text:
\section{Introduction}

In order to approach the fundamental problem of classifying the irreducible representations of a given finite-dimensional algebra, the concept of cellularity, defined by Graham and Lehrer \cite{GL}, has proven extremely useful. A cellular algebra comes, by definition, with a finite chain of ideals, whose subquotients, denoted by cells, decompose as left modules into a direct sum of copies of a certain module, called the cell module. This cell module comes with a bilinear form. Factoring out the radical with respect to this form leads to a simple module or zero, and in this way, one obtains a complete set of isomorphism classes of simple modules for the given cellular algebra. Examples of cellular algebras include many finite-dimensional Hecke algebras \cite{geck5}.

Recently, Koenig and Xi \cite{KX} have generalised this concept to algebras over a principal ideal domain $k$ of not necessarily finite dimension, by introducing the notion of an affine cellular algebra. Keeping the idea of having a filtration by a finite chain of ideals, the ideals are now allowed to be of infinite dimension. Where before a cell was isomorphic to a matrix ring over $k$ with a twisted multiplication, it is now isomorphic to a matrix ring (still of finite rank) over a quotient of a polynomial ring over $k$, with a twisted multiplication. The most important class of examples in \cite{KX} of affine cellular algebras is given by the extended affine Hecke algebras of type $A$.

In this article, we prove the following theorem, providing the first examples of affine cellularity for affine Hecke algebras with unequal parameters.

\begin{Th}\label{mainthmintro}
Let $\mathcal{H}$ be an affine Hecke algebra of rank two defined over $\mathbb{C}[v,v^{-1}]$ with generic parameters. Then $\mathcal{H}$ is affine cellular in the sense of Koenig and Xi.\end{Th}

Theorem \ref{mainthmintro} is proved by explicit construction of the associated twisted matrix rings and the isomorphism between these and the corresponding cells.

\section{Affine cellular algebras} \label{affcell}

Let $k$ be a principal ideal domain. For a $k$-algebra $A$, a $k$-linear anti-automorphism $i$ of $A$ satisfying $i^2 = id_A$ is called a $k$-involution on $A$. For two $k$-modules $V,W$ denote by $\sigma$ the map $V \otimes_k W \rightarrow W \otimes_k V$ given by $\sigma(v \otimes w) = w \otimes v$. If $B=k[t_1, \dots, t_r]/I$ for some ideal $I$ in a polynomial ring in finitely many variables over $K$, then $B$ is called an affine $k$-algebra.
\begin{Def}\cite[Definition 2.1]{KX}
Let $A$ be a unitary $k$-algebra with a $k$-involution $i$. A two-sided ideal $J$ in $A$ is called an \emph{affine cell ideal} if and only if the following two conditions are satisfied.
\begin{enumerate}
\item We have $i(J) = J$.
\item There exists an affine $k$ algebra $B$ with a $k$-involution $\nu$, and a free $k$-module $V$ of finite rank such that $\Delta:=V \otimes_k B$ is an $A$-$B$-bimodule, where the right $B$-structure is induced by the regular $B$-module $B_B$.
\item There is an $A$-$A$-bimodule isomorphism $\alpha: J \rightarrow \Delta \otimes_B\Delta'$, where $\Delta' = B \otimes_k V$ is the $B$-$A$-bimodule with left $B$-structure induced by ${}_BB$ and right $A$ structure defined by $(b\otimes v)a = \sigma(i(a)(v \otimes b))$, such that the following diagram commutes:

$$\xymatrix{ J \ar^{\alpha}[rr]  \ar^{i}[d]&& \Delta \otimes_B\Delta' \ar^{v \otimes b \otimes_B b' \otimes w \mapsto w \otimes \nu(b') \otimes_B \nu(b) \otimes v }[d] \\ J \ar^{\alpha}[rr]&&\Delta \otimes_B\Delta'.}$$
\end{enumerate}

The algebra $A$ together with its $k$-involution $i$ is called \emph{affine cellular} if and only if there is a $k$-module decomposition $A= J_1' \oplus J_2' \oplus \cdots \oplus J_n'$ for some $n$ with $i(J_l')=J_l$ for $1 \leq l \leq n$, such that, setting $J_m:= \bigoplus_{l=1}^m J_l'$, we obtain a filtration
$$0=J_0 \subset J_1 \subset J_2 \subset \cdots \subset J_n=A$$
of $A$ by two-sided ideals, where each $J_m' = J_m/J_{m-1}$ is an affine cell ideal of $A/J_{m-1}$ (with respect to the involution induced by $i$ on the quotient).
\end{Def}

For an affine $k$-algebra $B$ with a $k$-involution $\nu$, a free $k$-module $V$ of finite rank and a $k$-bilinear form $\varphi:V\times V \rightarrow B$, denote by $\mathbb{A}(V,B,\varphi)$ the (possibly non-unital) algebra given as a $k$-module by $V\otimes_k B \otimes_k V$, on which we impose the multiplication $(v_1 \otimes b_1\otimes w_1)(v_2 \otimes b_2 \otimes w_2) := v_1 \otimes b_1 \varphi(w_1,v_2) b_2 \otimes w_2$.

The description of affine cell ideal we are going to use is the following:

\begin{Prop}\cite[Proposition 2.3]{KX} \label{descrip}
Let $k$ be a principal ideal domain, $A$ a unitary $k$-algebra with a $k$-involution $i$. A two-sided ideal $J$ in $A$ is an affine cell ideal if and only if there exists an affine $k$-algebra $B$ with a $k$-involution $\nu$, a free $k$-module $V$ of finite rank and a bilinear form $\varphi: V \otimes V \rightarrow B$, and an $A$-$A$-bimodule structure on $V \otimes_k B\otimes_kV$, such that $J \cong \mathbb{A}(V,B,\varphi)$ as an algebra and an $A$-$A$-bimodule, and such that under this isomorphism the $k$-involution $i$ restricted to $J$ corresponds to the $k$-involution given by $v \otimes b\otimes w \mapsto w \otimes \nu(b) \otimes v$.
\end{Prop}

Let now $A$ be an affine cellular algebra with a cell chain $0=J_0 \subset J_1 \subset J_2 \subset \cdots \subset J_n=A$, such that each subquotient $J_i/J_{i-1}$ is an affine cell ideal in $A/J_{i-1}$. Then  $J_i/J_{i-1}$ is isomorphic to $\mathbb{A}(V_i,B_i,\varphi_i)$ for some finite-dimensional vector space $V_i$, a commutative $k$-algebra $B_i$ and a bilinear form $\varphi_i: V_i \times V_i \rightarrow B_i$. Let $(\phi^i_{st})$ be the matrix representing the bilinear form $\phi$ with respect to some choice of basis of $V_i$.
Then Koenig and Xi obtain a parameterisation of simple modules over an affine cellular algebra by establishing a bijection between isomorphism classes of  simple $A$-modules and the set
$$\{ (j, \mathrm{m}) \mid 1 \leq j\leq n,\mathrm{m} \in \mathrm{MaxSpec}(B_j) \textrm{ such that some } \phi^j_{st} \not\in \mathrm{m}\}$$
where $\mathrm{MaxSpec}(B_j)$ denotes the maximal ideal spectrum of $B_j$.
Furthermore, assume that $J_i/J_{i-1}$ is an idempotent ideal in $A/J_{i-1}$ (meaning $(J_i/J_{i-1})^2 = J_i/J_{i-1}$),  that it contains an idempotent in $A/J_{i-1}$ and that the radical of every $B_j$  is zero. Then,  Koenig and Xi show that $A$ has finite global dimension if and only if every $B_i$ has finite global dimension.

\section{Hecke algebras and Kazhdan-Lusztig cells}
In this section $(W,S)$ denotes an arbitrary Coxeter system (with $|S|<\infty$) together with a positive weight function $L$.  A positive weight function is a function $L:W\rightarrow\mathbb{N}$ such that $L(ww')=L(w)+L(w')$ whenever $\ell(ww')=\ell(w)+\ell(w')$ where $\ell$ denotes the usual length function on $W$. The main reference is \cite{bible}.

\subsection{Hecke algebras and Kazhdan-Lusztig basis}
Let $\mathcal{A}=\mathbb{C}[v,v^{-1}]$ where $v$ is an indeterminate. Let $\mathcal{H}$ be the Iwahori-Hecke algebra associated to $W$, with $\mathcal{A}$-basis  $\{T_{w}|w\in W\}$ and multiplication rule given by
\begin{equation*}
T_{s}T_{w}=
\begin{cases}
T_{sw}, & \mbox{if } \ell(sw)>\ell(w),\\
T_{sw}+(v^{L(s)}-v^{-L(s)})T_{w}, &\mbox{if } \ell(sw)<\ell(w),
\end{cases}
\end{equation*}
for all $s\in S$ and $w\in W$. Let $\bar\ $ be the ring involution of $\mathcal{A}$ which takes $v$ to $v^{-1}$. It can be extended to a ring involution of $\mathcal{H}$ via
$$\overline{\sum_{w\in W} a_{w}T_{w}}=\sum_{w\in W} \bar{a}_{w}T_{w^{-1}}^{-1}\quad (a_{w}\in\mathcal{A}).$$
We set
$$\begin{array}{ccccccc}
\mathcal{A}_{<0}&=v^{-1}\mathbb{Z}[v^{-1}] &\text{ and }& \mathcal{H}_{<0}=\bigoplus_{w\in W} \mathcal{A}_{<0}T_{w}.
\end{array}$$
For each $w\in W$ there exists a unique element $C_{w}\in\mathcal{H}$ (see \cite[Theorem 5.2]{bible}) such that
\begin{enumerate}
\item $\bar{C}_{w}=C_{w}$
\item $C_{w}\equiv T_{w} \mod \mathcal{H}_{<0}$.
\end{enumerate}
For any $w\in W$ we set
$$C_{w}=T_{w}+\sum_{y\in W } P_{y,w} T_{y}\quad \text{where $P_{y,w}\in \mathcal{A}_{< 0}$}.$$
It is well known (\cite[\S 5.3]{bible}) that $P_{y,w}=0$ whenever $y\nleq w$  (here $\leq$ denotes the Bruhat order). It follows that $\{C_{w}|w\in W\}$ forms an $\mathcal{A}$-basis of $\mathcal{H}$ (the ``Kazhdan-Lusztig basis''). The coefficients $P_{y,w}$ are known as the Kazhdan-Lusztig polynomials.

\begin{Def}\label{flat}
Following Lusztig \cite[\S 3.4]{bible}, there exists a unique involutive antiautomorphism, i.e. an $\mathcal{A}$-involution, $\flat:\mathcal{H}\longrightarrow\mathcal{H}$ which carries $T_{w}$ to $T_{w^{-1}}$.
\end{Def}

\begin{Rem}\label{flataction}
Using this map, we obtain right handed version of the multiplication of $\mathcal{H}$:
\begin{equation*}
T_{w}T_{s}=
\begin{cases}
T_{ws}, & \mbox{if } \ell(ws)>\ell(w),\\
T_{ws}+(v^{L(s)}-v^{-L(s)})T_{w}, &\mbox{if } \ell(ws)<\ell(w).
\end{cases}
\end{equation*}
Further, since $\flat$ sends $\mathcal{H}_{<0}$ to itself it can be shown that \cite[\S 5.6]{bible} that $C_{w}^{\flat}=C_{w^{-1}}$, from where it follows that
$$P_{y,w}=P_{y^{-1},w^{-1}}.$$
\end{Rem}

\subsection{Kazhdan-Lusztig cells}
We denote by $h_{x,y,z}$ the structure constant with respect to the Kazhdan-Lusztig basis. That is,  we set

$$C_{x}C_{y}=\sum_{z\in W} h_{x,y,z}C_{z}.$$

Note that $\bar{h}_{x,y,z}=h_{x,y,z}$ and, similarly to Remark \ref{flataction}, we have  $h_{x,y,z}=h_{y^{-1},x^{-1},z^{-1}}$. We write $z\leftarrow_{\mathcal{L}} y$ if there exists some $s\in S$ such that $h_{s,y,z}\neq 0$, that is $C_{z}$ appears with a non-zero coefficient in the expression of $C_{s}C_{y}$ in the Kazhdan-Lusztig basis. The Kazhdan-Lusztig left pre-order $\leq_{\mathcal{L}}$ on $W$ is the transitive closure of this relation.
The equivalence relation associated to $\leq_{\mathcal{L}}$ will be denoted by $\sim_{\mathcal{L}}$, that is

$$x\sim_{\mathcal{L}}y \Longleftrightarrow x\leq_{\mathcal{L}}y\text{ and } y\leq_{\mathcal{L}} x\quad (x,y\in W).$$

The corresponding equivalence classes are called the left cells of $W$. Similarly, we can define a pre-order $\leq_{\mathcal{R}}$ multiplying on the right in the defining relation. The associated equivalence relation will be denoted by $\sim_{\mathcal{R}}$ and the corresponding equivalence classes are called the right cells of $W$. Using the antiautomorphism $\flat$, we have (see \cite[\S 8]{bible})

$$x\leq_{\mathcal{L}} y \Longleftrightarrow x^{-1}\leq_{\mathcal{R}} y^{-1}.$$

Finally we write $x\leq_{\mathcal{LR}} y$ if there exists a sequence $x=x_{0},x_{1},...,x_{n}=y$ of $W$ such that for each $0\leq i\leq n-1$ we have either $x_{i}\leftarrow_{\mathcal{L}} x_{i+1}$ or $x_{i}\leftarrow_{\mathcal{R}} x_{i+1}$. The equivalence relation associated to $\leq_{\mathcal{LR}}$ will be denoted by $\sim_{\mathcal{LR}}$ and the equivalence classes are called the two-sided cells of $W$. \\
The preorders $\leq_{\mathcal{L}},\leq_{\mathcal{R}},\leq_{\mathcal{LR}}$ induce partial orders on the left, right and two-sided cells, respectively.

\subsection{Kazhdan-Lusztig cell modules}
\label{cell-mod}
We will follow the notation of \cite{semi}. Let $?\in\{\mathcal{L},\mathcal{R},\mathcal{LR}\}$. We define a $?$-ideal to be a left ideal if $?=\mathcal{L}$, a right ideal if $?=R$ and a two-sided ideal if $?=\mathcal{LR}$. Similarly, an $\mathcal{H}$-?-module is a left $\mathcal{H}$-module if $?=\mathcal{L}$, a right $\mathcal{H}$-module if $?=R$ and an $\mathcal{H}$-$\mathcal{H}$-bimodule if $?=\mathcal{LR}$.

Let $\Gamma$ be a $?$-cell of $W$.  We set

$$\begin{array}{rllcc}
\mathcal{H}_{\leq_{?}\Gamma}&=\mathcal{h} C_{y}\mid y\leq_{?}w, w\in\Gamma \mathcal{i}_{\mathcal{A}} \text{ and }\\
\mathcal{H}_{<_{?}\Gamma}&=\mathcal{h} C_{y}\mid y<_{?}w, w\in\Gamma \mathcal{i}_{\mathcal{A}}.\\
\end{array}$$

Then by definition of  $\leq_{?}$, we see that  $\mathcal{H}_{\leq_{?}\Gamma}$ and $\mathcal{H}_{<_{?}\Gamma}$ are $?$-ideals of $\mathcal{H}$. Therefore
$$\mathcal{M}^{?}_{\Gamma}:=\mathcal{H}_{\leq_{?}\Gamma}/\mathcal{H}_{<{?}\Gamma}$$  is naturally an $\mathcal{H}$-$?$-module. It is called the Kazhdan-Lusztig cell module associated to $\Gamma$. Note that it is a free $\mathcal{A}$-module with basis the images of the elements $C_{w}$ for $w\in \Gamma$.

Let $\Gamma$ be a two-sided cell of $W$ and let
$$\Gamma=\bigcup^{m}_{i=1} \Gamma_{i}$$
be its decomposition into left cells. We denote by $[\ .\ ]$ the natural projection onto $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$. (Then $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ has an $\mathcal{A}$-basis $\{[C_{w}]\mid w\in \Gamma\}$.) As an $\mathcal{A}$-module we have

$$\mathcal{M}^{\mathcal{LR}}_{\Gamma}\cong\bigoplus_{i=1}^{m} \mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}.$$
Further if one assumes that the Lusztig conjecture
\begin{equation*}
x\leq_{\mathcal{L}} y \text{ and } x\sim_{\mathcal{LR}} y \Longrightarrow x\sim_{\mathcal{L}}y\tag{$\ast$}
\end{equation*}

holds, then the $\mathcal{A}$-submodules $\mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}$ are left $\mathcal{H}$-submodules of $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ for all $1\leq i\leq m$. Indeed, for all $h\in\mathcal{H}$ and $y \in \Gamma_{i}$, we have
$$hC_{y}=\sum_{z\leq_{\mathcal{L}}y} a_{z} C_{z}=\sum_{z\leq_{\mathcal{L}} y, z\in \Gamma}a_{z} C_{z}+\sum_{z\leq_{\mathcal{L}}y,z\notin \Gamma}a_{z} C_{z}\quad\text{for some $a_{z}\in \mathcal{A}$}$$
which, using $(\ast)$, yields that $h[C_{y}]\in\mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}$.

\begin{Rem}
Conjecture $(\ast)$ is known to hold in the equal parameter case. In \cite{jeju4} it is shown that it holds in affine Weyl groups of rank 2 for all choices of parameters.
\end{Rem}

Finally, since the $\mathcal{H}$-$\mathcal{H}$-bimodule $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is a two-sided ideal in $\mathcal{H}/ \mathcal{H}_{<_{\mathcal{LR}} \Gamma}$, it  can be viewed as an algebra (possibly without identity element). The multiplication is given by
$$[C_{x}][C_{y}]=[C_{x}C_{y}]=\sum_{z\in\Gamma} h_{x,y,z}[C_{z}].$$
If one assumes that $(\ast)$ holds, then the $\mathcal{A}$-submodules $\mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}$ are left ideals of $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ for all $1\leq i\leq m$.

\begin{Rem}
Note that the involution $\flat$ fixes each $\mathcal{H}_{\leq_{\mathcal{LR}}\Gamma}$ and $\mathcal{H}_{<_{\mathcal{LR}}\Gamma}$ hence induces an involution on $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$. We will still denote this involution by $\flat$.
\end{Rem}

\subsection{Generic parameters}
\label{generic}
Let $\bar{S}=\{\bar{{\bf s}}_{1},...,\bar{{\bf s}}_{r}\}$ be the set of conjugacy classes in $S$. Any weight function on $W$ is completely determined by its values on $\bar{S}$. Let $V=\mathbb{R}^{r}$ be the Euclidean space of dimension $r$ and let $\omega_{1},...,\omega_{r}$ be the standard basis of $V$.  We identify the set of weight functions on $W$ with the set of points in $V$ with integer coordinates via
$$L\longrightarrow (L(s_{1}),...,L(s_{r}))\in V$$
where $s_{i}\in \bar{{\bf s}}_{i}$ for all $i$. The element of the $r$-tuple $(L(s_{1}),...,L(s_{r}))$ are called the parameters. To any choice of parameters one can associate a partition of $W$ into left, right and two-sided cells.

According to Bonnaf\'e's semicontinuity conjecture  \cite{semi}, there exists a minimal finite set of hyperplanes $\mathfrak{H}$ in $V$ such that the partition of $W$ into cells is the same for all parameters $\mathcal{P},\mathcal{P}'\in\mathbb{N}^{r}$ belonging to the same $\mathfrak{H}$-facet (we refer to \cite{Bourbaki} for the definition of facets). The elements of this minimal set are called essential hyperplanes. The conjecture also states that the partition into cells for non-generic parameters can be recovered from the partition with respect to generic parameters. We refer to \cite{semi} for details on this conjecture.  In the following definition we assume that Bonnaf\'e's conjecture holds.
\begin{Def}
The parameters $\mathcal{P}:=(a_{1},\ldots,a_{r})\in\mathbb{N}^{r}$ are called generic if they do not belong to any essential hyperplane for $W$.
\end{Def}
In this paper, we are only concerned with affine Weyl groups of rank 2 where the semicontinuity conjecture is known to hold and where the generic parameters have been determined in \cite{jeju4}.

\begin{Exa}
\label{G2}
Let $(W,S)$ be the affine Weyl group of type $\tilde{G}_{2}$ with diagram and weight function given by
\begin{center}
\begin{picture}(150,32)
\put( 40, 10){\circle{10}}
\put( 44,  7){\line(1,0){33}}
\put( 45,  10){\line(1,0){30.5}}
\put( 44, 13){\line(1,0){33}}
\put( 81, 10){\circle{10}}
\put( 86, 10){\line(1,0){29}}
\put(120, 10){\circle{10}}
\put( 38, 20){$a$}
\put( 78, 20){$b$}
\put(118, 20){$b$}
\put( 38, -3){$s_{1}$}
\put( 78, -3){$s_{2}$}
\put(118,-3){$s_{3}$}
\end{picture}
\end{center}
\vspace{.05cm}
where $a,b$ are positive integers. Then $(a,b)\in\mathbb{N}^{2}$ is generic if and only if $a/b\neq 1,\ 3/2,\ 2$. The corresponding partition into cells can be found in \cite{jeju4}.
\end{Exa}

\begin{Exa}
\label{B2}
Let $(W,S)$ be the affine Weyl group of type $\tilde{B}_{2}$ with diagram and weight function given by
\begin{center}
\begin{picture}(150,32)
\put( 40, 10){\circle{10}}
\put( 44.5,  8){\line(1,0){31.75}}
\put( 44.5,  12){\line(1,0){31.75}}
\put( 81, 10){\circle{10}}
\put( 85.5,  8){\line(1,0){29.75}}
\put( 85.5,  12){\line(1,0){29.75}}
\put(120, 10){\circle{10}}
\put( 38, 20){$a$}
\put( 78, 20){$b$}
\put(118, 20){$c$}
\put( 38, -3){$s_{1}$}
\put( 78, -3){$s_{2}$}
\put(118,-3){$s_{3}$}
\end{picture}
\end{center}
\vspace{.05cm}
where $a,b,c$ are positive integers. We define the following hyperplanes in $\mathbb{R}^{2}$
\begin{center}
\psset{xunit=1.5cm}
\psset{yunit=1.5cm}
\begin{pspicture}(0,-0.3)(5.5,5.3)
\psgrid[subgriddiv=1,griddots=10,gridlabels=8 pt](0,0)(5,5)

\psline(0,0)(5,5)
\psline(1,0)(1,1)
\psline(1,1)(5,1)
\psline(1,0)(0.5,.5)
\psline(1,0)(5,4)
\psline(2,0)(1,1)
\psline(2,0)(5,3)

\psline(0,1)(1,1)

\psline(.5,.5)(0,1)

\psline(1,1)(1,5)

\psline(1,1)(0,2)

\psline(0,2)(3,5)
\psline(0,1)(4,5)

\rput(3.5,.5){$A_{1}$}
\rput(2,.5){$A_{2}$}
\rput(1.5,.2){$A_{3}$}
\rput(1.2,.5){$A_{4}$}
\rput(1.5,.8){$A_{5}$}

\rput(4.5,1.5){$C_{3}$}
\rput(3,1.5){$C_{2}$}
\rput(2,1.5){$C_{1}$}

\rput(.5,.25){$B_{2}$}
\rput(.8,.5){$B_{1}$}

\psline{->}(0,0)(0,5)
\psline{->}(0,0)(5,0)
\end{pspicture}
\end{center}

Then  $(a,b,c)\in\mathbb{N}^{3}$ is generic if and only if $(a/b,c/b)$ does not belong to any hyperplanes on the picture above \cite{jeju4,comp}. The corresponding partition into cells can be found in \cite{comp}.
\end{Exa}

\subsection{On the induction of Kazhdan-Lusztig cells}
In this section we introduce the relative Kazhdan-Lusztig polynomials as in \cite{geck}. Let $S'\subsetneq S$. We denote by $W'$ the subgroup of $W$ generated by $S'$ and by $X'$ the set of distinguished left coset representative of $W'$ in $W$. Every element of $w\in W$ can be written uniquely $w=xu$ where $x\in X'$ and $u\in W'$. Note that $\ell(w)=\ell(x)+\ell(u)$. \\
Let $\preceq'$ be the Kazhdan-Lusztig left preorder relation defined with respect to the Coxeter group $(W',S')$ and the corresponding Hecke algebra. We define a relation $\sqsubseteq$ on $W$ as follows. Let $x,y\in X'$ and $u,v\in W'$. We write $xu\sqsubset yv$ if $x<y$ and $u\preceq v$. We write $xu\sqsubseteq yv$ if $xu\sqsubset yv$ or $x=y$ and $u=v$. Then we have
\begin{Prop}(\cite[Proposition 3.3]{geck})
\label{induction-st}
For any $y\in X'$ and $v\in W'$ we have
$$C_{yv}=\underset{xu\sqsubseteq yv}{\sum_{x\in X',u\in W'}}p^{*}_{xu,yv}T_{x}C_{u}$$
where $p^{*}_{yv,yv}=1$ and $p_{xu,yv}\in\mathbf{A}_{<0}$ if $xu\sqsubset yv$.
\end{Prop}

\begin{Rem}
Recall the $\mathcal{A}$-involution $\flat:\mathcal{H}\rightarrow\mathcal{H}$ from Definition \ref{flat}, which can be used to obtain a right-handed version of the above result. First $Y'=X'^{-1}$ is the set of distinguished right coset representative of $W'$ in $W$. Any $w\in W$ can be uniquely written $w=ux$ where $u\in W'$ and $x\in Y'$. We get for all $y\in Y'$ and $v\in W'$:
$$C_{vy}=\underset{ux\sqsubseteq_{\mathcal{R}} vy}{\sum_{x\in Y',u\in W'}}p^{\ast,r}_{ux,vy}C_{u}T_{x}$$
where $p^{\ast,r}_{ux,vy}=p^{\ast}_{(ux)^{-1},(yv)^{-1}}$.
\end{Rem}
Using this theorem, Geck obtained the following result (see \cite[Section 4]{geck}).
\begin{Cor}
\label{left-ideal}
Let $\mathfrak{b}$ be a left ideal of $W'$ with respect to $\leq'_{\mathcal{L}}$. Then the set $X'\mathfrak{b}$ is a left ideal of $W$ with respect to $\leq_{\mathcal{L}}$.
\end{Cor}

\subsection{Generalised induction of Kazhdan-Lusztig cells}
\label{gid}
We now introduce the Generalised Induction Theorem. The idea is to generalise the construction above to some subsets of $W$ which may not be parabolic subgroups. We refer to \cite{jeju3,jeju4} for details.\\

We consider a subset $U\subseteq W$ and a collection $\{X_{u}\ |\ u\in U\}$ of subsets of $W$ satisfying the following conditions
\begin{enumerate}
\item[{\bf I1}.] for all $u\in U$, we have $e\in X_{u}$,
\item[{\bf I2}.] for all $u\in U$ and $x\in X_{u}$ we have $\ell(xu)=\ell(x)+\ell(u)$,
\item[{\bf I3}.] for all $u,v\in U$ such that $u\neq v$ we have $X_{u}u\cap X_{v}v=\emptyset$,
\item[{\bf I4}.] the submodule $\mathcal{M}:=\mathcal{h} T_{x}C_{u}|\ u\in U,\ x\in X_{u}\mathcal{i}_{\mathcal{A}}\subseteq \mathcal{H}$ is a left ideal.
\end{enumerate}
One can easily see that the set $\mathcal{B}:=\{T_{x}C_{u}|u\in U,x\in X_{u}\}$ is an $\mathcal{A}$-basis of $\mathcal{M}$. Thus for all $y\in W$ and all $v\in U$, we can write
$$T_{y}C_{v}=\sum_{u\in U,x\in X_{u}}a_{x,u}T_{x}C_{u}\quad \text{for some $a_{x,u}\in \mathcal{A}$}.$$
Let $\preceq$ be the relation on $U$ defined as follows. Let $u,v\in U$. We write $u\preceq v$ if there exist $y\in W$ and $x\in X_{u}$ such that $T_{x}C_{u}$ appears with a non-zero coefficient in the expression of $T_{y}C_{v}$ in the basis $\mathcal{B}$. We still denote by $\preceq$ the pre-order induced by this relation (i.e. the transitive closure). For $u,v\in U$, $x\in X_{u}$ and $y\in X_{v}$ we write $xu\sqsubset yv$ if $u\preceq v$ and $xu< yv$. We write $xu\sqsubseteq yv$ if $xu\sqsubset yv$ or $x=y$ and $u=v$.
\begin{Prop}
(\cite[Proposition 3.8]{jeju3})
\label{Gind}
For any $v\in U$ and $y\in X_{u}$, there exist a unique family of polynomials $(p^{*}_{xu,yv})_{xu\sqsubset yv}$ in $\mathbf{A}_{<0}$ such that
$$\tilde{C}_{yv}:=T_{y}C_{v}+\underset{xu\sqsubset yv}{\sum_{u\in U, x\in X_{u}}}p^{*}_{xu,yv}T_{x}C_{u}$$
is stable under the $\bar{\ }$ involution.
\end{Prop}
\noindent
Now if one assumes that
\begin{enumerate}
\item[{\bf I5}.] for all $v\in U$, $y\in X_{v}$ we have
$$T_{y}C_{v}\equiv T_{yv} \mod \mathcal{H}_{<0} $$
\end{enumerate}
then we have $\tilde{C}_{yv}=C_{yv}$.
\begin{Rem}
This is really a generalisation of Geck' s result. If we set $U=W'$ and $X_{u}=X'$ for all $u\in U$ then condition {\bf I1}--{\bf I4} are satisfied. For condition {\bf I5} we have for $v\in W'$ and $y\in X'$:
\begin{align*}
\tilde{C}_{yv}&=T_{y}C_{v}+\underset{xu\sqsubset yv}{\sum_{u\in U, x\in X_{u}}}p^{*}_{xu,yv}T_{x}C_{u}\\
&=T_{y}\big( T_{v}+\sum_{v_{1}<v}P_{v_{1},v}T_{v_{1}} \big)+
\underset{xu\sqsubset yv}{\sum_{u\in U, x\in X_{u}}}p^{*}_{xu,yv}T_{x}\sum_{u_{1}\leq u}P_{u_{1},u}T_{u_{1}}\\
&=T_{yv}+\big(\sum_{v_{1}<v}P_{v_{1},v}T_{y}T_{v_{1}} \big)+
\underset{xu\sqsubset yv}{\sum_{u\in U, x\in X_{u}}}\sum_{u_{1}\leq u}p^{*}_{xu,yv}P_{u_{1},u}T_{x}T_{u_{1}}\\
&=T_{yv}+\big(\sum_{v_{1}<v}P_{v_{1},v}T_{yv_{1}} \big)+
\underset{xu\sqsubset yv}{\sum_{u\in U, x\in X_{u}}}\sum_{u_{1}\leq u}p^{*}_{xu,yv}P_{u_{1},u}T_{xu_{1}}\\
&\equiv T_{yv} \mod \mathcal{H}_{<0}.
\end{align*}
Since $\tilde{C}_{yv}$ is stable under the involution  $\bar{\ }$ we get $\tilde{C}_{yv}=C_{yv}$.
\end{Rem}

\section{Main result}
Throughout this section, $W$ denotes an affine Weyl group of rank 2 and $L$ a generic weight function. We fix a two-sided cell $\Gamma$ of $W$ and wish to show that $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is an affine cell ideal in $\mathcal{H}/\mathcal{H}_{<_{\mathcal{LR}}\Gamma}$. In Sections \ref{description}--\ref{main-result}, we assume that, in the case where $W$ is of type $\tilde{G}_{2}$, the two-sided cell $\Gamma$ is either infinite or does not intersect the group generated by $s_{2},s_{3}$. The remaining cases will be treated in Section \ref{finite}.\\

\subsection{Description of $\Gamma$}
\label{description}
In this section we present a very nice description of $\Gamma$. It is rather surprising that most of the two-sided cells in $W$ can be described in such a uniform way. Note that this description is vital in the proof of affine cellularity. We refer to the next section for examples of this description.\\

By inspection of the different partitions into cells given in \cite{comp,jeju4}, one can show  that there exist two subsets $\mathcal{T}$ and $\mathcal{Z}=\{z_{1}=e,z_{2},\ldots,z_{m}\}$ of $W$ and an element $w_{\Gamma}\in W$ in a parabolic subgroup $W'$ such that for all  $1\leq i,j\leq m$ and all $\tau\in\mathcal{T}$ we have
$$\ell(z_{i}^{-1}\tau w_{\Gamma}z_{k})=\ell(z_{i}^{-1})+\ell(\tau)+\ell(w_{\Gamma})+\ell(z_{j})$$
 and the following map is bijective
$$\begin{array}{ccccccc}
&\mathcal{Z}\times \mathcal{T}\times\mathcal{Z} &\longrightarrow & \Gamma&\\
& (z_{i},\tau,z_{j})&\longmapsto& z_{i}^{-1}\tau w_{\Gamma}z_{j}&\in\Gamma_{i}^{-1}\cap \Gamma_{j}.
\end{array}$$
The left cells lying in $\Gamma$ are of the form
$$\Gamma_{j}=\{z_{i}^{-1}\tau w_{\Gamma}z_{j}\mid \tau \in \mathcal{T}, 1\leq i\leq m\}.$$
and we have
$$(\Gamma_{i})^{-1}\cap \Gamma_{j}=\{z_{i}^{-1}\tau w_{\Gamma}z_{j}\mid \tau\in\mathcal{T}\}.$$
The set $\mathcal{T}$ can be expressed in one of the following forms:
\begin{enumerate} \label{cases}
\item $\mathcal{T}=\{t_{1}^{n}t_{2}^{m}\mid n,m\in\mathbb{N}\}$
\item $\mathcal{T}=\{t^{n}\mid n\in\mathbb{N}\}$
\item $\mathcal{T}=\{e,t\}$
\item $\mathcal{T}=\{e\}$
\end{enumerate}
In case (1) we set $T=\{t_{1},t_{2}\}$, in case (2) and (3) we set $T=\{t\}$ and in case (4) we set $T=\{e\}$. In all cases it can be checked that for $\tau \in \mathcal{T}$, we have
\begin{equation*}
\tau w_{\Gamma}=w_{\Gamma}\tau^{-1}.
\end{equation*}
It is clear that if $\Gamma$ is infinite, then we must be in case (1) or (2) and if $\Gamma$ is finite then we must be in case (3) or (4).
We refer to Section 5 for examples of the different cases: when $\Gamma=\tilde{c}_{0}$ we are in case (1), when $\Gamma=\tilde{c}_{1}$ or $\tilde{c}_{2}$ we are in case (2), when $\Gamma=\tilde{c}_{3}$ we are in case (3) and when $\Gamma=\tilde{c}_{4}$ we are in case (4).
\begin{Rem}
This description is no longer true for all cells if the parameters are non-generic which is why in this paper we are only considering generic parameters.
\end{Rem}

\noindent
\subsection{The element $\mathbf{P}$}
\label{Ppol}

Let $X'$ be the set of distinguished left coset representative of $W'$ in $W$ and
let $\leq^{'}_{\mathcal{L}}$ be the Kazhdan-Lusztig left preorder relation defined with respect to $W'$. Then explicit computations show that either
\begin{enumerate}
\item $w_{\Gamma}$ is the longest element in $W'$;
\item $w_{\Gamma}=sw'$ where $w'$ is the longest element of $W'$, $w_{\Gamma}^{2}=1$
and   $\{w_{\Gamma},w'\}$ is a left ideal of $W'$ with respect to $\leq'_{\mathcal{L}}$.
\end{enumerate}

\medskip

\emph{The element $\mathbf{P}$ in Case (1).} Assume we are in Case (1). By well-known properties of the longest element in a Coxeter group, for all $u\in W'$ we have $u\leq'_{\mathcal{L}} w_{\Gamma}$ implies that $u=w_{\Gamma}$. Further, one can easily check from the definition of $\sqsubset$ that this implies that if $w\in W$ satisfies $w\sqsubseteq yw_{\Gamma}$ then $w=xw_{\Gamma}$ with $x<y$ and $x\in X'$.
Thus by Proposition \ref{induction-st} we get for all $y\in X'$
\begin{align*}
C_{yw_{\Gamma}}&=T_{y}C_{w_{\Gamma}}+\sum_{x<y, x\in X'} p^{*}_{xw_{\Gamma},yw_{\Gamma}}T_{z}C_{w_{\Gamma}}\\
&=\big( T_{y}+\sum_{x<y, x\in X'} p^{*}_{xw_{\Gamma},yw_{\Gamma}}T_{z}\big)C_{w_{\Gamma}}.
\end{align*}
We set for $y\in\mathcal{Z}^{-1}\cup T$
$$\mathbf{P}(y)=\sum_{x\leq y, x\in X'}p^{\ast}_{xw_{\Gamma},yw_{\Gamma}}T_{x},\quad \mathbf{P}_{\mathbf{R}}(y^{-1})=(\mathbf{P}(y))^{\flat},$$
so that we have
$$\mathbf{P}(y)C_{w_{\Gamma}}=C_{yw_{\Gamma}}\text{ and } C_{w_{\Gamma}}\mathbf{P}_{\mathbf{R}}(y^{-1})=C_{w_{\Gamma}y^{-1}}.$$
The last equality holds since
$$C_{w_{\Gamma}}\mathbf{P}_{\mathbf{R}}(y^{-1})=C_{w_{\Gamma}}(\mathbf{P}(y))^{\flat}=(\mathbf{P}(y)C_{w_{\Gamma}})^{\flat}=(C_{yw_{\Gamma}})^{\flat}=C_{w_{\Gamma}^{-1}y^{-1}}=C_{w_{\Gamma}y^{-1}}.$$

\medskip

\emph{The element $\mathbf{P}$ in Case (2).}  Assume that we are in Case (2). We set $U=\{w_{\Gamma}\}$ and $X=X'\cup X's$.

\emph{Claim.} The set $U$ together with $X$ satisfy condition {\bf I1}--{\bf I5}.

 Assume for now that it is the case, then for all $y\in X$ we have (note that in the sum below the element $x$ is chosen in $X$ and not in $X'$ as above!)
\begin{align*}
C_{yw_{\Gamma}}&=T_{y}C_{w_{\Gamma}}+\sum_{x<y, x\in X} p^{*}_{xw_{\Gamma},yw_{\Gamma}}T_{x}C_{w_{\Gamma}}\\
&=\big( T_{y}+\sum_{x<y, x\in X_{c}} p^{*}_{xw_{\Gamma},yw_{\Gamma}}T_{z}\big)C_{w_{\Gamma}}.
\end{align*}
For $y\in\mathcal{Z}^{-1}\cup T$ we set
$$\mathbf{P}(y)=\sum_{x\leq y, x\in X}p^{\ast}_{xw_{\Gamma},yw_{\Gamma}}T_{x},\quad \mathbf{P}_{\mathbf{R}}(y^{-1})=(\mathbf{P}(y))^{\flat},$$
so that we have
$$\mathbf{P}(y)C_{w_{\Gamma}}=C_{yw_{\Gamma}}\text{ and } C_{w_{\Gamma}}\mathbf{P}_{\mathbf{R}}(y^{-1})=C_{w_{\Gamma}y^{-1}}.$$
The last equality holds since
$$C_{w_{\Gamma}}\mathbf{P}_{\mathbf{R}}(y^{-1})=C_{w_{\Gamma}}(\mathbf{P}(y))^{\flat}=(\mathbf{P}(y)C_{w_{\Gamma}})^{\flat}=(C_{yw_{\Gamma}})^{\flat}=C_{w_{\Gamma}^{-1}y^{-1}}=C_{w_{\Gamma}y^{-1}}$$ (here we need the fact that $w_{\Gamma}^{2}=1$).

\emph{Proof of Claim.} In order to verify that $U=\{w_{\Gamma}\}$ together with  $X=X'\cup X's$ satisfies condition {\bf I1}--{\bf I5}, note that $Xw_{\Gamma}=X'w'\cup X'w_{\Gamma}$, from where
conditions {\bf I1}--{\bf I3} follow easily. Next we know that $\{w',w_{\Gamma}\}$ is a left ideal of $W'$, thus by Corollary \ref{left-ideal} we get that
$$\mathfrak{B}=\mathcal{h} C_{w}\mid w\in X'w_{\Gamma}\cup X'w'\mathcal{i}_{\mathcal{A}}=\mathcal{h} C_{xw_{\Gamma}}\mid x\in X\mathcal{i}$$ is a left ideal of $\mathcal{H}$. Since
$$T_{x}C_{w_{\Gamma}}=C_{xw_{\Gamma}}+\sum_{z<xw_{\Gamma},z\in \mathfrak{B}} C_{z},$$
it follows that $\mathfrak{B}=\mathcal{h} T_{x}C_{w_{\Gamma}}\mid x\in X\mathcal{i}$ and {\bf I4} follows. Let $x\in X$. If $x\in X'$ Condition {\bf I5} is clearly satisfied since $w_{\Gamma}\in W'$. So assume that $x\in X's$, that is $x=x's$ for some $x'\in X'$. Then we have
\begin{align*}
T_{x}C_{w_{\Gamma}}&=T_{x'}T_{s}C_{w_{\Gamma}}\\
&=T_{x'}(C_{w'}-v^{-L(s)}C_{w_{\Gamma}})\\
&=T_{x'}C_{w'}-v^{-L(s)}T_{x'}C_{w_{\Gamma}}.
\end{align*}
The claim follows since $T_{x'}C_{w'}\equiv T_{x'w'} \mod \mathcal{H}_{<0}$ and $T_{x'}C_{w_{\Gamma}}\equiv T_{x'w_{\Gamma}} \mod \mathcal{H}_{<0}$.

\bigskip

Finally, in both Cases (1) and (2), we set
$$\begin{array}{rllc}
\mathbf{P}(t_{1}^{m}t_{2}^{n})&=\mathbf{P}(t_{1})^{m}\mathbf{P}(t_{2})^{n}&\mbox{if $T=\{t_{1},t_{2}\}$}\\
\mathbf{P}(t^{n})&=\mathbf{P}(t)^{n}& \mbox{otherwise}.
\end{array}$$

\subsection{Properties of the element $\mathbf{P}$}
\label{properties}
Recall that $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ can be viewed as an algebra and that the $\mathcal{A}$-modules
$$\mathcal{h} [C_{w}]\mid w\in \Gamma_{j}\mathcal{i}_{\mathcal{A}}$$
are left ideals in $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ (see Section \ref{cell-mod}).

\begin{Lem}
\label{left-right}
Let $\tau\in \mathcal{T}$. We have
$$\mathbf{P}(\tau)C_{w_{\Gamma}}=C_{w_{\Gamma}}\mathbf{P}_{R}(\tau^{-1}).$$
\end{Lem}

\begin{proof}
Using the fact that for all $\tau\in\mathcal{T}$ we have $\tau w_{\Gamma}=w_{\Gamma}\tau^{-1}$ we get
$$\mathbf{P}(\tau)C_{w_{\Gamma}}=C_{\tau w_{\Gamma}}=C_{w_{\Gamma}\tau^{-1}}=C_{w_{\Gamma}}\mathbf{P}_{R}(\tau^{-1}).$$
\end{proof}

We define the following $\mathcal{A}$-submodule of  $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$:
$$\mathcal{M}_{\mathcal{T}}=\mathcal{h} [C_{\tau w_{\Gamma}}]\mid \tau\in\mathcal{T}\mathcal{i}_{\mathcal{A}}.$$

\begin{Lem}
\label{mod-tau}
We have
$$\mathcal{h} [C_{w}]\mid w\in\Gamma_{1}\mathcal{i}_{\mathcal{A}}\cap \mathcal{h} [C_{w}]\mid w\in (\Gamma_{1})^{-1}\mathcal{i}_{\mathcal{A}}=\mathcal{M}_{\mathcal{T}}$$
\end{Lem}

\begin{proof}
This follows directly from the fact that $\mathcal{h} [C_{w}]\mid w\in\Gamma_{1}\mathcal{i}_{\mathcal{A}}$ (respectively \\ $\mathcal{h} [C_{w}]\mid w\in (\Gamma_{1})^{-1}\mathcal{i}_{\mathcal{A}}$) is a left ideal of $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ (respectively a right ideal of $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$) and the equality
$$(\Gamma_{1})^{-1}\cap \Gamma_{1}=\{\tau w_{\Gamma}\mid \tau\in\mathcal{T}\}.$$
\end{proof}

\begin{Lem}
\label{basis-tau}
The set $\{[\mathbf{P}(\tau)C_{w_{\Gamma}}]\mid \tau\in\mathcal{T}\}$ is an $\mathcal{A}$-basis of $\mathcal{M}_{\mathcal{T}}$.
\end{Lem}

\begin{proof}
Since $\mathbf{P}(\tau)C_{w_{\Gamma}}=C_{w_{\Gamma}}\mathbf{P}_{R}(\tau^{-1})$ and
$$\mathbf{P}(\tau)C_{w_{\Gamma}}\in \mathcal{H} C_{w_{\Gamma}}\subseteq \sum_{z\leq_{\mathcal{L}} w_{\Gamma}} \mathcal{A} C_{z}\text{ and } C_{w_{\Gamma}}\mathbf{P}_{R}(\tau^{-1})\in C_{w_{\Gamma}}\mathcal{H}\subseteq \sum_{z\leq_{\mathcal{R}} w_{\Gamma}}\mathcal{A} C_{z}$$
we have $[\mathbf{P}(\tau)C_{w_{\Gamma}}]\in \mathcal{M}_{\mathcal{T}}$ by the previous lemma.  Then the result follows easily from the fact that
$$\mathbf{P}(\tau)C_{w_{\Gamma}}=C_{\tau w_{\Gamma}}+\sum_{z<\tau w_{\Gamma}} \mathcal{A} C_{z}.$$
\end{proof}

\begin{Lem}
\label{basis-m1}
The set  $\{[\mathbf{P}(z_{j}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}]\mid 1\leq j\leq m, \tau\in\mathcal{T}\}$ is an $\mathcal{A}$-basis of $\mathcal{M}^{\mathcal{L}}_{\Gamma_{1}}$.
\end{Lem}

\begin{proof}
Since
$$\mathbf{P}(z_{j}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\in \mathcal{H} C_{w_{\Gamma}}\subseteq \sum_{z\leq_{\mathcal{L}} w_{\Gamma}}\mathcal{A} C_{z}$$
we have $[\mathbf{P}(z_{j}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}]\in\mathcal{M}^{\mathcal{L}}_{\Gamma_{1}}$. Then the results follows easily from the fact that
$$\mathbf{P}(z_{j}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}=C_{z_{i}^{-1}\tau w_{\Gamma}}+\sum_{z<z_{i}^{-1}\tau w_{\Gamma}}C_{z}.$$
\end{proof}

\subsection{Main result}
\label{main-result}

We are now ready to define the different ingredients needed in order to show that each cell ideal is affine cellular. Recall the definition of $T$ and $\mathcal{T}$ in Section \ref{description}. As our principal ideal domain $k$, we choose $\mathcal{A}$. We set
$$B=
\begin{cases}
\mathcal{A}[t_{1},t_{2}] &\mbox{if $T=\{t_{1},t_{2}\}$}\\
\mathcal{A}[t]&\mbox{if $T=\{t\}\qquad$}\\
\mathcal{A}[t]/(t^{2}-1)&\mbox{if $T=\{e,t\}$}\\
\mathcal{A}&\mbox{if $T=\{e\}$.}
\end{cases}$$

Note that the monomials in $B$ corresponds to the elements of $\mathcal{T}$ and we will use this identification freely. \\
For all $1\leq i,j\leq m$ we know, by Lemma \ref{mod-tau}, that
$$[C_{w_{\Gamma}z_{j}}][C_{z_{i}^{-1}w_{\Gamma}}]\in\mathcal{M}_{\mathcal{T}},$$
thus, by Lemma \ref{basis-tau}, we have
$$[C_{w_{\Gamma}z_{j}}][C_{z_{k}^{-1}w_{\Gamma}}]=\sum_{\tau \in\mathcal{T}} a^{j,k}_{\tau}[\mathbf{P}(\tau)C_{w_{\Gamma}}]\text{ where $a^{j,k}_{\tau}\in \mathcal{A}$}.$$
Let $V$ be the free $\mathcal{A}$-module of rank $m$ on basis $v_1, \dots , v_m$ and define the $\mathcal{A}$-bilinear form $\varphi$ by
$$\begin{array}{ccccccc}
\varphi:&V\times V &\longrightarrow & B\\
& (v_{j},v_{k})&\longmapsto& \underset{\tau\in \mathcal{T}}{\sum}a^{j,k}_{\tau}\tau.
\end{array}$$

This defines an algebra $\mathbb{A}(V,B,\varphi)\cong V \otimes_{\mathcal{A}} B\otimes_{\mathcal{A}} V$ with multiplication $\mathcal{A}$-bilinearly extended from $(v_i \otimes \tau \otimes v_j)(v_k \otimes \tau' \otimes v_l) = v_i \otimes \tau \varphi(v_j,v_k)  \tau' \otimes v_l$  as in Section \ref{affcell}.

We now define a map $$\tilde\Phi: \mathbb{A}(V,B,\varphi) \rightarrow \mathcal{H}$$ by $$v_i \otimes \tau \otimes v_j \mapsto \mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})$$
for basis elements $v_i,v_j $ of V and $\tau \in \mathcal{T}$.

We have
$$\mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})\in \mathcal{H} C_{w_{\Gamma}}\mathcal{H}\subseteq \sum_{z\leq_{\mathcal{LR}} w_{\Gamma}} \mathcal{A} C_{z}.$$
Hence, the image of $\tilde \Phi$ is contained in $\mathcal{H}_{\leq_{\mathcal{LR}} \Gamma}$ and we can compose $\tilde{\Phi}$ with the natural projection $[\ .\ ]$ onto $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$. We obtain a map
$$\begin{array}{ccccccc}
\Phi:&\mathbb{A}(V,B,\varphi) &\longrightarrow & \mathcal{M}^{\mathcal{LR}}_{\Gamma}\\
& v_{i}\otimes \tau \otimes v_{j}&\longmapsto& [\mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})].
\end{array}$$
Note that when $B=\mathcal{A}[t_{1},t_{2}]$ we have $t_{1}^{m}t_{2}^{n}=t_{2}^{n}t_{1}^{m}$ in $B$ thus for $\Phi$ to be well-defined we need to have
$$\Phi(v_{i}\otimes t_{1}^{m}t_{2}^{n}\otimes v_{j})=\Phi(v_{i}\otimes t_{2}^{n}t_{1}^{m}\otimes v_{j}),$$
that is
$$[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t_{1})^{m}\mathbf{P}(t_{2})^{n}C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})]=[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t_{2})^{n}\mathbf{P}(t_{1})^{m}C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})].$$
To prove this, it is enough to show that
$$[\mathbf{P}(t_{1})\mathbf{P}(t_{2})C_{w_{\Gamma}}]=[\mathbf{P}(t_{2})\mathbf{P}(t_{1})C_{w_{\Gamma}}].$$
This is checked by {\it explicit computation with} GAP.

\begin{Prop}\label{affine-cellular-ideal}

\begin{enumerate}
\item \label{isom}
The map $\Phi:\mathbb{A}(V,B,\varphi) \rightarrow \mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is an isomorphism of $\mathcal{A}$-algebras.
\item \label{bimod} Using \eqref{isom} to define left and right $\mathcal{H}$-module structures on  $\mathbb{A}(V,B,\varphi)$ by letting  $h \in \mathcal{H}$ act, for $v,w \in V$ and $b \in B$, as $h(v\otimes b \otimes w) = \Phi^{-1}(h\Phi(v \otimes b\otimes w)$ and $(v\otimes b\otimes w)h = \Phi^{-1}(\Phi(v\otimes b\otimes w)h)$ respectively, $\Phi$ is an isomorphism of $\mathcal{H}$-$\mathcal{H}$-bimodules.
\item \label{invol} We have $\Phi(v\otimes b \otimes w)^{\flat}=\Phi(w \otimes b \otimes v)$ for $v,w \in V$ and $b \in B$.
\end{enumerate}
\end{Prop}

\begin{proof}
The map $\Phi$ is $\mathcal{A}$-linear by definition. We have, for basis elements $v_i,v_j,v_k,v_l$ of $V$ and $t,t' \in \mathcal{T}$,
\begin{align*}
&\Phi\big(v_{i}\otimes t\otimes v_{j}\big)\Phi\big(v_{k}\otimes t'\otimes v_{l}\big)\\
&=[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})][\mathbf{P}(z_{k}^{-1})\mathbf{P}(t')C_{w_{\Gamma}}\mathbf{P}_{R}(z_{l})]\\
&=[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t)C_{w_{\Gamma}z_{j}}\mathbf{P}(z_{k}^{-1})C_{w_{\Gamma}}\mathbf{P}_{R}(t'^{-1})\mathbf{P}_{R}(z_{l})]\\
&=[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t)C_{w_{\Gamma}z_{j}}C_{z_{k}^{-1}w_{\Gamma}}\mathbf{P}_{R}(t'^{-1})\mathbf{P}_{R}(z_{l})]\\
&=[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t)\big(\sum_{\tau \in\mathcal{T}} a^{j,k}_{\tau}\mathbf{P}(\tau)\big)C_{w_{\Gamma}}\mathbf{P}_{R}(t'^{-1})\mathbf{P}_{R}(z_{l})]\\
&=[\mathbf{P}(z_{i}^{-1})\mathbf{P}(t)\big(\sum_{\tau \in\mathcal{T}} a^{j,k}_{\tau}\mathbf{P}(\tau)\big)\mathbf{P}(t')C_{w_{\Gamma}}\mathbf{P}_{R}(z_{l})]\\
&=\Phi\big(v_{i}\otimes t\varphi(v_{j},v_{k})t'\otimes v_{l}\big).
\end{align*}
So $\Phi$ is indeed a morphism of $\mathcal{A}$-algebras.
\begin{Rem}
In the case where $\mathcal{T}=\{e,t\}$ we quotient out by $t^{2}-1$ in $B$ because we have
$$(\mathbf{P}(t))^{2}C_{w_{\Gamma}}=C_{w_{\Gamma}}.$$
\end{Rem}

The fact that $\Phi$ is bijective follows easily from the fact that
$$\mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})=C_{z_{i}^{-1}\tau w_{\Gamma}z_{j}}+\sum_{z<z_{i}^{-1}\tau w_{\Gamma}z_{j}}\mathcal{A} C_{z}.$$ This completes the proof of \eqref{isom}.

Claim \eqref{bimod} follows directly from the definition and the fact that $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is an $\mathcal{H}$-$\mathcal{H}$-bimodule.

To prove Claim \eqref{invol}
we let $v_i,v_j$ be basis elements of $V$ and $\tau \in \mathcal{T}$, and check
\begin{align*}
\Phi(v_{i}\otimes\tau\otimes v_{j})^{\flat}&=[(\mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j}))]^{\flat}\\
&=[(\mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j}))^{\flat}]\\
&=[(\mathbf{P}_{R}(z_{j}))^{\flat}(C_{w_{\Gamma}})^{\flat}(\mathbf{P}(\tau))^{\flat}(\mathbf{P}(z_{i}^{-1}))^{\flat}]\\
&=[\mathbf{P}(z_{j}^{-1})C_{w_{\Gamma}}\mathbf{P}_{R}(\tau^{-1})\mathbf{P}_{R}(z_{i})]\\
&=[\mathbf{P}(z_{j}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{i})]\\
&=\Phi(v_{j}\otimes\tau\otimes v_{i}).
\end{align*}
The claim follows from $\mathcal{A}$-linearity.
\end{proof}

\begin{Th}
\label{maintheorem}
$\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is an affine cell ideal in $\mathcal{H}/\mathcal{H}_{<_{\mathcal{LR}} \Gamma}$ with the $\mathcal{A}$-involution induced by $\flat$.
\end{Th}

\begin{proof}
According to Proposition \ref{descrip}, this follows from Proposition \ref{affine-cellular-ideal} by choosing the $\mathcal{A}$-involution on $B$ to be the identity.
\end{proof}

\subsection{Remaining cases}
\label{finite}
Assume that $W$ is of type $\tilde{G}_{2}$ (as in Example \ref{G2}) and that $\Gamma$ be a finite two-sided cell which intersect the group generated by $s_{2},s_{3}$. Let
$$\Gamma=\bigcup_{i=1}^{m} \Gamma_i$$
be the decomposition of $\Gamma$ into left cells. \\

Assume that $a>b$. Then it can be checked by inspection that for all $i,j$ we have that $(\Gamma_{i})^{-1}\cap\Gamma_{j}$
only contains one element: we will denote it by  $w^{i,j}$.

Note that this implies that each left cell contains $m$ elements. Let $V$ be a $m$-dimensional $\mathcal{A}$-module on basis $v_1,\dots, v_m$ and let $B=\mathcal{A}$. Let

$$\begin{array}{ccccccc}
\varphi:&V\times V &\longrightarrow & B\\
& (v_{j},v_{k})&\longmapsto& a_{j,k}
\end{array}$$

where $a_{j,k}\in B$ is such that

$$[C_{w^{1,j}}][C_{w^{k,1}}]=a_{j,k}[C_{w^{1,1}}].$$

Then it can be checked in each case that the map
$$\begin{array}{ccccccc}

\Phi:&\mathbb{A}(V,B,\varphi) &\longrightarrow & \mathcal{M}^{\mathcal{LR}}_{\Gamma}\\
& v_{i}\otimes 1 \otimes v_{j}&\longmapsto& [C_{w^{i,j}}]
\end{array}$$

satisfies the required properties. This is proved by explicit computation. \\

\medskip

Assume that $a<b$. Then $\Gamma=\Gamma_{1}\cup\Gamma_{2}\cup \Gamma_{3}$ where
\begin{align*}
\Gamma_{1}&=\{s_{3},s_{2}s_{3},s_{1}s_{2}s_{3},s_{2}s_{1}s_{2}s_{3},s_{3}s_{2}s_{1}s_{2}s_{3},s_{1}s_{2}s_{1}s_{2}s_{3}\},\\
\Gamma_{2}&=\{s_{2},s_{3}s_{2}, s_{1}s_{2},s_{2}s_{1}s_{2},s_{3}s_{2}s_{1}s_{2},s_{1}s_{2}s_{1}s_{2}\},\\
\Gamma_{3}&=\{s_{2}s_{1},s_{3}s_{2}s_{1}, s_{1}s_{2}s_{1},s_{2}s_{1}s_{2}s_{1},s_{3}s_{2}s_{1}s_{2}s_{1},s_{1}s_{2}s_{1}s_{2}s_{1}\}.
\end{align*}
Let $B=\mathcal{A}[t]/(t^{2}-1)$ and let $V$ be a 3-dimensional $\mathcal{A}$-module with basis $E:=\{v_{1},v_{2},v_{3}\}$. Let

$$\phi: E\times \{1,t\}\times E\rightarrow \Gamma$$

be such that $\phi(v_{i},1,v_{j})$ (respectively $\phi(v_{i},t,v_{j})$) is the element of minimal (respectively of maximal) length in $(\Gamma_{i})^{-1}\cap \Gamma_{j}$.
Then we define $\varphi: V\times V\rightarrow B$ by the following matrix
$$
\left(\begin{array}{ccc}
a_{1}&a_{2}+t&1\\
a_{2}+t&a_{3}+a_{4}t&0\\
1&0&a_{1}
 \end{array}\right)
$$
where
$$\begin{array}{llllll}
a_{1}&=v^{b}+v^{-b}& a_{2}&=v^{a-b}+v^{b-a}\\
a_{3}&=v^{-b}+v^{2a-b}+v^{b-2a}+v^{b} & a_{4}&=v^{-a}+v^{a}.
\end{array}$$
Finally we set
$$\begin{array}{ccccccc}
\Phi:&\mathbb{A}(V,B,\varphi) &\longrightarrow & \mathcal{M}^{\mathcal{LR}}_{\Gamma}\\
& v_{i}\otimes 1 \otimes v_{j}&\longmapsto& [C_{\phi(v_{i},1,v_{j})}]\\
& v_{i}\otimes t \otimes v_{j}&\longmapsto& [C_{\phi(v_{i},t,v_{j})}].
\end{array}$$
Then it can be checked by explicit computations that $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is affine cellular in the quotient $\mathcal{H}/\mathcal{H}_{<_{\mathcal{LR}}\Gamma}$.

\subsection{Proof of Theorem \ref{mainthmintro}}

Theorem \ref{mainthmintro} now follows from the results in Subsections \ref{main-result} and \ref{finite}. Indeed, let $W$ be an affine Weyl group of rank 2 together with a generic weight function $L$ and let $\mathcal{H}$ be the associated Hecke algebra. Consider the filtration of $\mathcal{H}$ by two-sided ideals $\mathcal{H}_{\leq_{\mathcal{LR}}\Gamma}$ given by the partial order $\leq_{\mathcal{LR}}$ on two-sided cells $\Gamma$ of $W$. We need to show that the Kazhdan-Lusztig cell modules $\mathcal{M}_{\Gamma}^{\mathcal{LR}}$ are isomorphic to some $\mathbb{A}(V,B,\varphi)$. First we show that ``most'' of the two-sided cells $\Gamma$ of $W$ can be described as
$$\Gamma=\{z^{-1}\tau w_{\Gamma}z'\mid \tau \in \mathcal{T}, z,z'\in\mathcal{Z}\}$$
for some subsets $\mathcal{T},\mathcal{Z}$ of $W$ and $w_{\Gamma}\in W$ (see  Sections \ref{description}, Section \ref{examples} and Appendix \ref{appendix}). Then, using the Generalised Induction Theorem of Kazhdan-Lusztig cells (see Section \ref{gid}), we define polynomials
$\mathbf{P}(\tau)\in\mathcal{A}$ for all $\tau\in\mathcal{T}$ (see Section \ref{Ppol}). Finally, we show that the map
$$\begin{array}{ccccccc}
\Phi:&\mathbb{A}(V,B,\varphi) &\longrightarrow & \mathcal{M}^{\mathcal{LR}}_{\Gamma}\\
& v_{i}\otimes \tau \otimes v_{j}&\longmapsto& [\mathbf{P}(z_{i}^{-1})\mathbf{P}(\tau)C_{w_{\Gamma}}\mathbf{P}_{R}(z_{j})]
\end{array}$$
is an isomorphism of $\mathcal{A}$-algebras (see Section \ref{main-result}). We then treat the case of the two-sided cells which cannot be described as above in Section \ref{finite}.   \\

Applying the results in Section \ref{affcell}, we obtain a parameterisation of simple modules of $\mathcal{H}$: For each cell we obtain a simple module for every maximal ideal of the corresponding $\mathcal{A}$-algebra $B$, defined at the start of Subsection \ref{main-result}. If we specialise to $\mathbb{C}(v) \otimes_{\mathcal{A}}\mathcal{H}$, the parametrisation is simply given by tuples $(a,b) \in \mathbb{C}(v)^2$ if $T=\{t_{1},t_{2}\}$, $a \in\mathbb{C}(v)$ if $T=\{t\}$, $\pm1$ if $T=\{e,t\}$, and $1$ if $T=\{e\}$.
Also it is clear that the affine $\mathcal{A}$-algebras $B$ that appear in our construction satisfy $\mathrm{rad}(B)=0$ and have finite global dimension. Thus in order to prove that the affine Hecke algebra $\mathcal{H}$ has finite global dimension, using the affine cellular structure, one would need to show that every $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is an idempotent ideal in $\mathcal{H}/\mathcal{H}_{<_{\mathcal{LR}}\Gamma}$ and that it contains an idempotent element in $\mathcal{H}/\mathcal{H}_{<_{\mathcal{LR}}\Gamma}$.

\begin{Rem}
The problem of finiteness of global dimension of affine Hecke algebras have already been addressed by Opdam and Solleveld in \cite{OS}. Using methods of harmonic analysis, they determined the (finite) global dimension of affine Hecke algebras in the case where $v$ is specialized to a positive real number.
\end{Rem}

\section{Examples}
\label{examples}
The aim of this section is to provide some explicit examples of the sets $\mathcal{Z}$ and $\mathcal{T}$ as defined in Section \ref{description}.
Let $W$ be an affine Weyl group of type $\tilde{G}_{2}$ as in Example \ref{G2} together with some generic parameters $a,b$ such that $a/b>2$.
The partition into cells in this case is shown in the following figure: the left cells are formed by the alcoves lying in the same connected component after removing the thick lines and the two-sided cells are the unions of all the left cells whose names share the same subscript. The alcove corresponding to the identity is denoted by $\tilde{c}_{6}^{1}$. We use the geometric presentation of $W$ as defined in \cite{Lus1}.
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We first have a look at the lowest two-sided cell $\tilde{c}_{0}$. In Figure \ref{c0},  we show the elements of the set $\mathcal{Z}=\{z_{1},\ldots,z_{12}\}$: these are the elements which correspond to the alcoves in dark gray. The alcoves lying in the box in light gray correspond to the elements in $\mathcal{Z}^{-1}$. We set  $t_{1}=s_{2}s_{1}s_{2}s_{1}s_{2}s_{3}$, $t_{2}=s_{1}s_{2}s_{1}s_{2}s_{3}s_{1}s_{2}s_{1}s_{2}s_{3}$ and
$$\mathcal{T}:=\{t_{1}^{n}t_{2}^{m}|n,m\in \mathbb{N}\}.$$
Then we have
\begin{align*}
\tilde{c}_{0}&=\{z_{i}^{-1}\tau w_{0}z_{j}\mid \tau \in \mathcal{T}, 1\leq i,j\leq 12\}\\
\tilde{c}_{0}^{j}&=\{z_{i}^{-1}\tau w_{0}z_{j}\mid \tau \in \mathcal{T}, 1\leq j\leq 12\}.
\end{align*}
where $w_{0}=w_{1,2}$. This should be understood in the following way. Let $w=z_{i}^{-1}\tau w_{0}z_{j}\in\tilde{c}_{0}$. The element $z_{j}$ indicate in which connected component of $\tilde{c}_{0}$ the element $w$ lies. Then $\tau$ indicates in which translate of the box $w$ lies and finally $z_{i}^{-1}$ indicates where in the translate of box $w$ lies. This is explained in Figure \ref{c0}.
\begin{figure}[h!]
\caption{Description of $\tilde{c}_{0}$}
\label{c0}
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$\ $\\

\vspace{7.3cm}
\noindent
We now have a look at the two-sided cell $\tilde{c}_{1}$. We set $w_{1}=s_{1}s_{2}s_{1}s_{2}s_{1}$. In Figure \ref{c1}, we show the elements of the set $\mathcal{Z}=\{z_{1},\ldots,z_{6}\}$: these are the elements which correspond to the alcoves in dark gray. We set  $t=s_{1}s_{2}s_{1}s_{2}s_{3}$ and
$$\mathcal{T}:=\{t^{n}|n\in \mathbb{N}\}.$$
Then we have
\begin{align*}
\tilde{c}_{1}&=\{z_{i}^{-1}\tau w_{1}z_{j}\mid \tau \in \mathcal{T}, 1\leq i,j\leq 6\}\\
\tilde{c}_{1}^{j}&=\{z_{i}^{-1}\tau w_{1}z_{j}\mid \tau \in \mathcal{T}, 1\leq j\leq 6\}.
\end{align*}
This is explained in Figure \ref{c1}.

\newpage

\begin{figure}[h!]
\caption{Description of the two-sided cell $\tilde{c}_{1}$}
\label{c1}
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$\ $\\

\vspace{5.7cm}
We now have a look at the two-sided cell $\tilde{c}_{2}$. We set $w_{2}=s_{1}s_{3}$. In Figure \ref{c2}, we show the elements of the set $\mathcal{Z}=\{z_{1},\ldots,z_{6}\}$: these are the elements which correspond to the alcoves in dark gray. We set  $t=s_{1}s_{3}s_{2}$ and
$$\mathcal{T}:=\{t^{n}|n\in \mathbb{N}\}.$$
Then we have
\begin{align*}
\tilde{c}_{2}&=\{z_{i}^{-1}\tau w_{2}z_{j}\mid \tau \in \mathcal{T}, 1\leq i,j\leq 6\}\\
\tilde{c}_{2}^{j}&=\{z_{i}^{-1}\tau w_{2}z_{j}\mid \tau \in \mathcal{T}, 1\leq j\leq 6\}.
\end{align*}
This is explained in Figure \ref{c2}.

\begin{figure}[h!]
\caption{Description of the two-sided cell $\tilde{c}_{2}$}
\label{c2}
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\vspace{7cm}
\noindent
We now have a look at $\tilde{c}_{3}$. We set $w_{\tilde{c}_{3}}=s_{1}$, $\mathcal{Z}=\{z_{1},z_{2},z_{3}\}=\{e,s_{2},s_{2}s_{3}\}$ and
$$\mathcal{T}:=\{e,s_{1}s_{2}\}.$$
Then we have
\begin{align*}
\tilde{c}_{3}&=\{z_{i}^{-1}\tau w_{1}z_{j}\mid \tau \in \mathcal{T}, 1\leq i,j\leq 3\}\\
\tilde{c}_{3}^{j}&=\{z_{i}^{-1}\tau w_{1}z_{j}\mid \tau \in \mathcal{T}, 1\leq j\leq 3\}.
\end{align*}

\bigskip

Finally, we have a look at $\tilde{c}_{4}=\{s_{2}s_{3}s_{2}\}$. In this case, if  we set $w_{\tilde{c}_{4}}=s_{2}s_{3}s_{2}$ and $\mathcal{Z}=\mathcal{T}=\{e\}$, the description in Section \ref{description} clearly holds.

redpajama_set_name: RedPajamaArXiv


text:
SOLAR POLICY FOR Dadra & Nagar Haveli
1. GROUND MOUNTED (GRID CONNECTED) SOLAR PV SYSTEMS
System size:
= 500 kWp
Voltage level:
Upto 10 kW, 230V- single phase
Above 10 kW and upto 25 kW, 400 V-Three phase
Above 25 kW and upto 100 kW, 400 V-Three phase
Above 100 kW, 11 kV-Three phase (HT)
Cumulative capacity of all solar systems installed in your area shall not exceed 30% of distribution transformer capacity2 in your area.
Ownership options
Self ownership (CAPEX3 model)
Third party ownership (RESCO4 model)
Half yearly (Sep & March)
Any unadjusted electricity credits shall be paid as per Solar Energy Tariff Order by JERC
Exempted from wheeling, banking charges & cross subsidy surcharges
2. ROOFTOP (GRID CONNECTED) SOLAR PV SYSTEMS
Min 1 kWp
Max = 500 kWp
40% Accelerated Depreciation
10 year Tax holiday on solar projects
Loans available for up to Rs.15 Crore for renewable energy projects under Priority sector lending
Dadra & Nagar Haveli Solar Policy and it's Solar Highlights
Dadra & Nagar Haveli is a Union territory located in the Western part of India and is close to the states of Gujarat and Maharashtra. The capital city of Dadra and Nagar Haveli is Silvassa, a rather popular tourist destination. Dadra & Nagar Haveli has been chosen as one of the Union Territories for the Power for All Programme run by the Government of India, with the objective to supply 24x7 uninterrupted power to all the category of consumers in the territory. It is interesting to note that Dadra & Nagar Haveli has the highest per capita power consumption in the country at 13,769 units as per FY 2015 against the national average of 1010 units. The primary reason behind this huge difference is that 97% of the power consumption in Dadra & Nagar Haveli is from the industries in the area. The UT has already planned to setup or procure 94.27MW solar energy and will be investing Rs 126.39 crores in order to generate this. Dadra & Nagar Haveli has got a target to install 449MW of solar capacity by the year 2022 as a part of the 100GW target of 2022. So far only, .60MW has been commissioned in 2017. Due to the small size of the Union Territory, the UT can setup PPA agreements with solar plants outside in order to achieve the target of 449MW that has been set.
Solar State Policy of Daman & Diu
Solar State Policy of Maharashtra
Solar State Policy of Gujrat
Solar State Policy of Goa

redpajama_set_name: RedPajamaCommonCrawl


A weather vane (or wind vane) is a tool for measuring wind instructions. It is an easy tool, containing an arrow, or any figure, mounted on a rod. When there is wind, this pole relocates in the instructions of the wind. The front part of the vane is more or less pointed as well as the backside is large, so that also the slightest breeze will removal the vane. The arrow transforms till the wind is similarly distributed on either side of the weather vane, making the front component suggest the exact instructions of the wind. Wind vanes are positioned in high and also open locations to capture the wind.
Weather vanes are one of the earliest wind tools made use of by guy. At one factor of time, these weather vanes were most likely utilized as flags, as the origin of the word "vane" appears to have come from the Anglo-Saxon word "fane", which implies flag.
Weather vanes believe been used to represent patriotic feelings as depicted by the Federal Eagle and also the Goddess of Freedom vanes. Weather vanes, also today, are used much more as ornaments on buildings than for any other certain purpose. Even office complex are showing off weather vanes with corporate logos engraved on them. They can be custom-made created, from the dimension of the vane, to the product utilized, and obviously the style. Modern wind vanes are likewise placed with an anemometer, a gadget for gauging the rate of the wind. They may also be linked to a remote analysis terminal. Modern weather vanes are corrosion free and also maintenance free.
Vintage weather vanes are so much in demand that there are some dealers who specialize in recreating weather vanes to look like vintages. Weather vanes are much more preferred for the "nation" or "ranch" look, they could add an elegant touch to any type of building. Weather vanes are additionally used as decorative products inside the residence or also on desktops.
Weather vanes are instruments made use of to identify the direction of the wind. Another vane made by the same master vane-maker was a huge copper Indian vane and also it was mounted in 1740 on Boston's Old North Church.
Vintage weather vanes are much and really rare sought after. Handmade ones are really tough to confirm. There is a terrific demand for those made in the manufacturing facilities which go back to the mid-to-late nineteenth century. Several of the most preferred weather vane makers were A.L. Jewell, J.W.Fiske, E.G.Washburne and Cushing and also White. Some unscrupulous dealers duplicate weather vanes to resemble antiques by coating them with pet manure, positioning them in dirt-filled boxes with seawater and also hiding them below ground for a couple of months. The antique appearance could likewise be replicated by treating them with copper sulphate, acetic acid as well as potassium sulphide.
Vintage weather vanes were also extremely experimented upon, and so we locate many one-of-a-kind items like the one made by Thomas Jefferson which had a shaft that went through the roofing right into the space below. Antique iron weather vanes are also offered in numerous kinds like: deck or fencing weather vanes, grass weather vanes, roof leading weather vanes, table top weather vanes, as well as extra.
There are certain standards for recognizing whether a weather vane is really an antique or not. Genuine antique vanes are much better than phony ones. Because antique vanes were normally covered with steel sheets, rather than being paintinged on, they may have some square boundaries of gilt staying. They would certainly have a smooth burnished finish rather of a shiny look. Because weather vanes were generally made use of for capturing practice in olden days, some genuine antique vanes may have bullet holes in them. There must be some dirt accumulated inside if it has crevices or holes. A good shake must disclose this. These are not fail-safe examinations for antique weather vanes, they can aid to understand at a look if the vane is truly antique or not. Antique weather vanes, like various other antique items, are in high need by enthusiasts.
A weather vane (or wind vane) is a tool for gauging wind instructions. Vintage weather vanes are so much in need that there are some dealerships who specialize in recreating weather vanes to look like vintages. Antique iron weather vanes are additionally readily available in numerous kinds like: deck or fence weather vanes, yard weather vanes, roof covering leading weather vanes, table leading weather vanes, as well as more. Because weather vanes were normally used for shooting practice in olden days, some real antique vanes might have bullet openings in them. These are not sure-fire examinations for antique weather vanes, they could help to recognize at a glimpse if the vane is really antique or not.


text:
First Breakthrough for Future Air-Breathing Magneto-Plasma Propulsion Systems
By Berkant Göksel - IB Göksel Electrofluidsystems, Managing Director
IB Göksel Electrofluidsystems announced the first breakthrough for future air-breathing magneto-plasma propulsion systems with the release of a new Star Wars movie in December 2015 and presented a first prototype at the ILA Berlin Air Show in June 2016. The first scientific paper was published in the Journal of Physics in April 2017. New results will be presented at the ILA Berlin Air Show in April 2018. It is envisioned that air-breathing plasma propulsion systems will enable future 'stratollite' airships and aircrafts to easily reach altitudes from the ground to 50 km and even beyond.
The working principle of the new pulsed plasma engine is based on Lorentz forcing through a set of arc discharges with self-induced magnetic fields. Arc discharges can have very high currents which again generate high magnetic fields. Now in our case, there are arc discharges running from at least six outer bar anodes to a center cathode. The right-hand rule for current-carrying wires might be known to most readers from our high-school physics class:
The thumb is the arc current from each of the bar anodes to the center cathode. The trigger finger is the induced tangential magnetic field. The middle finger is the direction of the Lorentz force which is pushing the ionized air towards a pinching focus point where all six force vectors meet and compress the ionized air plasma to pressures up to 100 bar and beyond. After the compression zone, there is an expansion to very high speeds. At ground level conditions, the speed is in the range of 3,300 ft/s (1,000 m/s). At higher altitudes, the plasma exhaust speed can reach values of 5-20 km/s. So, the final effect is very similar to what we have in a rocket (Laval) nozzle that we deal in principle with a kind of an electromagnetic pulse detonation rocket engine analogue.
The total current is very high and can reach values up to several tens of Kiloamperes, whereby the self-induced tangential magnetic fields are as strong as naturally generated by the best rare-earth neodymium magnets with 1.6-1.8 Tesla. With the innovative ignition method based on nanosecond fast high voltage pulse excitations to induce multiple self-organized plasma channels, it is now possible to drive the plasma engine from ground level up to 50 km and beyond. From this point it is a first breakthrough for high-thrust plasma engines which can be often seen in science-fiction movies.
Now imagine a hybrid subsonic flying wing with two types of engines: 1. Two electric aircraft engines with each 50 kilograms weight and 260 kilowatts of power as recently developed by Siemens, and 2. A multi-array of 100 trailing edge distributed air-breathing magneto-plasma flux compression thrusters with combined jet focusing nozzles and a total thrust of 112.4 lbf (500N) to be developed by IB Göksel Electrofluidsystems.
The two classical electric motors from Siemens with plasma noise and separation flow controlled propellers from IB Göksel Electrofluidsystems would be used to start and climb up to 50,000 ft (15.2 km). The second more futuristic engine would be used for a short but fast climb operation from 50,000 ft (15.2 km) to stratospheric altitudes from 85,000 ft (25.9 km) to even 100,000 ft (30.5 km) reaching speeds of 500 mph (800 km/h). The "shining" Plasma Stingray named "Sirius" would bring six space tourists to the edge of near space in less than an hour, switch-off the plasma pulse detonation engine in near space and come back as a glider or lifting body aerospace shuttle. Today, the most popular subsonic aircrafts which could fly at 80,000-85,000 ft are the Lockheed U-2 and Grob Strato-2C. In August 2001, the solar-powered Helios reached an official world record altitude of 96,863 ft (29.5 km) during a maximum-altitude flight for a non-rocket powered aircraft.
The Sirius could take off from any small airfield and would be driven by a compact aircraft fuel cell system with a power density of less than 2.2 lb/kW (1kg/kW) and 660 kW. Wing mounted flexible solar cells based on optical rectenna technology would provide additional 80-100 kW or at least 40 kW power using state-of-the-art solar cells with 30% efficiency.
Future supersonic and hypersonic Sirius systems with 50-100 MW plasma engines would use compact plasmas fusion reactors which will be the most environmentally friendly power units in the future and are already in development by leading aerospace companies. In principle, it is also possible to imagine future single-stage-to-orbit (SSTO) vehicles or aerospace planes as the magneto-plasma thrusters can be fed by gases like Argon and thus would also operate in deep space. For re-entry heat protection and high speed flight modes a second multi-array of thrusters distributed along the leading edge or nose section could work as counterflow magneto-plasma jet actuators comparable to NASA's LPM system in the Paradoxal Concept but without the need of compressed bleed air from mechanical engines. Let's imagine a future with plasma fusion powered plasma aircrafts and airships.
As Einstein told:
"Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand."


Actor Alden Ehrenreich was already having a huge year in Hollywood, with featured roles in the Coen Bros' hit Hail, Caesar! and the big Warren Beatty comeback drama Rules Don't Apply. But his profile shot through the stratosphere when he won the highly coveted role of young Han Solo in an upcoming Star Wars spin-off from Disney and Lucasfilm.
Now, the rising superstar is talking about his prospective co-stars and hopes for the film. The new movie, which is said to be the part of a potential trilogy within the "Star Wars Stories" subseries that arrive every two years, will take place several years prior to events of the original Star Wars: A New Hope and flesh out the backstory of the popular but often mysterious character originally portrayed by Harrison Ford.
Still unknown is what the storyline of the new film will follow. While Han, Lando and Chewbacca have their backgrounds kept largely vague in the original films, fans will recall that Solo and Calrissian were close friends at one point and that Lando was the original owner of the Millenium Falcon. Chewbacca is also described as owing Han a "life debt" for an unspecified act on his behalf, which many fans had expected to be touched on in the feature. Some details of Chewbacca's earlier life and extended family were offered by the Star Wars Holiday Special, but were later declared not part of the franchise's official canon.


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Q: Check if FORMAT is valid based on a pattern list Using URLs..
I have a list of acceptable url formats, like this:

*

*domain.com/test

*domain.com/test/

*domain.com/test/?test=1

*domain.com/test/?test=1&bla=2

User input this anydomain.com/any/?test=1 will return True based on my #3 entry at url format list..
The user input something like anydomain.com/any/asd/?test=1
return false.. domain.com/ return false, because does not match with any entry of my format list.
whats the best way to do this?
edit: im not trying to check if url is valid! i'm trying to check if FORMAT is valid based on my pattern list..

A: You can use filter_var function of php for valid format of url
if (filter_var($url, FILTER_VALIDATE_URL) === FALSE) {
    return FALSE:
}

NOTE
Valid url start from http:// OR https://

A: You probably need to use regular expressions to do that:
$validFormats = Array(
    'domain.com/test',
    'domain.com/test/',
    'domain.com/test/?test=(\d+)',
    'domain.com/test/?test=(\d+)&bla=(\d+)'
);

$urlToCheck = 'domain.com/test/?test=1';

$isValid = false;
foreach($validFormats as $format) {
    if(preg_match('#^'.$format.'$#i', $urlToCheck)) {
        $isValid = true;
        break;
    }
}

if($isValid) {
    echo 'Forbidden!';
} else {
    echo 'Welcome!';
}

A: You must use a regex like this :
function checkUrl($var) {
    if(is_array($var) || is_object($var)) {
        return false;
    }
    return (preg_match("`^([a-z0-9]+)\.([a-z0-9]+){2,4}/([a-z0-9]+)/?(\?[a-z0-9A-Z&=]+)?$`", $var)) ? true : false;
}
checkUrl("anydomain.com"); // false
checkUrl("anydomain.com/any"); // true
checkUrl("anydomain.com/any/"); // true
checkUrl("anydomain.com/any/?test=1"); // true
checkUrl("anydomain.com/any/?test=1&bla=2"); // true
checkUrl("anydomain.com/any/asd/?test=1"); // false

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