\mu)$ over $k$ is ${\mathcal S}^\pm(\lambda+\mu)$.
\end{theorem}
\begin{proof}
This statement is the main result of \cite{RR85}.\;Note that the difficulty lies in the case of positive characteristic: since ${\mathcal S}^\pm$ is an integral domain, if
${\rm char}\,k=0$, then the claim immediately follows from \eqref{Sm}(i) and
the inclusions in \eqref{grS+-} because then ${\mathcal S}^\pm(\lambda+\mu)$ is a simple
$G$-module.
\end{proof}

\begin{theorem}\label{fg}\

\begin{enumerate}[\hskip 2.2mm\rm(i)]
\item If ${\mathscr G}$ is a generating set of the semigroup $\XT+$, then the $k$-algebra ${\mathcal S}^\pm$ is generated
by the subspace $\bigoplus_{\lambda\in {\mathscr G}}{\mathcal S}^\pm(\lambda)$.
\item The $k$-algebras $\mathcal {\mathcal S}^+$ and ${\mathcal S}^-$ are finitely generated.
\end{enumerate}
\end{theorem}
\begin{proof} Part (i) follows from \eqref{grS+-} and
Theorem \ref{RR}.\;Being the intersection of the lattice $\Xt$ with a convex cone in
$\Xt\otimes_{\mathbb Z} \mathbb Q$ generated by finitely many vectors,
the semigroup $\XT+$ is finitely ge\-ne\-ra\-ted.\;This, (i), and the inequality
$\dim_k{\mathcal S}^\pm(\lambda)<\infty$ imply (ii).
\end{proof}

Now we are ready to turn to the proof of Conjecture (S).

\begin{theorem}\label{sur}
The homomorphism
$\mu$ is surjective.
\end{theorem}

Our proof of Theorem \ref{sur} is based on two general results.\;The first is the following
well-known
surjectivity criterion:
\begin{lemma}\label{ce} The following properties of a morphism $\varphi\colon X\to Y$
of affine algebraic varieties are equivalent:
\begin{enumerate}[\hskip 4.2mm \rm(a)]
 \item
 $\varphi$ is a closed embedding;
 \item
$\varphi^*\colon k[Y]\to k[X]$ is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
See, e.g.,\;\cite[1.5]{St74}.
\end{proof}

The second  is the closedness criterion for
orbits of
connected solvable affine algebraic groups
that generalizes
Rosenlicht's classical theorem on closedness of orbits of unipotent groups
\cite[Thm.\,2]{Ro61}.

\begin{theorem}\label{clos} Let a connected solvable affine algebraic group $S$
act on an affine algebraic variety $Z$.\;Let $x$ be a point
of $Z$.\;Consider the orbit morphism $\tau: S\rightarrow Z$, $s\mapsto s\cdot z$.\;Then
the following
properties
are equivalent:
\begin{enumerate}[\hskip 4.2mm\rm(a)]
\item the orbit $S\cdot z$ is closed in $Z$;
\item the semigroup
$\{\lambda \!\in\! {\rm X}(S)\mid \mbox{
\hskip -1mm the function
$S\!\to\! k$, $s\!\mapsto\! s^{\lambda}$ lies in $\tau^{*}(k[Z])$}\!\}$
is a group.
\end{enumerate}
\end{theorem}
\begin{proof} This is proved in \cite[Thm.\,4]{Po88}
\end{proof}

\begin{remark}\label{group} Since ${\rm X}(S)$ in Theorem \ref{clos} is a finitely generated free abelian group, it can be naturally regarded as a lattice in
 ${\rm X}(S)\otimes_{\mathbb Z}\mathbb Q$.\;Hence
the following general criterion is applicable
for verifying condition (b).

Let $M$ be a nonempty subset of a finite
dimensional vector space $V$ over $\mathbb Q$.\;Let ${\mathbb Q}_{\geqslant 0}M$,
${\rm conv} M$, and $\mathbb QM$ be, respectively, the convex cone generated by $M$,
 the convex hull of $M$, and  the linear span of $M$ in $V$.\;Then
the
following properties are equivalent (see \cite[p.\,386]{Po88}):
\begin{enumerate}[\hskip 2.2mm \rm(i)]
\item $0$ is an interior point of ${\rm conv} M$,
\item $\mathbb Q_{\geqslant 0} M=\mathbb Q M$.
\end{enumerate}
If $M$ is a subsemigroup of $V$, then
(i) and (ii) are equivalent to
\begin{enumerate}[\hskip -6.0mm \rm(iii)]
\item $M$ is a group.
\end{enumerate}
\end{remark}

\begin{proof}[Proof of Theorem {\rm \ref{sur}}]\

1. We consider the action of $G$ on its underlying algebraic variety by left
translations.\;By Theorem \ref{fg}, there is
an irreducible affine algebraic variety $X$ endowed with an action
of $G$ and a $G$-equivariant dominant morphism
\begin{equation}\label{alpha}
\alpha \colon G\to X\;\;\mbox{such that $\alpha^*$ is an isomorpshism $k[X]\xrightarrow{\cong}{\mathcal S}^+$.}
\end{equation}

Let $x:=\alpha(e)$.\;Since $\alpha$ is $G$-equivariant, we have
\begin{equation}\label{ag}
\alpha(g)=g\cdot x \;\;\mbox{for every $g\in G$},
\end{equation}
 and since $\alpha$ is dominant,
the orbit $G\cdot x$
is open and dense in $X$.\;Consider the canonical projection
$\pi \colon G\to G/U^+$.\;It is the geometric quotient for the action of $U^+$ on $G$ by right translations.\;Therefore, \eqref{S+} yields the isomorphism
\begin{equation}\label{pi*}
\pi^*\colon k[G/U^+]\xrightarrow{\cong}{\mathcal S}^+
\end{equation}
and,  since $\alpha$ is constant on the fibers $\pi$,
there exists a $G$-equivariant
morphism $\iota\colon G/U^+\to X$ such that
\begin{equation}\label{aip}
\alpha=\iota\circ\pi.
\end{equation}

From \eqref{aip} we infer that the image of $\iota$ is $G\cdot x$.\;Since the group $U^+$ is unipotent, the algebraic variety $G/U^+$ is quasiaffine
(see \cite[Thm.\,3]{Ro61}). There\-fore, $k(G/U^+)$ is the field of fractions
of $k[G/U^+]$.\;On the other hand, $k(X)$ is the field of fractions of $k[X]$ inasmuch as $X$ is affine.\;Using that \eqref{aip} and isomorphisms \eqref{alpha}, \eqref{pi*} yield the isomorphism
$\iota^*\colon k[X]\xrightarrow{\cong} k[G/U^+],$
we conclude that $\iota $ is a birational isomorphism.\;Therefore, for
a point $z$ in ge\-ne\-ral position in $G\cdot x$ the fiber $\iota^{-1}(z)$ is a single point.\;Being $G$-equivariant, $\iota $ is then injective.\;Finally, since $G$ is smooth, $k[G]$ is integrally closed; therefore, ${\mathcal S}^\pm$ is integrally closed as well in view of \eqref{S+} (see, e.g., \cite[Thm.\,3.16]{PV94}).\;Thus $X$ is normal, and hence by \;Zariski's Main Theorem, $\iota\colon G/U^+\to G\cdot x$ is an isomorphism.\;Using that $\pi$ is separable (see, e.g., \cite[II.6.5]{Bor91}), from this we infer that the following properties hold:
    \begin{enumerate}[\hskip 3.2mm\rm (i)]
      \item[$({\rm i}_1)$] $G_x=U^+$;
      \item[$({\rm ii}_1)$] $G\to G\cdot x$, $g\mapsto \alpha(g)=g\cdot x$, is a separable morphism.
    \end{enumerate}

\vskip 1mm

2. Let $y:={\overset{.}{w}}_0\cdot x$.\;Consider the $G$-equivariant morphism
\begin{equation}\label{beta}
    \beta\colon G\to X,\;\;g\mapsto g\cdot y.
\end{equation}
From \eqref{ag}, \eqref{beta}, $({\rm i}_1)$, and $({\rm ii}_1)$ we infer that the following properties hold:
 \begin{enumerate}[\hskip 4.8mm\rm (i)]
      \item[$({\rm i}_2)$] $G_y=U^-$;
      \item[$({\rm ii}_2)$] $G\to G\cdot y$, $g\mapsto \beta(g)=g\cdot y$, is a separable morphism;
      \item[$({\rm iii}_2)$] $\beta^*$ is an isomorphism  $k[X]\xrightarrow{\cong} {\mathcal S}^-$.
    \end{enumerate}

    \vskip 1mm

3. Now consider the $G$-equivariant  morphism
\begin{equation}\label{gamma}
    \gamma:=\alpha\times\beta\colon G\to X\times X,\quad g\mapsto g\cdot z,\;\;\mbox{where $z:=(x, y)$.}
\end{equation}
From \eqref{gamma} and $({\rm i}_1)$, $({\rm i}_2)$ we obtain
\begin{equation}\label{trivial}
G_z=G_x\cap G_y
=U^+\cap U^-
=\{e\},
\end{equation}
hence $\gamma$ is injective.
We claim that $\gamma$ is a closed embedding, i.e.,
\begin{enumerate}[\hskip 4.2mm\rm(a)]
  \item $G\to G\cdot z$, $g\mapsto g\cdot z$ is an isomorphism;
  \item $G \cdot z$ is closed in $X\times X$.
\end{enumerate}

If this claim is proved, then the proof of Theorem \ref{sur}
is completed as follows.\;Consider the isomorphism
\begin{equation}\label{theta}
   k[X]\otimes_kk[X]\to k[X\times X],\quad f\otimes h\mapsto fh.
\end{equation}
Then \eqref{mu}, \eqref{alpha}, $({\rm iii}_3)$, \eqref{gamma}, \eqref{theta} imply that $\mu$ is the composition of the homomorphisms
\begin{equation}\label{ep}
    {\mathcal S}^+\otimes_k{\mathcal S}^-\xrightarrow[\cong]{(\alpha^*)^{-1}\otimes (\beta^*)^{-1}}
k[X]\otimes_kk[X]
\xrightarrow[\cong]{\eqref{theta}}
k[X\times X]\xrightarrow{\gamma^*}k[G].
\end{equation}
Hence the surjectivity of $\mu$ is equivalent to the surjectivity of $\gamma^*$.\;By Lemma
\ref{ce},
the latter is equivalent to the property that $\gamma$ is a closed embedding, i.e., that properties (a) and (b) hold.

Thus the proof of Theorem \ref{sur} is reduced to proving properties (a) and (b).

\vskip 1mm

4. First, we shall prove property (a).\;Since $\gamma$ is injective, this is reduced to proving separability of $\gamma$.\;In turn, in view of \eqref{gamma}, the latter is reduced to proving that ${\rm ker}(d\gamma)_e$ is contained in ${\rm Lie}\,G_z$, i.e., that ${\rm ker}(d\gamma)_e=\{0\}$
because of \eqref{trivial} (see \cite[II.6.7]{Bor91}).\;Using
loc.cit., from \eqref{ag}, \eqref{beta},  $({\rm i}_1)$, $({\rm ii}_1)$,  $({\rm i}_2)$, $({\rm ii}_2)$ we infer that ${\rm ker}(d\alpha)_e\subseteq {\rm Lie}\,U^+$, ${\rm ker}(d\beta)_e\subseteq {\rm Lie}\,U^-$.\;In view of \eqref{gamma}, we then have
${\rm ker}(d\gamma)_e\!=\!{\rm ker}(d\alpha)_e\cap {\rm ker}(d\beta)_e\!\subseteq\!
{\rm Lie}\,U^+\cap {\rm Lie}\,U^-=\{0\}$.\;This proves property (a).

\vskip 1mm

5. Now we shall prove property (b).\;Actually, we shall prove the stronger property that the orbit $B^+\cdot z$ is closed in $X\times X$: since the algebraic variety $G/B^+$ is complete, this stronger property implies property (b) (see \cite[Sect.\,2.13, Lemma\,2]{St74}).\;Using that $B^+$ is connected solvable, to this end we shall apply Theorem \ref{clos}.

Namely, consider the morphism $\tau\colon B^+\to X\times X$, $b\mapsto b\cdot z$ and
the following subsemi\-group $M$ in ${\rm X}(B^+)$:
\begin{equation*}\label{M}
    M:=\{\lambda \!\in\! {\rm X}(B^+)\mid \mbox{
    the function $B^+\to k$, $b\mapsto b^{\lambda}$ lies in $\tau^{*}(k[X\times X])$.}\}
\end{equation*}
We identify ${\rm X}(B^+)$ with the lattice in $L:={\rm X}(B^+)\otimes _{\mathbb Z}\mathbb Q$.\;In view of Theorem \ref{clos} and Remark \ref{group}, the orbit $B^+\cdot z$ is closed if and only if
\begin{equation}\label{=}
\mathbb Q_{\geqslant 0} M=\mathbb Q M.
\end{equation}
Given this, the problem is reduced to proving that property \eqref{=} holds.\;This is done below.

\vskip 1mm

6. Since  $\tau=\gamma|^{\ }_{B^+}$, the algebra $\tau^*(k[X\times X])$ is the image of the homomorphism $\gamma^*(k[X\times X])\to k[B^+]$, $f\mapsto f|^{\ }_{B^+}$.\;From \eqref{ep} we see that
$\gamma^*(k[X\times X])$ contains ${\mathcal S}^+$ and ${\mathcal S}^-$.\;Hence the restrictions of
${\mathcal S}^+$ and ${\mathcal S}^+$ to $B^+$ lie in $\tau^*(k[X\times X])$.\;We
shall exhibit some characters of $B^+$ lying in these restrictions.

First consider the restriction of ${\mathcal S}^+(\lambda)$ to $B^+$ for $\lambda\in \XT+$.\;Note that ${\mathcal S}^+(\lambda)$ contains a function $f$ such that $f(e)\neq 0$.\;Indeed, in view of \eqref{Sm}(i) and Borel's fixed point theorem, ${\mathcal S}^+(\lambda)$ contains a $B^-$-stable line $\ell$.\;The group $B^-$ acts on $\ell$ by means of
a character $\nu\in {\rm X}(B^-)$.\;Take a nonzero function $f\in \ell$.\;For every $b\in B^-$, $u\in U^+$, we then have $f(b^{-1}u)=b^\nu f(u)\overset{\eqref{S+}}{=}b^\nu f(e)$; whence $f(e)\neq 0$ because
$B^-U^+$ is dense in $G$.\;This proves the existence of $f$.\;Multiplying $f$ by $1/f(e)$, we may assume
that $f(e)=1$.\;Then  for every $b\in B^+$, we deduce from $\eqref{S+}$, $\eqref{S+la}$ that
$f(b)=b^\lambda f(e)=b^{\lambda}$, i.e., $f|^{\ }_{B^+}$ is the character $B^+\!\to k$, $b\mapsto b^\lambda$.
This proves the inclusion
 \begin{equation}\label{M+}
{\rm X}(B^+)^{\ }_+\subseteq M.
 \end{equation}

Now consider the restriction of ${\mathcal S}^-(\lambda)$ to $B^+$ for $\lambda\in \XT+$.\;In view of \eqref{Sm}(ii), there is a $B^+$-stable line $\ell$  in ${\mathcal S}^-(\lambda)$, on which $B^+$ acts by the character $\lambda^*\in {\rm X}(B^+)$.\;Take a nonzero function $f\in \ell$.\;We may assume that
$f(e)=1$: this is proved as above with $\nu=\lambda$, replacing $B^-$ by $B^+$, and $U^+$ by
$U^-$.\;For every $b\in B^+$, we then have $f(b^{-1})=b^{\lambda^*}$, i.e., $f|^{\ }_{B^+}$ is the character $B^+\!\to k$, $b\mapsto b^{-\lambda^*}=b^{w_0(\lambda)}$.\;This proves the inclusion
 \begin{equation}\label{M-}
-{\rm X}(B^+)^{\ }_+\subseteq M.
 \end{equation}

 Since ${\mathbb Q}^{\ }_{\geqslant 0}({\rm X}(B^+)^{\ }_+)-{\mathbb Q}^{\ }_{\geqslant 0}({\rm X}(B^+)^{\ }_+)=L$, the inclusions \eqref{M+}, \eqref{M-} imply the equality ${\mathbb Q}^{\ }_{\geqslant 0}M=L$; whence a fortiori the equality \eqref{=} holds.
This completes the proof of Theorem \ref{sur}.
\end{proof}

\section{Proof of Conjecture (K)}

We now intend to describe  the ideal ${\rm ker}\,\mu$ in ${\mathcal S}^+\otimes_k {\mathcal S}^-$.\;This is done in Sections 3 and 4 in several steps: first in Theorem \ref{ker} we prove that ${\rm ker}\,\mu$
is generated by $({\rm ker}\,\mu)^G$, then in Theorem \ref{vect} we describe ${\rm ker}\,\mu$ as a vector space, and finally in Theorem \ref{gi} we find a standard finite generating set of ${\rm ker}\,\mu$.

The first step is based on the following general statement:
\begin{theorem}\label{Luna}
Let $Z$ be an affine algebraic variety endowed with an action of a reductive algebraic group $H$.\;Let $a\in Z$ be a point such that the orbit morphism
\begin{equation*}
\varphi\colon H\to Z,\;\;h\mapsto h\cdot a
\end{equation*}
 is a closed embedding.\;Then the ideal ${\rm ker}\,\varphi^*$ in $k[Z]$ is generated by
 $({\rm ker}\,\varphi^*)^H$.
\end{theorem}

For the proof of Theorem \ref{Luna} we need the following

\begin{lemma}\label{idealm}
Let $\psi\colon Y\to Z$ be a morphism of irreducible affine algebraic varieties and let $z\in \psi(Y)$ be a smooth point of
$Z$.\;Assume that for each point $y\in \psi^{-1}(z)$
the following hold{\rm:}
\begin{enumerate}[\hskip 2.2mm\rm(i)]
\item $y$ is a smooth point of $\,Y$;
\item
the differential
$d_y\psi$ is surjective.
\end{enumerate}
 Then the ideal
$\{f\in k[Y]\mid f|_{\psi^{-1}(z)}=0\}$
of $k[Y]$ is generated by $\psi^*(\mathfrak m)$, where
$\mathfrak m:=\{h\in k[Z]\mid h(z)=0\}$.
\end{lemma}

\begin{proof}
Given  a nonzero function $f\in k[Y]$, below we denote by $Y_f$  the principal open subset $\{y\in Y\mid f(y)\neq 0\}$ of $Y$;\;it is affine and  $k[Y_f]=k[Y]_f$.

\vskip 1mm

1.\;Let $s_1,\ldots, s_d$ be a system of generators of the ideal $\mathfrak m$ of $k[Z]$.\;Put $t_i:=\psi^*(s_i)$.\;Then we have
\begin{equation}\label{fifi}
\{y\in Y\mid t_1(y)=\cdots=t_d(y)=0\}=\psi^{-1}(z).
\end{equation}

We claim  that, for every point $a\in Y$, there is a function $h_a\in k[Y]$ such that the principal open subset $U=Y_{h_a}$ is a neighborhood of $a$ and  $I^{\ }_U:=\{f\in k[U]\mid
f|_{\psi^{-1}(z)\cap U}=0\}$ is the ideal of $k[U]$
generated by $t_1|_U,\ldots, t_d|_U$.

Proving this, we consider two cases.

First,
 consider the case where $a\notin \psi^{-1}(z)$.\;Then any principal open neighborhood of $a$ not intersecting
$\psi^{-1}(z)$ may be taken as $U$ because in this case $I^{\ }_U=k[U]$ and, in view of \eqref{fifi} and Hilbert's Nullstellensatz,
$k[U]=
k[U]t_1|_U+\cdots+ k[U]t_d|_U$.

Second,
 consider the case where $a\in \psi^{-1}(z)$.\;Let $n=\dim Y, m=\dim Z$. Since $a$ and $z$ are the smooth points, the assumption (ii) yields the equality
\begin{equation}\label{n-m}
\dim {\rm ker}\,d_a\psi=n-m.
\end{equation}
The functions $s_1,\ldots, s_d$ generate the maximal ideal of ${\mathcal O}_{z, Z}$.\;Therefore,
re\-num\-bering them if necessary, we may (and shall) assume that $s_1,\ldots, s_m$ is a system of local parameters of $Z$ at $z$, i.e., $\bigcap_{i=1}^m{\rm ker}\,d_zs_i=\{0\}$.\;Since $d_at_i=d_a\psi\circ d_zs_i$, we then infer from (ii)  that $\bigcap_{i=1}^m{\rm ker}\,d_at_i={\rm ker}\,d_a\psi$.\;In view of \eqref{n-m}, the latter equality implies the existence of functions $f_1,\ldots, f_{n-m}\in {\mathcal O}_{a, Y}$ such that
$t_1,\ldots, t_m, f_1,\ldots, f_{n-m}$ is a system of local parameters of $Y$ at $a$.\;Let
\begin{equation}\label{part}
F:=\{y\in Y\mid t_1(y)=\cdots=t_m(y)=0\}.
\end{equation}
By \cite[Chap.\,II, \S3, Sect.\,2, Thm.\,4]{Sh13}, there is a principal open neighborhood $U$ of $a$ such that
$F\cap U$ is an irreducible smooth $(n-m)$-dimensional closed subvariety of $U$ whose ideal in $k[U]$ is generated by $t_1|_U,\ldots, t_m|_U$.\;On the other hand, \eqref{fifi} and \eqref{part} yield $\psi^{-1}(z)\subseteq F$ and, by the fiber dimension theorem,  every irreducible component of
$\psi^{-1}(z)$ has dimension $\geqslant n-m$.\;Hence $U\cap F=\psi^{-1}(z)\cap U$.\;This and \eqref{fifi}
prove the claim.

\vskip 1mm

2.\;Using this claim, the proof of Lemma \ref{idealm} is completed as follows. Since $Y=\bigcup_{a\in Y} Y_{h_a}$ and $Y$ is quasi-compact, there exists a finite set of points $a_1,\ldots, a_r\in Y$ such that
\begin{equation}\label{covering}
Y=\displaystyle  \bigcup_{i=1}^r Y_{h_{i}},\;\;\mbox{where $h_i:=h_{a_i}$.}
\end{equation}

Now, let $f\in k[Y]$ be a function such that $f|_{\psi^{-1}(z)}=0$.\;Then, in view of the definition of $h_a$, for every $i=1,\ldots, r$,  we have
\begin{equation}\label{decomps}
fh_i^{b_i}=c_{i,1}t_1+\cdots +c_{i,d}t_d\quad\mbox{for some $c_{i,j}\in k[Y]$ and $b_i\in{\mathbb N}$}.
\end{equation}
From \eqref{covering} and Hilbert's Nullstellensatz we infer that there are functions $q_1,\ldots, q_r\in k[Y]$
such that
\begin{equation}\label{pe}
1=q_1h_1^{b_1}+\cdots + q_sh_s^{b_r}.
\end{equation}
From \eqref{decomps} and \eqref{pe} we then deduce that
\begin{equation*}
f=\displaystyle \bigg(\sum_{i=1}^rq_ic_{i,1}\bigg)t_1+\cdots+\bigg(\sum_{i=1}^rq_ic_{i,d}\bigg)t_d\in
k[Y]t_1+\cdots+k[Y]t_d.
\end{equation*}
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem {\rm \ref{Luna}}]
There is a closed equivariant embedding of $Z$ in an affine space on which $H$ operates linearly (see \cite[Lemma 2]{Ro61}, \cite[Thm.\,1.5]{PV94}).\;Hence we may (and shall) assume that $Z$ is an irreducible smooth affine algebraic variety.

Since $G$ is reductive, $k[Z]^G$ is a finitely generated $k$-algebra
(see, e.g., \cite[Thm.\,A.1.0]{MF82} and the references therein).\;Denote by
 $Z/\!\!/H$ the affine algebraic variety ${\rm Specm}(k[Z]^G)$ and by
 $\pi
 \colon Z\to Z/\!\!/H$ the morphism corresponding to the inclusion homomorphism
 $k[Z]^G\hookrightarrow k[Z]$.

The condition on the point $a$ implies that its $H$-stabilizer is trivial,
\begin{equation}\label{Ha}
H_a=\{e\}.
\end{equation}
Hence $H\cdot a$ is a closed  $H$-orbit of maximal dimension.\;Taking into account that in
 every fiber of $\pi$ there is a unique closed
 orbit lying in the closure of every orbit contained in this fiber (see
 \cite[Cors.\,1.2, A.1.0]{MF82}), from this we deduce the equality
\begin{equation}\label{of}
\pi^{-1 }
(\pi
(a))=H\cdot a.
\end{equation}

Since  the group $\{e\}$ is linearly reductive, from
\eqref{Ha} and
the separability of $\varphi$ we infer
by \cite[Prop.\,7.6]{BR85}  that there is  a smooth affine subvariety $S$ of the $H$-variety $Z$, which is an \'etale slice at $a\in S$.\;In view of \eqref{Ha}, this means the following:
\begin{enumerate}[\hskip 2.2mm\rm(i)]
\item $\pi|^{\ }_S\colon S\to Z/\!\!/H$ and $\psi\colon H\times S\to Z$, $(h, s)\mapsto h\cdot s$  are the    \'etale
morphisms;
\item the diagram
\begin{equation*}\label{cd}
\begin{matrix}
\xymatrix{ H\times S\ar[r]^{\hskip 3mm \psi}\ar[d]_{{\rm pr}^{\ }_2}& Z\ar[d]^{\pi}\\
S\ar[r]^{\pi|^{\ }_S}&Z/\!\!/H
}
\end{matrix}
\end{equation*}
is a Cartesian square, i.e., it is commutative and the morphism $H\times S \to S\times_{Z/\!\!/H} Z$ determined by $\psi$ and ${\rm pr}_2$ is an isomorphism.
\end{enumerate}

From (i) and (ii) we deduce  that $\pi(a)$ is a smooth point of $Z/\!\!/H$ and the differentials $d_{(e,a)}\psi$, $d_a(\pi|^{\ }_{S})$ are isomorphisms.\;Since $d_{(e,a)}{{\rm pr}^{\ }_2}$ is clearly surjective, (ii)
then implies that
$d_a\pi$ is surjective, too.

Now, in view of \eqref{of} and transitivity of the action of $H$ on $H\cdot a$, we conclude that $d_z\pi$ is surjective for every point $z\in \pi^{-1}(\pi(a))$.\;In view of Lemma \ref{idealm}, this implies the claim of Theorem \ref{Luna}.
\end{proof}

\begin{theorem}\label{ker}
The ideal ${\rm ker}\,\mu$ in ${\mathcal S}^+\otimes_k {\mathcal S}^-$
is generated by $({\rm ker}\,\mu)^G$.
\end{theorem}

\begin{proof} In the proof of Theorem \ref{sur} we have shown that
\begin{enumerate}[\hskip 2.2mm ---]
\item the homomorphism $\mu$ is the composition of the homomorphisms \eqref{ep};
\item the morphism $\gamma$
is a closed
embedding.
\end{enumerate}
In view of these facts, Theorem \ref{ker} is equivalent to the claim that the ideal
${\rm ker}\,\gamma^*$ in $k[X\times X]$ is generated by $({\rm ker}\,\gamma^*)^G$.\;This
claim
follows from Theorem\;\ref{Luna}.
\end{proof}

\section{Structure of $({\rm ker}\,\mu)^G$}

We shall use the following lemma for describing $({\rm ker}\,\mu)^G$ as a vector space.
\begin{lemma}\label{otimes}
\begin{align}
\dim({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\nu)\!)^G&=
\Big\{\hskip -1.5mm\begin{array}{ll}
1& \mbox{if $\;\nu=\lambda^*$,}\\
0& \mbox{if $\;\nu\neq\lambda^*$}
\end{array}
\;\;
\mbox{for every $\lambda, \nu\in \XT+$},\label{10}\\
({\mathcal S}^+\otimes_k{\mathcal S}^-)^G&=\displaystyle \bigoplus_{\lambda\in \XT+}({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G.\label{G=}
\end{align}
\end{lemma}
\begin{proof}  In view of \eqref{grS+-}, the equality \eqref{G=} follows from \eqref{10}.\;To prove \eqref{10}, we note that
\begin{equation*}
({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\nu)\!)^G\cong {\rm Hom}_G
({\mathcal S}^+(\lambda)^*, {\mathcal S}^-(\nu))
\end{equation*}
and, in view of \eqref{H}, the $G$-module
${\mathcal S}^+(\lambda)^*$ is
the universal highest weight module of weight $\lambda$
(the Weyl mo\-du\-le);
in particular,
for each $G$-module $M$, there is an isomorphism
\begin{equation}\label{we}
{\rm Hom}_G({\mathcal S}^+(\lambda)^*, M))\xrightarrow{\cong} (M^{U^+})_\lambda,
\end{equation}
where the right-hand side of \eqref{we} is the weight space of $T$  (see \cite[II.2.13, Lemma]{Ja87}).\;Sin\-ce ${\mathcal S}^-(\nu)^{U^+}$\;is a line
on which $B^+$ acts by means of $\nu^*$ (see \cite[II.2.2, Prop.]{Ja87}), this proves \eqref{10}.
\end{proof}

We identify $k[G]\otimes_k k[G]$ with $k[G\times G]$ by the isomorphism
\begin{equation}\label{ide}
k[G]\otimes_k k[G]\to k[G\times G],\quad
f_1\otimes f_2
\mapsto
((a,b)
\mapsto f_1(a)f_2(b)).
\end{equation}
Thus ${\mathcal S}^+\otimes_k {\mathcal S}^-$ is regarded as a subalgebra of $k[G\times G]$,
and \eqref{mu}, \eqref{ide} yield the equality
\begin{equation}\label{rst}
f(a,a)=\mu(f)(a)\;\;\mbox{for every $f\in {\mathcal S}^+\otimes_k {\mathcal S}^-$ and $a\in G$.}
\end{equation}

\begin{theorem}\label{vect}\

\begin{enumerate}[\hskip 2.2mm \rm (i)]
  \item If $f\in ({\mathcal S}^+\otimes_k {\mathcal S}^-)^G$, then
  $f-f(e,e)\in ({\rm ker}\,\mu)^G$.
  \item Every $h\in  ({\rm ker}\,\mu)^G$\;can be uniquely written in the form
  \begin{equation}\label{decom}
    h=\displaystyle \sum
    (h_{\lambda}-h_{\lambda}(e,e)),\quad\mbox{
    $h_\lambda\in ({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G$,}
  \end{equation}
the sum is taken over a finite set of nonzero elements $\lambda\in {\rm X}\hskip -.2mm (T)\hskip -.2mm_+\hskip -.2mm.$
  \end{enumerate}
\end{theorem}
\begin{proof}
(i) Since $\mu$
is $G$-equivariant, its restriction to $({\mathcal S}^+\otimes_k{\mathcal S}^-)^G$ is a homomorphism to $k[G]^G=k$.\;Hence $\mu(f)$ is
a constant.\;In view of \eqref{rst}, this implies (i).

(ii) If \eqref{decom} holds, then the decomposition \eqref{G=} implies that $h_\lambda$ is the natural projection of
$h$ to $({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G$ determined by this decomposition; whence the uniqueness of \eqref{decom}.\;To prove the existence, let $h_\lambda$ be
the aforementioned projection of $h$ to $({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G$.\;Then
$h=\sum_{\lambda\in F} h_\lambda$ for a finite set $F\subset\XT+$.\;Hence
$0=\mu(h)=\sum_{\lambda\in F}\mu(h_\lambda)$.\;As above, $\mu(h_\lambda)=h_{\lambda}(e,e)$; this implies the equality \eqref{decom}, where the sum is taken over all $\lambda\in F$.\;Since $h_0$ is a constant, we may assume that $F$ does not contain $0$.\;This proves\;(ii).
\end{proof}

In
the next lemma, for brevity,  we put (cf.\;\cite{Ja87})
\begin{equation}\label{notatio}
\begin{gathered}
V(\lambda):={\mathcal S}^-(\lambda)^*\cong {\mathcal S}^+(\lambda)^*,\quad  L(\lambda):=V(\lambda)/{\rm rad}_GV(\lambda),\\[-.5mm]
\pi_\lambda\colon V(\lambda)\to L(\lambda)\;\;\mbox{is the canonical projection.}
\end{gathered}
\end{equation}
The $G$-module $V(\lambda)$ (hence $L(\lambda)$ as well) is generated by a $B^+$-stable line of weight $\lambda$ (see \cite[II, Sect.\;2.13, Lemma]{Ja87}); whence $V(\lambda)$ is also ge\-ne\-rated by a $B^-$-stable line of weight $-\lambda^*$.

Also, for the $G$-modules $P$ and $Q$, we denote by
${\mathscr B}(P\times Q)$
the $G$-module of all bilinear maps $P\times Q\to k$;\;we then have
the isomorphism of $G$-modules
\begin{equation}\label{bd}
P^*\otimes_k Q^*\xrightarrow{\cong} {\mathscr B}(P\!\times\!Q),\quad f\otimes h\mapsto fh.
\end{equation}

\begin{lemma}\label{bili}For all elements $\lambda, \nu\in \XT+$, the following hold{\rm:}
\begin{enumerate}[\hskip 4.2mm\rm(a)]
  \item $\dim {\mathscr B}(V(\lambda)\!\times\!V(\nu)\!)^G\hskip -1mm =\Big\{\hskip -1.5mm\begin{array}{ll}
1& \mbox{if $\;\nu=\lambda^*\!,$}\\
0& \mbox{if $\;\nu\neq\lambda^*\!.$}
\end{array}$
  \item $\dim {\mathscr B}(L(\lambda)\!\times\!L(\nu)\!)^G\hskip -1mm =\Big\{\hskip -1.5mm\begin{array}{ll}
1& \mbox{if $\;\nu=\lambda^*\!,$}\\
0& \mbox{if $\;\nu\neq\lambda^*\!.$}
\end{array}$
  \item  Every nonzero element $\theta\in {\mathscr B}(L(\lambda)\times
  L(\lambda^*)\!)^G$ is a nondegenerate pair\-ing $
      L(\lambda)\!\times\!L(\lambda^*)\to k${\rm.}
  \item If
  $\,l^+\in L(\lambda)$, $l^-\in L(\lambda^*)$ are the nonzero semi-invariants of, res\-pec\-ti\-vely, $B^+$ and $B^-$, then $\theta(l^+, l^-)\neq 0$ for $\theta$ from {\rm (c)}.\;For every nonzero ele\-ment $\epsilon\in k$, there exists a unique $\theta$ such that $\theta(l^+, l^-)=\epsilon$.
  \item Every element $\vartheta\in {\mathscr B}(V(\lambda)\!\times\!V(\lambda^*)\!)^G$\;vanishes on ${\rm ker}\,\pi^{\ }_{\lambda}\times {\rm ker}\,\pi^{\ }_{{\lambda}^*}$.\;If $\vartheta\neq 0$, then $\vartheta$ is a nondegenerate pairing $V(\lambda)\times V(\lambda^*)\to k$.
  \item Let $v^+\in V(\lambda)$
  and $v^-\in V(\lambda^*)$ be, respectively,
  the non\-zero $B^+$\!\!\!-
and $B^-$\!\!\!-semi-invari\-ants of weights $\lambda$ and $-\lambda$
  that generate the $G$-modules $V(\lambda)$ and $V(\lambda^*)$.\;Then
  $\vartheta(v^+, v^-)\neq 0$ for every nonzero
  element
  $\vartheta\in {\mathscr B}
  (V(\lambda)\times V(\lambda^*)\!)^G$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (a) follows from \eqref{10}, \eqref{bd}, \eqref{notatio}.\;Part (b) is proved similarly, using that $L(\lambda)$ is a simple $G$-module with highest weight $\lambda$ (see \cite[II.2.4]{Ja87}).\;The simplicity of $L(\lambda)$  implies (c) because the left and right kernels of $\theta$ are $G$-stable.

Proving (d), take a basis $\{p_1,\ldots, p_s\}$ of $L(\lambda)$ such that
$p_1=l^+$
and eve\-ry  $p_i$ is a weight vector of $T$.\;Let $\{p_1^*,\ldots, p_s^*\}$ be the basis of $L(\lambda^*)$ dual to $\{p_1,\ldots, p_s\}$ with respect to $\theta$.\;Let
$L(\lambda)'$ be
the linear span over $k$ of all $p_i$'s with $i>1$.\;Then $L(\lambda)'$ is $B^-$-stable, and, for every element $u\in U^-$, we have $u\cdot p_1=
p_1+p'$, where $p'\in L(\lambda)'$ (see, e.g.,\;\cite[Sect.\;3.3, Prop.\;2 and p.\,84]{St74}).\;Then, for every elements $\alpha_1,\ldots,\alpha_s\in k$, we have
\begin{align*}
(u\cdot p^*_1)\bigg(\displaystyle \sum_{i=1}^s\alpha_ip_i\bigg)&=
p^*_1\bigg(\displaystyle \sum_{i=1}^s\alpha_i(u^{-1}\cdot p_i)\!\bigg)\\
&=p^*_1(\alpha_1p_1+\mbox{an element of $L(\lambda)'$})\\
&=\alpha_1=p^*_1\bigg(\displaystyle \sum_{i=1}^s\alpha_ip_i\!\bigg);
\end{align*}
whence $u\cdot p^*_1\!=\!p^*_1$.\;Therefore,  $l^-\!=\!\lambda p_1^*$ for a nonzero $\lambda\in k$, hence
$\theta(l^+, l^-)=\lambda\neq 0$.\;This and (b) prove (d).

It follows from \eqref{notatio}, (a), (b) that the embedding
\begin{equation*}
    {\mathscr B}(L(\lambda)\!\times\!L(\lambda^*))^G\to
    {\mathscr B}(V(\lambda)\!\times\!V(\lambda^*))^G,\quad \theta\mapsto \theta\circ (\pi_{\lambda}\times \pi_{\lambda^*})
\end{equation*}
is an isomorphism.\;Part (e) follows from this and (c).

Part (f) follows from (d) and (e), because $\pi_\lambda(v^+)$ and $\pi_{\lambda^*}(v^-)$ are, in view of \eqref{notatio}, the
nonzero semi-invariants of, respectively, $B^+$ and $B^-$.
\end{proof}

\begin{lemma}\label{covv}
Let an algebraic group $H$ act on a algebraic variety $Z$ and let $V$ be a finite-dimensional
submodule of the $H$-module $k[Z]$.\;Then the morphism
\begin{equation}\label{cova}
    \varphi\colon Z\to V^*,\quad \mbox{$\varphi(a)(f)=f(a)$ for every $a\in Z, f\in V$}
\end{equation}
has the following properties:
\begin{enumerate}[\hskip 2.2mm\rm(i)]
  \item $\varphi$ is $H$-equivariant;
  \item the restriction of $\;\varphi^*$ to $(V^*)^*$ is
  an isomorphism $(V^*)^*\to V$;
  \item $\varphi^*$ exercises an isomorphism between $k[\overline{\varphi(Z)}]$ and the subalgebra of $k[Z]$ generated by $V$.
\end{enumerate}
\end{lemma}
\begin{proof} Part (i) is proved by direct verification.

Every function $f\in V$
determines an element $l_f\in (V^*)^*$ by the formula $l_f(s)=s(f)$, $s\in V^*$.\;It is immediate that $V\to (V^*)^*$, $f\mapsto l_f$ is a vector space isomorphism and that \eqref{cova} implies $\varphi^*(\l_f)=f$.\;This proves (ii).

Let $\iota\colon (V^*)^*\to  k[\overline{\varphi(Z)}]$ be the restriction homomorphism.\;The $k$-algebra $k[\overline{\varphi(Z)}]$ is generated by $\iota((V^*)^*)$.\;Part (iii) now follows from the fact that
$\varphi^*$ exercises an embedding of $k[\overline{\varphi(Z)}]$ in $k[Z]$ and, in view of (ii), the image of $\iota((V^*)^*)$
under this embedding is $V$.
\end{proof}

\begin{corollary}\label{coro1} In the notation of Lemma {\rm \ref{covv}},
let $V\neq \{0\}$ and let the orbit $H\cdot a$ be dense in $Z$.\;Then  $\varphi(a)\neq 0$.
\end{corollary}

We call the morphism  \eqref{cova} the {\it covariant determined by the submodule $V$}.

\begin{lemma}\label{phipm} Let $\lambda$ be an element of $\XT+$ and let
\begin{equation*}
\varphi^+\colon G\to {\mathcal S}^+(\lambda)^*, \quad
\varphi^-\colon G\to {\mathcal S}^-(\lambda^*)^*
\end{equation*}
be the covariants determined by the submodules ${\mathcal S}^+(\lambda)$ and ${\mathcal S}^-(\lambda^*)$ of
the $G$-module
$k[G]$.\;Then $v^+:=\varphi^+(e)$ and $v^-:=\varphi^-(e)$ are, respectively, the nonzero $B^+$\!\!- and $B^-$\!\!-se\-mi\-invariants of weights $\lambda$ and $-\lambda$.
\end{lemma}
\begin{proof}
First, we have
$v^+\neq 0$, $v^-\neq 0$ by Corollary \ref{coro1}.\;Next, for every  $f\in {\mathcal S}^+(\lambda)$, $b\in B^+$, we have
\begin{align*}(b\cdot v^+)(f)&=\varphi^+(e)(b^{-1}\cdot f)\overset{\eqref{cova}}{=\hskip -1mm=}
(b^{-1}\cdot f)(e)\\[-1.5mm]
&=f(b)\overset{\eqref{S+la}}{=}b^{\lambda}f(e)\overset{\eqref{cova}}{=\hskip -1mm=}(b^{\lambda}v^+)(f);\end{align*}
whence $b\cdot v^+=b^{\lambda}v^+$, i.e., $v^+$ is a nonzero $B^+$-semi-invariant of weight $\lambda$, as claimed.\;For $v^-$ the proof is similar.
\end{proof}
\begin{theorem}\label{iiss}
The restriction of $\mu$ to $({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda)\!)^G$
for every $\lambda\in \XT+$
is an isomorphism $({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G\xrightarrow{\cong} k[G]^G=k$.
\end{theorem}
\begin{proof}
In view of
\eqref{rst}
and Lemma \ref{otimes}, the proof is reduced to showing that there is a function $f\in
({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G$ such that $f(e,e)\neq 0$.

Consider the covariants
$\varphi^+$ and $\varphi^-$ from Lemma \ref{phipm} and the $G$-equivariant morphism
\begin{equation*}
 \varphi:=   \varphi^+\times \varphi^-\colon G\times G\to {\mathcal S}^+(\lambda)^*\times {\mathcal S}^-(\lambda^*)^*.
\end{equation*}
Lemma \ref{bili}(a) and \eqref{notatio} imply that ${\mathscr B}({\mathcal S}^+(\lambda)^*\times {\mathcal S}^-(\lambda^*)^*)^G$ contains a nonzero element $\vartheta$.\;By Lemma \ref{covv}, the function
$f:=\vartheta\circ\varphi\colon G\times G\to k$ is contained  in $({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G$.\;For this $f$, using Lemmas \ref{phipm} and \ref{bili}(f), we obtain
\begin{equation}\label{varth}
f(e,e)=\vartheta(\varphi(e,e))=\vartheta(\varphi^+(e), \varphi^-(e))\neq 0.
\end{equation}
This completes the proof.
\end{proof}

\begin{corollary}\label{corrr}
For every element $\lambda\in\XT+$, there exists a unique element
\begin{equation}\label{s}
s_\lambda\in ({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G\subseteq k[G\times G] \;\;\mbox{such that
$s_{\lambda}(e,e)=1$.}
\end{equation}
If $\{f_1,\ldots, f_d\}$ and $\{h_1,\ldots , h_d\}$ are the bases of ${\mathcal S}^+(\lambda)$ and ${\mathcal S}^-(\lambda^*)$ dual with respect to a nondegenerate $G$-invariant pairing
${\mathcal S}^+(\lambda)\times {\mathcal S}^-(\lambda^*)\to k$ {\rm(}the latter exists by {\rm \eqref{bd}} and Lemma {\rm\ref{bili})}, then $\varepsilon:=\sum_{i=1}^d f_i(e)h_i(e)\neq 0$ and
\begin{equation*}
s_\lambda=\varepsilon^{-1}\bigg(\displaystyle \sum_{i=1}^d f_i\otimes h_i\bigg).
\end{equation*}
\end{corollary}
\begin{proof} First, note that if $P$, $Q$ are the finite dimensional $kG$-modules,  $\theta\in {\mathscr B}(P, Q)^G$ is a nondegenerate pairing $P\times Q\to k$, and
$\{p_1,\ldots, p_m\}$ and $\{q_1,\ldots, q_m\}$ are the bases of $P$ and $Q$ dual with respect to $\theta$, then $\sum_{i=1}^m p_i\otimes q_i$ is a nonzero element of $(P\otimes_k Q)^G$ (not depending on the choice of these bases). Indeed, $\theta$ determines the isomorphism
of $G$-modules
\begin{equation}\label{mmmm}
\begin{split}
    \phi\colon
    P\otimes_k Q&\to {\rm Hom}(P, P),\\
    (\phi(p\otimes q))(p')&=\theta(p', q)p,\;\mbox{where $p, p'\in P,\, q\in Q$}.
    \end{split}
\end{equation}
From \eqref{mmmm} we then obtain
\begin{align*}
\bigg(\phi\bigg(\displaystyle  \sum_{i=1}^mp_i\otimes q_i\bigg)\!\!\bigg)(p_j)&=
\displaystyle \sum_{i=1}^m\theta(p_j, q_i)p_i=\displaystyle \sum_{i=1}^m\delta_{ij} p_i=p_j,
\end{align*}
therefore, $\phi\big(\sum_{i=1}^mp_i\otimes q_i\big)={\rm id}_{P}$; whence the claim.

For $P={\mathcal S}^+(\lambda)$, $Q={\mathcal S}^-(\lambda^*)$, it yields that $\sum_{i=1}^d f_i\otimes h_i$ is a nonzero element of $({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*))^G$.\;Theorem \ref{iiss} and \eqref{rst} then complete the proof.
\end{proof}

\begin{remark}\label{000} For ${\rm char}\,k=0$, there is another characterization of $s_\lambda$.\;Namely, let
${\mathscr U}$ be the universal enveloping algebra of ${\rm Lie}\,G$.\;Every
${\mathcal S}^\pm(\lambda)$ is endowed with the natural ${\mathscr U}$-module structure.\;Let
$\{x_1,\ldots, x_n\}$ and $\{x_1^*,\ldots, x_n^*\}$ be the bases of ${\rm Lie}\,G$ dual with respect to the Killing form $\Phi$.\;Identify ${\rm Lie}\,T$ with its dual space by means of $\Phi$.\;Let $\sigma$ be the sum of all positive roots.\;For every $\lambda\in \XT+$, put
\begin{equation}\label{cl}
c_\lambda:=\Phi(\lambda + \sigma, \lambda)+\Phi(\lambda^* + \sigma, \lambda^*)
\end{equation}
and consider on the space ${\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)$
the linear operator
\begin{equation}\label{Delta}
\Delta:=\displaystyle \sum_{i=1}^{n}(x_i\otimes x_i^*+x_i^*\otimes x_i).
\end{equation}
\begin{proposition} The following properties of an element $t\in {\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)$ are equivalent:
\begin{enumerate}[\hskip 4.2mm \rm(i)]
\item $t=s_\lambda$;
\item $\Delta (t)=-c_\lambda t$ and $t(e,e)=1$.
\end{enumerate}
\end{proposition}
\begin{proof} By \cite[Chap.\,VIII, \S6, Sect.\,4, Cor.]{Bou75}, the Casimir element $\Omega:=\sum_{i=1}^nx_ix_i^*\in {\mathscr U}$ acts on any simple ${\mathscr U}$-module with the highest weight $\gamma$ as scalar multiplication by
$\Phi(\gamma+\sigma, \gamma)$.\;Since $\Phi(\gamma+\sigma, \gamma)>0$ if $\gamma\neq 0$, the kernel of $\Omega$ in any finite dimensional ${\mathscr U}$-module $V$ coincides with $V^G$.\;We apply this to $V={\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)$.\;For any elements $f\in {\mathcal S}^+(\lambda)$, $h\in {\mathcal S}^-(\lambda^*)$, we deduce from \eqref{cl}, \eqref{Delta} the following:
\begin{equation*}
\begin{split}
\Omega(f\!\otimes\! h)\!&=\!\displaystyle  \sum_{i=1}^n\!\big(x_ix_i^*(f)\!\otimes\! h\!+\!x_i^*(f)\!\otimes\! x_i(h)\!
+\!x_i(f)\!\otimes\! x_i^*(h)\!+\!f\!\otimes\! x_ix_i^*(h)\!\big)\\
&=\Omega(f)\!\otimes\! h\!+\! f\!\otimes\! \Omega(h)\!+\!\Delta(f\!\otimes\! h)=c_\lambda (f\!\otimes\!h)\!+\!\Delta(f\!\otimes\! h).
\end{split}
\end{equation*}
Now Corollary \ref{corrr} and the aforesaid about ${\rm ker}\,\Omega$ complete the proof.
\end{proof}
\end{remark}

\begin{theorem}\label{iG}
Let $\lambda_1,\ldots, \lambda_m$ be a system of generators of the monoid
$\XT+$. Then $({\rm ker}\,\mu)^G$ is the linear span over $k$ of all monomials of the form
\begin{gather*}
    (s_{\lambda_1}-1)^{d_1}\cdots (s_{\lambda_m}-1)^{d_m}\!,\;\mbox{where $d_i\in {\mathbb N}$,  $d_1+\cdots+d_m>0$,}
\end{gather*}
where $s_{\lambda_i}$ is defined in
Corollary {\rm \ref{corrr}}.
\end{theorem}
\begin{proof}
By Theorem \ref{vect}(i), the linear span $L$ referred to in Theorem \ref{iG} is contained in $({\rm ker}\,\mu)^G$.\;In view of Theorem \ref{vect}, to prove the converse inclusion $({\rm ker}\,\mu)^G\subseteq L$, we have to show that, for every function
\begin{equation}\label{fff}
f\in ({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G,
\end{equation}
we have $f-f(e,e)\in L$.\;Since $\lambda_1,\ldots, \lambda_m$ is a system of generators of
$\XT+$, there are the integers $d_1,\ldots, d_m\in {\mathbb N}$ such that $\lambda=\sum_{i=1}^md_i\lambda_i$.\;From \eqref{s} and \eqref{grS+-} we then infer
that $h:=\prod_{i=1}^ms_{\lambda_i}^{d_i}
\in ({\mathcal S}^+(\lambda)\otimes_k {\mathcal S}^-(\lambda^*)\!)^G$ and $h(e,e)=1$.\;This, \eqref{fff}, and \eqref{10} imply that $f=f(e,e)h$.\;Therefore,
\begin{equation}\label{ef}
f-f(e,e)=f(e,e)(h-1)=f(e,e)\bigg(\!\displaystyle \prod_{i=1}^m((s_{\lambda_i}-1)+1)^{d_i}-1\!\bigg).
\end{equation}
The right-hand side of \eqref{ef} clearly lies in $L$. This completes the proof.
\end{proof}

\begin{theorem}\label{gi}
Let $\lambda_1,\ldots, \lambda_m$ be a system of generators of the monoid
$\XT+$.\;Then the ideal ${\rm ker}\,\mu$ in ${\mathcal S}^+\otimes_k {\mathcal S}^-$
is generated by $s_{\lambda_1}-1,\ldots, s_{\lambda_m}-1$, where $s_{\lambda_i}$ is defined in
Corollary {\rm \ref{corrr}}.
\end{theorem}
\begin{proof} This follows from Theorems \ref{ker} and \ref{iG}.
\end{proof}

\section{Presentation of ${\mathcal S}^\pm$}

If the group $G$ is semisimple, then the semigroup $\XT+$ has no units other than $0$.\;Hence
the set ${\mathscr H}$ of all indecomposable elements of $\XT+$ is finite,
\begin{equation}\label{Hilb}
{\mathscr H}=\{\lambda_1,\ldots, \lambda_d\},
\end{equation}
generates $\XT+$, and
every generating set of $\XT+$ contains ${\mathscr H}$ (see, e.g., \cite[Lemma 3.4.3]{Lo05}).\;Note that
${\mathscr H}$, called the {\it Hilbert basis of $\;\XT+$}, in general  is not a free generating system of $\XT+$ (i.e., it is not true that every element
$\alpha\in \XT+$ may be uniquely expressed in the form
$\alpha=\sum_{i=1}^d c_i\lambda_i$, $c_i\in{\mathbb N}$).\;Namely, it is free if and only if
$G=G_1\times\cdots\times G_s$ where
every $G_i$ is either a simply connected simple
algebraic group or isomorphic to ${\rm SO}_{n_i}$ for an
odd $n_i$ (see \cite[\S3]{St75}, \cite[Prop.\;4.1]{Ri79}, \cite[Prop.\;13.3]{Ri82}, \cite[Remark\;3.16]{Po11}).\;In
particular,
 if $G$ is simply connected, then ${\mathscr H}$
 coincides with the set of all fundamental weights
 and generates $\XT+$ freely.\;Note that $\lambda^*_i\in \mathscr H$ for every $i$.

To understand presentation of ${\mathcal S}^\pm$, denote respectively by ${\rm Sym}\,{\mathcal S}^\pm(\lambda_i)$ and ${\rm Sym}^m\,{\mathcal S}^\pm(\lambda_i)$  the symmetric algebra and the $m$th symmetric power of ${\mathcal S}^\pm(\lambda_i)$.\;The naturally ${\mathbb N}^d$-graded free commutative $k$-algebra
\begin{equation}\label{falga}
{\mathcal F}^\pm:={\rm Sym}\,{\mathcal S}^\pm(\lambda_1)\otimes_k\cdots\otimes_k {\rm Sym}\,{\mathcal S}^\pm(\lambda_d)
 \end{equation}
 may be viewed as the algebra of regular functions $k[L^\pm]$ on the vector space
 \begin{equation*}
 L^\pm:={\mathcal S}^\pm(\lambda_1)^*\oplus\cdots \oplus {\mathcal S}^\pm(\lambda_d)^*.
 \end{equation*}
 Let $e_{i}$ be the $i$th unit vector of ${\mathbb N}^d$ and let ${\mathcal F}^\pm_{p,q}$ be the homogeneous component of ${\mathcal F}^\pm$ of degree $e_p+e_q$.\;We have the natural isomorphisms of $G$-modules
\begin{equation}\label{ssss}
\varphi_{p,q}^\pm\colon {\mathcal F}^\pm_{p,q}\xrightarrow{\cong}{\mathcal S}^\pm_{p,q}:=\begin{cases}
{\mathcal S}^\pm(\lambda_p)\otimes_{k}{\mathcal S}^\pm(\lambda_q)
&\mbox{if $p\neq q$},\\
{\rm Sym}^{2}\,{\mathcal S}^\pm(\lambda_p) &\mbox{if $p=q$.}\end{cases}
\end{equation}

By Theorems  \ref{RR}, \ref{fg} the natural multiplication homomorphisms
\begin{equation}\label{phipsi}
\phi^\pm\colon {\mathcal F}^\pm\to {\mathcal S}^\pm\quad\mbox{and}\quad
\psi_{p,q}^\pm\colon {\mathcal S}^\pm_{p,q}\to {\mathcal S}^\pm(\lambda_p+\lambda_q)
\end{equation}
 are surjective.\;Since ${\mathcal F}^\pm$ is a polynomial algebra, the surjectivity of $\phi^\pm$ reduces finding a presentation of ${\mathcal S}^\pm$ by generators and relations to describing ${\rm ker}\,\phi^\pm$.\;If
 $d=\dim T$, the following explicit description of ${\rm ker}\,\phi^\pm$ is available:

\begin{theorem}\label{grS} Let $G$ be a connected semisimple group such that the Hilbert basis
{\rm \eqref{Hilb}} freely generates the semigroup $\,\XT+$.\;Then
\begin{enumerate}[\hskip 2.2mm\rm(i)]
\item the ideal ${\rm ker}\,\phi^\pm$ of the ${\mathbb N}^d$-graded
$k$-algebra ${\mathcal F}^\pm$ is homogeneous;
\item this ideal is generated by the union of all its homogeneous components of the total degree $2$;
 \item the set of these homogeneous components coincides with the set of all
subspaces
$\big(\varphi_{p,q}^\pm\big)^{-1}\big({\rm ker}\,\psi_{p,q}^\pm\big)$,
$1\leqslant p\leqslant q\leqslant d$.
\end{enumerate}
\end{theorem}
\begin{proof} This is the main result of \cite{KR87}.
\end{proof}

\begin{remark}
 In characteristic $0$ for the first time the proof of Theorem \ref{grS}
 was
 obtained (but not published) by B. Kostant; his proof appeared in \cite[Thm.\;1.1]{LT79}.\;In this case, \eqref{ssss} and the surjectivity of $\psi_{p,q}^\pm$ yield
that $\psi_{p,q}^\pm$ is the projection of ${\mathcal S}^\pm_{p,q}$ to the Cartan component
of ${\mathcal S}^\pm_{p,q}$, and ${\rm ker}\,\psi_{p,q}^\pm$ is the unique $G$-stable direct complement to
this component.\;The subspace ${\rm ker}\,\psi_{p,q}^\pm$
admits the following description using the notation of Remark \ref{000}
(loc.\,cit.).\;Let
$\{x_1,\ldots, x_n\}$ and $\{x_1^*,\ldots, x_n^*\}$ be the dual bases of
${\rm Lie}\,G$
with respect to
$\Phi$.\;Then
${\rm ker}\,\psi_{p,q}^+$ is the ima\-ge of the linear transformation $\big(\!\sum _{s=1}^n(x_s\otimes x_s^*+x_s^*\otimes x_s)\!\big)-2\Phi(\lambda_p^*, \lambda_q^*){\rm id}$
of the vector space\;${\mathcal S}_{p,q}^\pm$.
\end{remark}

Summing up, if
$G$ is a connected semisimple group such that the Hilbert basis
{\rm \eqref{Hilb}} freely generates the semigroup $\,\XT+$, then
the sought-for cano\-ni\-cal presentation of $k[G]$ is given
by the surjective
homomorphism
\begin{equation}\label{phiphi}
\phi:=\phi^+\otimes \phi^-\colon {\mathcal F}:=\mathcal F^+\otimes_k \mathcal F^-
\to k[G]
\end{equation}
 of the polynomial $k$-algebra ${\mathcal F}$ and the following generating system $\mathscr R$ of the ideal
 $\ker\,\phi$.\;Identify
 ${\mathcal F}^+$ and ${\mathcal F}^-$ with subalgebras of $\mathcal F$
 in the natural way.\;Then

 ${\mathscr R}={\mathscr R}_1 \bigsqcup {\mathscr R}_2$, where
 \begin{equation}\label{R1}
 {\mathscr R}_1=\displaystyle
 \bigcup_{p, q}
 \big(\big(\varphi_{p,q}^+\big)^{-1}\big({\rm ker}\,\psi_{p,q}^+\big)\,\mbox{\Large $\cup$}\, \big(\varphi_{p,q}^-\big)^{-1}\big({\rm ker}\,\psi_{p,q}^-\big)\big)
 \end{equation}
(see the definition of $\varphi_{p,q}^\pm$, $\psi_{p,q}^\pm$ in  \eqref{ssss}, \eqref{phipsi})
and
 \begin{equation}\label{R2}
 {\mathscr R}_2=\{s_{\lambda_1}-1,\ldots, s_{\lambda_d}-1\}
   \end{equation}
   (see the definition of $s_{\lambda_i}$ in Corollary \ref{corrr}).\;The elements of
 ${\mathscr R}_1$ (respectively, ${\mathscr R}_2$) are the Pl\"ucker-type (respectively, the ${\rm SL}_2$-type) relations of the presentation.

The canonical presentation of $k[G]$
is redundant.\;To
reduce the size of ${\mathscr R}_1$, we may replace every space $\ker\,\psi^\pm_{p,q}$ in \eqref{R1} by a basis of
this space.\;Finding such a basis falls
within  the framework of Standard Monomial Theory.

\section{An example}\label{example}

As an illustration, here we explicitly describe  the canonical presentation of $k[G]$
for $G={\rm SL}_n$, $n\geqslant 2$, and ${\rm char}\,k=0$.

Let $T$ be the maximal torus of diagonal matrices in $G$, and let $B^+$ (respectively, $B^-$) be the Borel subgroup of lower (respectively, upper) triangular matrices in $G$.\;Then
\begin{gather*}
\mathscr H\!=\!\{\varpi_1,\ldots, \varpi_{n-1}\},\;\;\mbox{where}\\[-.5mm
\varpi_d\colon T\!\to\! k,\;\; {\rm diag}(a_1,\ldots, a_n)\mapsto a_{n-d+1}\cdots a_n.
\end{gather*}

Every pair $i_1, i_2\in [n]$
determines the function
\begin{equation}\label{fxij}
x_{i_1, i_2}\colon G\to k, \quad \begin{pmatrix}a_{1,1}&\ldots&a_{1,n}\\
\hdotsfor[2]{3}\\
a_{n,1}&\ldots&a_{n,n}
\end{pmatrix}
\mapsto a_{i_1,i_2}.
\end{equation}
The $k$-algebra generated by all functions
\eqref{fxij} is $k[G]$.

For every $d\in [n-1]$ and every sequence
$i_1,\ldots, i_d$ of $d$ elements of $[n]$, put
\begin{equation*}
f^-_{i_1,\ldots, i_d}:=\det\! \begin{pmatrix}x_{i_1,1}&\ldots&x_{i_1,d}\\
\hdotsfor[2]{3}\\
x_{i_d,1}&\ldots&x_{i_d,d}
\end{pmatrix}\!\!, \quad
f^+_{i_1,\ldots, i_d}:=\det \!\begin{pmatrix}x_{i_1,n-d+1}&\ldots&x_{i_1,n}\\
\hdotsfor[2]{3}\\
x_{i_d,n-d+1}&\ldots&x_{i_d,n}
\end{pmatrix}\!\!.
\end{equation*}

 For every fixed $d$, all functions $f^-_{i_1,\ldots, i_d}$ (respectively, $f^+_{i_1,\ldots, i_d}$) such that
 $i_1<\cdots< i_d$  are linearly independent over $k$ and their linear span over $k$ is
 the simple $G$-module ${\mathcal S}^-(\varpi_d)$ (respectively,  ${\mathcal S}^+(\varpi_d)$); see, e.g., \cite[Prop.\;3.2]{FT92}.\;Therefore, denoting by $x^\pm_{i_1,\ldots, i_d}$ the element $f^\pm_{i_1,\ldots, i_d}$ of the
 $k$-al\-geb\-ra ${\mathcal F}^\pm$ defined by  \eqref{falga}, we
identify ${\mathcal F}^\pm$ with  the polynomial $k$-algebra in variables $x^\pm_{i_1,\ldots, i_{d}}$, where $d$ runs
over $[n-1]$ and $i_1,\ldots, i_{d}$ runs over
$[n]_d$.\;Correspondingly, the $k$-algebra ${\mathcal F}$ is identified with
the polynomial $k$-algebra in the variables
$x^-_{i_1,\ldots, i_{d}}$ and $x^+_{i_1,\ldots, i_{d}}$,
the homomorphism \eqref{phiphi} takes the form
\begin{equation*}\label{phiphiphi}
\phi\colon {\mathcal F}
\to k[G],\quad x^+_{i_1,\ldots, i_{d}}\mapsto f^+_{i_1,\ldots, i_{d}},\;\; x^-_{i_1,\ldots, i_{d}}\mapsto f^-_{i_1,\ldots, i_{d}},
\end{equation*}
and $\phi^\pm=\phi|_{{\mathcal F}^\pm}$.\;Below the sets \eqref{R1} and \eqref{R2} are explicitly specified using this notation.

First, we will specify the Pl\"ucker-type relations.\;It
  is convenient to introduce the following elements of ${\mathcal F}^\pm$.\;Let  $i_1,\ldots, i_d$ be a sequence of $d\!\in\! [n\!-\!1]$ elements of $[n]$,  and let $j_1,\ldots, j_d$ be
the nondecreasing sequence obtained from $i_1,\ldots, i_d$ by permutation.\;Then we put
\begin{equation*}
x^\pm_{i_1,\ldots, i_d}=\begin{cases}
{\rm sgn}(i_1,\ldots, i_d)x^\pm_{j_1,\ldots, j_d}&\mbox{if $i_p\neq i_q$ for all $p\neq q$},\\
0& \mbox{otherwise}.
\end{cases}
\end{equation*}

The $k$-algebra $\mathcal S^\pm$ is the coordinate algebra of the affine multicone over the flag variety, see \cite{To79}.\;By the well-known classical Hodge's result \cite{Ho42}, \cite{Ho43} (see also \cite[p.\;434, Cor.\,1]{To79}) the ideal ${\rm ker}\,\phi^\pm$ is
  generated by all elements of the form
\begin{equation}\label{Plucker}
\displaystyle \sum_{l=1}^{q+1}(-1)^lx^\pm_{i_1,\ldots, i_{p-1}, j_l} x^\pm_{\hskip -.7mm j_1,\ldots, \widehat{j_l},\ldots, j_{q+1}},
\end{equation}
where $p$ and $q$ run over $[n-1]$, $p\leqslant q$, and $i_1,\ldots, i_{p-1}$ and $j_1,\ldots, j_{q+1}$ run over
$[n]_{p-1}$ and $[n]_{q+1}$ respectively.\;Since
every element \eqref{Plucker}
is homogeneous  of degree 2, this result together with
Theorem \ref{grS} imply that, for every fixed $p, q\in [n-1]$, the
set $\big(\varphi_{p,q}^\pm\big)^{-1}\big({\rm ker}\,\psi_{p,q}^\pm\big)$ in \eqref{R1} is the linear span of
all elements \eqref{Plucker}, where $i_1,\ldots, i_{p-1}$ and $j_1,\ldots, j_{q+1}$ run over
$[n]_{p-1}$ and $[n]_{q+1}$ respectively.\;This describes the Pl\"ucker-type relations \eqref{R1}.

Secondly,  we will describe $s_{\varpi_d}$.\;If
${\boldsymbol {i}}\in [n]_{n-d}$ is a sequence $i_1,\ldots, i_{n-d}$, we put $x^\pm_{\boldsymbol i}:=
x^\pm_{i_1,\ldots, i_{n-d}}$ and denote by
 ${\boldsymbol {i}}^*\in [n]_{d}$ the unique
 sequence $j_1,\ldots, j_{d}$
whose intersection with $i_1,\ldots, i_{n-d}$
is empty.\;Let ${\rm sgn}({\boldsymbol {i}}, {\boldsymbol {i}}^*)$ be the sign of the permutation
$(i_1,\ldots, i_{n-d},j_1,\ldots, j_{d})$.\;Then by \cite[Thm.\;3.1(b)]{FT92},
\begin{equation*}
s_{\varpi_d}=\displaystyle \sum_{\boldsymbol {i}\in [n]_{n-d}} {\rm sgn}({\boldsymbol {i}}, {\boldsymbol {i}}^*)x^-_{\boldsymbol {i}}x^+_{\boldsymbol {i}^*}.
\end{equation*}
This describes the ${\rm SL}_2$-type relations \eqref{R2}.

A similar description of the presentation of $k[G]$ may be given for
the classical groups $G$
 of several other types:\;for them, the Pl\"ucker-type (respectively, the ${\rm SL}_2$-type) relations are obtained
 using \cite{LT79}, \cite{LT85} (respectively,
\cite{FT92}).

redpajama_set_name: RedPajamaArXiv


text:
Coronavirus: John Magufuli declares Tanzania free of Covid-19
duration 8 June 2020
image copyrightAFP
image captionJohn Magufuli praised worshippers for not wearing masks and gloves
Tanzania's President John Magufuli has declared the country "coronavirus-free" thanks to prayers by citizens.
"The corona disease has been eliminated thanks to God," Mr Magufuli told worshippers in a church in the capital, Dodoma.
The World Health Organization (WHO) has expressed concern over the government's strategy on Covid-19.
The government has stopped publishing data on the number of coronavirus cases in the country.
On 29 April, the last day official data was released, there were 509 cases, with 21 deaths in Tanzania. However, Mr Magufuli said last week that only four patients were receiving treatment in the largest city, Dar es Salaam.
Last month, Tanzania's government dismissed a US embassy warning that hospitals in Dar es Salaam were "overwhelmed" and that the chances of contracting the virus was "extremely high".
Mr Magufuli has repeatedly said the health crisis has been exaggerated and urged people to attend services in churches and mosques, saying that prayers "can vanquish" the virus.
Africa Live: Updates on this and other stories from the continent
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Why lockdowns may not be the answer in Africa
What did Magufuli say?
"I want to thank Tanzanians of all faiths. We have been praying and fasting for God to save us from the pandemic that has afflicted our country and the world. But God has answered us.
"I believe, and I'm certain that many Tanzanians believe, that the corona disease has been eliminated by God," he told worshippers on Sunday in Dodoma.
His remarks were welcomed with applause and celebratory ululation.
Mr Magufuli also commended the priest and worshippers for not wearing gloves and masks to protect themselves from the virus.
What else has he said?
He made similar comments last Friday during a teachers' conference, declaring "corona is finished".
"The health minister told me that we only have four patients in Dar es Salaam but this will not stop rumours to the contrary," he said.
"We need to be careful because some of these donations to fight coronavirus could be used to transmit the virus.
"I want to urge you Tanzanians not to accept donations of masks, instead tell the donors to go and use them with their wives and children," he added.
media captionCoronavirus: How Covid-19 is affecting remittances to Africa
The Tanzanian president is among several African leaders who have received orders for a herbal tonic touted by the president of Madagascar as a cure for Covid-19. It is unclear if it has been given to patients in Tanzania.
There is no scientific proof that the tonic works and the WHO has urged people not to use unproven medication.
Opposition politicians have criticised Mr Magufuli in the past for promoting policies that could harm the health of Tanzanians.
The head of Tanzania's medical association, Elisha Osati, backed the government in a recent BBC interview, saying that hospitals were operating as normal and that the majority of coronavirus inpatients were suffering from mild symptoms.
Mr Magufuli has accused the country's health officials of exaggerating the coronavirus crisis. He has also ridiculed the strict measures neighbouring countries have imposed to fight the pandemic.
A SIMPLE GUIDE: How do I protect myself?
HOPE AND LOSS: Your coronavirus stories
FACE MASKS: Should I be wearing one?
HOW A VIRUS SPREADS: An explanation
John Magufuli
Coronavirus: The 'Wuhan shake' or the elbow bump?
A quick guide to Tanzania
Africa Today podcasts
duration1 hour ago
All 50 US states on alert for armed protests10

redpajama_set_name: RedPajamaCommonCrawl


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JCU in the News
Contact a JCU Expert
Featured News Tough laws prevent gun deaths
Thu, 18 Oct 2018
Image: Kony-Xyzx
A major global report confirms gun-related homicides, suicides and accidents are falling in Australia after the introduction of anti-gun laws, and that the effect of such tough laws is similar elsewhere.
James Cook University Associate Professor Richard Franklin joined hundreds of academics and organisations around the world to produce the report, that looked at firearm deaths outside of war zones between 1990 and 2016.
"In Australia we went from 614 firearm deaths in 1990 to 274 in 2016. That's a fall from 3.4 deaths per 100,000 people to 1 per 100,000 in 2016," he said.
Dr Franklin said the Australian National Firearms Agreement, enacted following the murder of 35 people in Tasmania in 1996 by a lone gunman armed with high-powered weapons, has been closely linked with declines in firearm deaths in Australia.
"We've seen a decline particularly in firearm suicides and an absence of mass shootings. It's a pattern we see in South Africa and Brazil, with tougher gun laws leading to a fall in firearm deaths in those places too," he said.
The researchers estimate that in 2016, over a quarter of a million-people died from firearm injuries. With six countries – Brazil, the United States, Mexico, Colombia, Venezuela and Guatemala - accounting for just over half of those deaths.
El Salvador recorded the worst results – its national age-standardised rate of gun deaths was 38.9 per 100,000 persons in 2016. By contrast, Singapore reported no firearms deaths that year.
Dr Franklin said that in all but one year between 1990 and 2016, the death toll from firearms used outside of war zones was higher than that within, with the majority being homicides.
"The numbers vary greatly from country to country, but globally we have seen no significant change in firearm homicide rates over this time period, though there has been a decrease in suicides, leading to a small fall in the number of firearm deaths overall."
Dr Franklin said men are at an overwhelmingly higher risk of dying from intentional or unintentional use of firearms than women.
"Men are the most likely victims of firearms and also the most likely perpetrators of firearm violence. This opens up the possibility of specific, targeted forms of intervention that address firearm use both by and against men," he said.
Dr Franklin said there were different patterns of firearm deaths, with homicide predominant in some countries and suicide in others, and both would require different strategies to combat them.
Link to paper here.
Globally, 64% of firearm injury deaths in 2016 were homicides and 27% were suicides. Around nine percent were accidents.
There was an annualised decrease of 0.9% in the global rate of age-standardised firearm deaths from 1990 to 2016.
Aggregate firearm injury deaths in 2016 were highest among persons aged 20 -24 years. There were an estimated 34,700 firearm-related deaths for men and 3580 for women in that age group.
Among children aged up to 14 years, there were an estimated 7220 deaths globally from firearm-related injuries in 2016.
Unintentional firearm injuries were 9.1% of the total, causing an estimated 22,900 deaths in 2016.
Globally, rates of firearm suicide decreased between 1990 and 2016 at an annualised rate of 1.6%, the fastest decreases were in the Philippines and Australia.
The researchers say that efforts to reduce the number of firearms in homes and supporting secure storage of existing firearms can reduce unintentional death, particularly for children, while also limiting immediate access to a means of harm that generally does not allow an opportunity for second thoughts.
Associate Professor Richard Franklin
E: Richard.franklin@jcu.edu.au


Hideo Kojima is a Japanese video game designer, screenwriter, director and game producer. Regarded as an auteur of video games, he founded Kojima Productions in 2005, after starting at Konami Digital Entertainment where he rose to vice president in 2011. In late 2015, Kojima became an independent game designer.
Scott Cawthon is an independent American video game designer, developer, animator, and writer, best known as the creator of the Five Nights at Freddy's game franchise. Cawthon has also created other games, such as Chipper & Son's Lumber Co., The Desolate Hope, and There is No Pause Button!, as well as Christian-based animations such as A Christmas Journey and The Pilgrim's Progress. He used Autodesk 3ds Max and Clickteam Fusion 2.5 to develop Five Nights at Freddy's and most of his other games.
Ernest Gary Gygax was an American game designer and author best known for co-creating the pioneering role-playing game Dungeons & Dragons (D&D) with Dave Arneson.
Theodore Robert Beale, professionally known as Vox Day, is an American writer, editor, video game designer, blogger and alt-right activist.
Peter Douglas Molyneux is an English video game designer and programmer. He created the god games Populous, Dungeon Keeper and Black & White, as well as Theme Park, the Fable series, Curiosity – What's Inside the Cube?, and Godus. He currently works at 22Cans as the founder.
William Ralph "Will" Wright is an American video game designer and co-founder of the former game development company Maxis, and then part of Electronic Arts (EA). In April 2009, he left EA to run Stupid Fun Club Camp, an entertainment think tank in which Wright and EA are principal shareholders.
Alexey Leonidovich Pajitnov is a Russian video game designer and computer engineer who developed Tetris while working for the Dorodnitsyn Computing Centre of the Soviet Academy of Sciences, a Soviet government-founded R&D center.
Jonathan Blow is an American video game designer and programmer, who is best known as the creator of the independent video games Braid (2008) and The Witness (2016), both of which were released to critical acclaim.
Richard Channing Garfield is an American mathematician, inventor, and game designer. Garfield created Magic: The Gathering, which is considered to be the first modern collectible card game (CCG). Magic debuted in 1993 and its success spawned many imitations. Garfield oversaw the successful growth of Magic and followed it with other game designs. Included in these are Netrunner, BattleTech(CCG), Vampire: The Eternal Struggle, Star Wars Trading Card Game, The Great Dalmuti, Artifact, and the board game RoboRally. He also created a variation of the card game Hearts called Complex Hearts. Garfield first became passionate about games when he played the roleplaying game Dungeons & Dragons, so he designed Magic decks to be customizable like roleplaying characters. Garfield and Magic are both in the Adventure Gaming Hall of Fame.
Philippe Poisson, better known as Phil Fish, is a French-Canadian former indie game designer best known for his work on the 2012 platform game Fez published by his company, Polytron Corporation. He was born and raised in Quebec, where his experiences with Nintendo games in his youth would later influence his game design. He studied game design at the Montreal National Animation and Design Centre, and worked at Ubisoft and Artificial Mind and Movement before starting Polytron in 2008.
Gunpei Yokoi , sometimes transliterated Gumpei Yokoi, was a Japanese video game designer. He was a long-time Nintendo employee, best known as creator of the Game & Watch handheld system, inventor of the Control Pad, the original designer of the Game Boy, and producer of a few long-running and critically acclaimed video game franchises, such as Metroid and Kid Icarus.
Chris Roberts is an American video game designer, programmer, film producer and film director. He created the Wing Commander series while at Origin Systems and is now working on the crowdfunded space simulator Star Citizen.
IceFrog is a pseudonymous video game designer, known for being the longest-serving designer of Defense of the Ancients (DotA), a custom Warcraft III: Reign of Chaos mod - and as the lead designer for the stand-alone sequel, Dota 2.
David Lance "Dave" Arneson was an American game designer best known for co-developing the first published role-playing game (RPG), Dungeons & Dragons, with Gary Gygax, in the early 1970s. Arneson's early work was fundamental to the development of the genre, developing the concept of the RPG using devices now considered to be archetypical, such as adventuring in "dungeons" and using a neutral judge who doubles as the voice and consciousness of all other characters to develop the storyline.


Mr. Waddington with co-counsel celebrating our acquittal with our client, her family, and her supporters.
Our client was charged with using cocaine on several occasions and with lying on a Top Secret security clearance form. She faced 10 years in prison, a Federal Felony Drug Conviction, and a Dismissal. The prosecution had a 2-hour video of her alleged confession to OSI and a supposed handwritten OSI "confession." OSI is shady, so we had to attack their investigation and show their bias and incompetence.
OSI then dug up dozens of her friends and was able to bully a few officers into testifying against her. The prosecution refused any type of plea deal as well as a resignation with an Other Than Honorable Discharge. They forced us to trial. The Government was so confident that they would win that the SJA was talking trash to Mr. Waddington before the trial.
Left with no options and ready to fight, Mr. Waddington and his ADC co-counsel, Capt Stuart Saylor, took the fight to the Government and fought all charges in front of an officer jury. At the end of the trial, Mr. Waddington and Capt Saylor went out for drinks, our client and her father included.
RESULT: NOT GUILTY OF ALL CHARGES. Click here to read what our client had to say about our representation.


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redpajama_set_name: RedPajamaGithub


The Mainstream Universe is the shared fictional universe where most comic book titles published by Mainstream Comics take place, including those featuring Mainstream's most popular titles, such as Cyberknight, Renegades, Flare, Nightveil and WestGuard.
The Mainstream Universe is further depicted as existing within a "multiverse" consisting of a number of separate universes (sometimes referred to as "realms", or "planes"), all of which are the creations of Mainstream Comics and all of which are, in a sense, "Mainstream universes". In this context, "Mainstream Universe" is taken to refer to the Mainstream continuity in which Earth exists (also called the "Earth realm").
This page was last modified on 18 June 2011, at 10:08. Content is available under Attribution-Share Alike 3.0 Unported unless otherwise noted.


text:
Origin of Symmetry
1 New Born 6:03
2 Bliss 4:12
3 Space Dementia 6:20
4 Hyper Music 3:20
5 Plug In Baby 3:40
6 Citizen Erased 7:19
7 Micro Cuts 3:38
8 Screenager 4:20
9 Darkshines 4:47
10 Feeling Good 3:19
11 Megalomania 4:40
Mushroom, 2001
In a nutshell, Origin of Symmetry details the surrealism of human isolation in a society powerless to stop the increasing dominance of technology in our lives. I'd praise its originality, if it were not for the fact that this record is not called OK Computer and the band is not called Radiohead. Not that Muse don't try, though: the sole strength of their 1999 debut Showbiz was how similar it was to old-skool guitars-and-drums (with the occasional keyboard) Radiohead, while their pseudo-mentors were busy alienating their fan base with the minimalist electronic adventures of Kid A and Amnesiac.
Vocalist Matt Bellamy sounds like Thom Yorke crossed with Jeff Buckley, right down to the impromptu falsettos and the stratospheric Freddie Mercury-esque pitch, while drummer Dominic Howard and bassist Chris Wolstenholme pound out rhythms at satisfactory decibel levels. Nothing really distinguishing the two, then, save for the fact that Radiohead had constructed two near-flawless records back to back, while all Muse had to their name was a less-than-perfect carbon copy of The Bends. So, upon its 2001 release, the best Origin of Symmetry could have hoped for was to help its audience forget that the band they'd rather be listening to instead had forsaken them.
That pre-conceived notion is the best thing about hearing the album for the first time. For it is in those first few moments that the band forever sheds the spectre of Radiohead and takes to the stars. Witness the opener New Born: the tinkling piano giving way to a chainsaw guitar so relentlessly catchy, it begs to be rewound repeatedly, just to relive that virgin adrenaline rush. It's one of the strongest first cuts I've heard in a while, superceding both 'Smells like Teen Spirit' and My Bloody Valentine's 'Only Shallow' in one fell swoop by combining the punk-guitar ferocity of the former, while maintaining the art-rock sensibility of the latter. The grandstanding continues on tracks like the magnificent Bliss, its more traditional guitars and less traditional electronic arpeggios making it a thoroughly compelling listen and richly earning its status as a mainstream hit.
Bellamy's budding keyboard skills, only fleetingly exhibited on Showbiz, are more fully fleshed out here; his Rachmaninoff-on-acid sound blends seamlessly with the harsh guitar wash of their now-trademark electro-rock style. The pitch of Bellamy's majestic voice can reach amplitudes Thom Yorke could never dream of, and he makes no effort to hide them in the (just barely) radio-ready singles 'Feeling Good' and 'Plug In Baby'. The clincher, however, is the sprawling, multi-segmented Citizen Erased: it's astonishing how Muse's combination of crashing guitars, vocal theatrics and the odd keyboard arrangement can make a seven-minute epic flit by like a three-minute pop jingle.
Though the baroque melodrama of 'Microcuts' almost steals the show with its deathrock-opera posturing and pounding thrash-punk conclusion, it should be noted that the record is not composed entirely of self-indulgent grandiosity. Perfectly positioned after the chilling 'Space Dementia' is 'Hyper Music', resplendent in its hi-octane, overdrive-pedal driven simplicity. Tucked away near the end is 'Dark Shines', the goth/surf guitar hybrid mixing repressed lust behind its melodic despair If you're still looking for the faintest hint of optimism here, you can stop reading; even a rendition of Nina Simone's cheery 'Feeling Good' is given a Portishead/Suede-esque subversively sexy melancholic polish.
The organ-heavy 'Megalomania' closes the album, giving it the hallmarks of a record we'd normally love to hate: laughably over-the-top, relentlessly histrionic angst from bored English country boys. But they've made a masterpiece out of something that history may have looked upon as an embarrassing sojourn into arty pretension. True, it's overblown, emotional and appallingly depressing. But you need to own it, unless you prefer not to writhe on the floor in orgasmic ecstasy to its devastating soundscapes, in which case I regret to inform you that you were never alive in the first place.
(First written in November 2004 for the New Age newspaper's entertainment section. Later published on Epinions on February 24, 2005).


The first Alliance missionaries entered Africa in 1884, serving in the Belgian Congo (now the Democratic Republic of the Congo). By 1892, nearly 30 Alliance missionaries were sharing the good news of Jesus throughout the continent.
Today, The Alliance has national churches in nine African countries, with an inclusive membership of more than 1 million believers. During the past decade, more than 400,000 people have come to faith in Christ through Alliance ministries in Africa. The advances experienced across Alliance fields testify to God's grace and sovereignty. God has used Alliance workers and fellow Africans as bridges to His redemption. The coming years will reveal shifting roles between the Alliance mission and local churches. As African churches become increasingly significant bearers of the gospel, reaching ethnic groups that have traditionally remained resistant, we will see Africa claimed for the gospel and for Jesus Christ.
Alliance ministry in Africa is primarily funded through the Great Commission Fund. Help fulfill Jesus' Great Commission and make a gift to the GCF today.


Does Waxman-Markey have emergency Presidential powers?
The Obama administration and the Democrats have begun building a nasty reputation for hiding draconian elements of legislation, from "comparative effectiveness" rationing to potential Internet shut-downs. Senator David Vitter says he's found another hidden within the Waxman-Markey bill that has already passed the House that would give Barack Obama the power to declare a national emergency on the basis of greenhouse-gas levels. If true, it would undermine the entire notion of a cap-and-trade system — and give the President dictatorial powers over energy production and manufacturing. But is it true?
Here's how: The bills require a federal declaration of a "climate emergency" if world greenhouse gas levels reach 450 parts per million. Guess what? The Pacific Northwest National Lab says it is a virtual certainty that level will be reached within a few months. The bill then requires the president to "direct all Federal agencies to use existing statutory authority to take appropriate actions…to address shortfalls" in achieving needed greenhouse gas reductions.
When Vitter asked EPA Administrator what would be done in such a situation, she refused to say. So it must be asked: Would the president be empowered to do things like nationalize whole sectors of industry, ban coal use, restrict private automobile use, or whatever else the "emergency" requires?
The Examiner's David Freddoso reports that Sen. David Vitter, R-LA, is holding a news conference later today concerning this provision. Vitter wonders if companies that support cap-and-trade in the hope they will profit from going green realize what could happen to them soon after enactment. More to the point, we wonder what the American people will do when they realize what is actually going on here.
The result is a a scenario in which the law not only permits but in fact requires the president to "direct all Federal agencies to use existing statutory authority to take appropriate actions…to address shortfalls" in emissions cutbacks.
The bill's language places an unusually broad mandate upon the president to act in the event of this "emergency" situation." In a letter to Vitter, EPA administration Lisa Jackson wrote that she does not know what her agency would do. "It is premature to describe exactly what additional actions EPA may take until such an analysis is conducted," she wrote.
But declaration of this "climate emergency" could result in federal agencies denying all discretionary permits for carbon-emitting industries, and EPA itself could impose non-attainment status upon the entire United States. "In that context, the carbon credits won't matter," Vitter told me yesterday.
If a future president tries to go easy on industry, environmental groups are sure to litigate based on the clear language in the law, forcing his hand.
"This provision was not focused on to any significant extent during the House debate," said Vitter. The climate change bill passed the House this summer, hours after hundreds of pages of amendments were added to it. Vitter said he plans to write industry leaders who are supporting the climate bill to ask whether they understand what the bill's language would do.
The one clearly objectionable part of this is the actual trigger. It hysterically assumes a disaster at an arbitrary level of greenhouse gases, all of which naturally occur in our atmosphere. It also assumes that federal control of manufacturing and energy production would cause less disruption and damage than a level of 450 parts per million of carbon dioxide and other gases. Where is the evidence of this cataclysmic occurrence? How did this level get chosen? As far as can be seen, it appears to have been selected simply because we will hit that level within a few months.
If what Vitter claims is true, it would have implications of creating a dictator out of the American presidency. Democrats used to shriek about the "unitary executive" philosophy during the Bush administration, which they poorly understood anyway, but claimed that Bush wanted to make himself into a dictator outside of the control of Congress. Have those same Democrats have rushed to give the White House vast, unchecked powers over private industry and private capital that might even make Hugo Chavez blush?
'SEC. 707. PRESIDENTIAL RESPONSE AND RECOMMENDATIONS.
(a) AGENCY ACTIONS.—The President shall direct relevant Federal agencies to use existing statutory authority to take appropriate actions identified in the reports submitted under sections 705 and 706, and to address any shortfalls identified in such reports, not later than July 1, 2015, and every 4 years thereafter.
(b) PLAN.—In the event that the Administrator or the National Academy of Sciences has concluded, in the most recent report submitted under section 705 or 706 respectively, that the United States will not achieve the necessary domestic greenhouse gas emissions reductions, or that global actions will not maintain safe global average surface temperature and atmospheric greenhouse gas concentration thresholds, the President shall, not later than July 1, 2015, and every 4 years thereafter, submit to Congress a plan identifying domestic and international actions that will achieve necessary additional greenhouse gas reductions, including any recommendations for legislative action.


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FUNimation Titles Distributed by Anime Limited Removed from Retailers or Listed as "Out of Stock"
In some rather interesting, and extremely surprising news, it appears that FUNimation titles distributed by Anime Limited, otherwise known as All The Anime, have been either listed as sold out or removed from online retailers; that is according to a report on Anime UK News and as pointed out by community members on the website. Read more of this post
Filed under Anime Limited, Anime News, FUNimation UK, News Tagged with Absolute Duo, All The Anime, Anime Limited, Assassination Classroom, Attack on Titan: Junior High, Barakamon, Blazblue: Alter Memory, Blood Blockade Battlefront, Brothers Conflict, Claymore, Daimidaler: Prince Vs. Penguin Empire, Eden of the East, Fairy Tail, Free! - Eternal Summer, FUNimation, Gangsta, Gonna Be the Twin-Tail, Heavy Object, High School DxD BorN, Hyouka, Lord Marksman and Vanadis, Lucky Star, Mikagura School Suite, News, No-Rin, Noein, Noragami Aragoto, Ouran High School Host Club, Overlord, Prison School, Psycho-Pass, Rage of Bahamut: Genesis, Selector Infected WIXOSS, SHIMONETA, Shomin Sample, Show By Rock, Snow white with the Red Hair, Tokyo ESP, Yona of the Dawn, Yurikuma Arashi
Unboxing: Noein: To Your Other Self – The Complete Series (Collector's Edition)
After a relatively short wait Noein has arrived on Blu-ray; but what can you expect from this Collectors Edition Blu-ray release of the series? Well let's take a look in our latest unboxing. Read more of this post
Filed under Unboxed Tagged with Anime Limited, Blu-ray, Collectors Edition, Noein, Noein: To Your Other Self, The Complete Series, Unboxed, Unboxing
Noein: To Your Other Self – The Complete Series (Blu-ray / DVD Combo Pack) – REGION FREE
Neoin: To Your Other Self – The Complete Series (Blu-ray / DVD Combo Pack)
BD Region: A & B (Confirmed Playable in the UK)
– Boxart states Region B Compatible
DVD Region: 1, 2 & 4 (Confirmed Playable in the UK)
– Boxart states Region 2 Compatible
Price: $43.79/ £30.69 (Excluding postage)
Episodes: 1 – 24 (3 Blu-ray's & 4 DVD's)
Plot: When summer break begins, Haruka is only thinking of hanging out with her friends – until one fateful night she learns that she's the keystone in an interdimensional battle for not only the Earth's future, but the very survival of all existence across the multiverse. She will have to trust not only herself, but multiple versions of her friends from parallel dimensions.
One of these visitors identifies himself as the future version of her troubled childhood friend Yuu. Karasu swears to protect her from the god-like Noein who wants to use her to collapse all reality. When Haruka learns she has the ability to not only see, but influence every possible future, will she be able to protect infinite worlds along with her own?
—-NOTE—-
Anime Limited will be releasing Noein: To Your Other Self as a Collectors Edition Blu-ray release later this year; and we assume (judging that this US release is UK compatible) that it will be the same content as this US release.
Filed under Region Free Tagged with Blu-ray / DVD Combo Pack, FUNimation, Noein, Noein: To Your Other Self, Region Free, The Complete Series
Anime Limited Confirms Delay's to Mobile Suit Gundam – Part 2, Noein and Fuse
As part of it's weekly newswire Anime Limited have revealed that a handful of their Q1 2016 releases have received minor delays; with both Mobile Suit Gundam – Part 2 and Noein – The Complete Series collection reciving the biggest set-backs.
To start with Mobile Suit Gundam – Part 2, which was previously confirmed for a 25th January 2016 release after it's Q4 2015 was delayed, will now be released on the 8th February 2016. The reason for the delay is due to production of the bonus booklet, which is only limited to 1000 units, taking longer than expected and being made out of a stronger material than previously designed. Read more of this post
Filed under Anime Limited, News Tagged with Alterations, Anime Limited, Delay, Fuse, Fuse - Memoirs of the Hunter Girl, Fuse: Memoirs of a huntress, Kill La Kill, Miss Hokusai, Mobile Suit Gundam, News, Noein, Release Date, Sword Art Online II
Anime Limited Confirm Amendments to its Release Schedule
As part of their weekly update Anime Limited have confirmed a selection of amendments to its release schedule; mainly those involving Sword Art Online II, Mobile Suit Gundam – Part 2 and Samurai Flamenco – Part 1, all three of which had had their release dates altered on various websites over the past few days.
As of now Sword Online Art II – Part 2 will be released during december, with its third installment released sometime in the new year, meanwhile Samurai Flamenco – Part 1 will also be released during december. It's also been confirmed that Mobile Suit Gundam – Part 2, which was scheduled for a late Q4 2015 release, has now been pushed back to nex year. Read more of this post
Filed under Anime Limited, News Tagged with Anime Limited, Blu-ray, Delay, Durarara!! x2 Shou, DVD, Mobile Suit Gundam, News, Noein, Ping Pong, Release Date, Samurai Flamenco, Selector Infected WIXOSS, Sword Art Online II, Tokyo ESP
Amazon UK List's Anime Limited's Q1 Release Schedule; Reveals Some Surprising New Licenses
It seems like online retailer Amazon UK may have 'jumped-the-gun' on posting Anime Limited's Q1 (January to March) 2016 pre-order listings onto their store as it reveals a seleciton of titles that have yet to be officially confirmed by the UK Distributor. Read more of this post
Filed under Anime Limited, News Tagged with All The Anime, Amazon UK, Anime Limited, Beck, Blu-ray, Brothers Conflict, Collectors Edition, Durarara!! x2 Shou, DVD, Expelled form Paradise, Listings, Lord Marksman and Vanadis, Miss Hokusai, News, Noein, Ping Pong, Ping Pong: The Animation, Pre-Order, Q1 2016, Release Schedule, Riddle Story of Devil, Selector Infected WIXOSS, Shinkai Twin Pack, The Place Promised in our Early Days, Tokyo ESP, Voices of a Distant Star