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\begin{document}
\def\smfbyname{}
\maketitle

\begin{abstract}
	We establish a generalised Springer correspondence for $\bfZ/m$-graded Lie algebras and degenerate double affine Hecke algebras, in the framework recently developed by Lusztig and Yun.
\end{abstract}

\section*{Introduction}
In the present chapter, we establish a generalised Springer correspondence in the case where the Lie algebra is $\bfZ / m$-graded, which was conjectured by Lusztig--Yun~\cite{LYIII}. The main result is~\autoref{theo:classification}, which confirms the multiplicity-one conjecture proposed in~\cite{LYIII} and can be viewed as a generalised Springer correspondence in the sense of Lusztig~\cite{lusztig84}\cite{lusztig88}, for certain degenerate double affine Hecke algebras (dDAHAs) with possibly unequal parameters. \par

In~\cite{LYIII}, the authors have attached for each admissible system $\xi$ an affine root system and a dDAHA $\bbH_{\xi}$. The main result of~\cite{LYIII} is the construction of an action of $\bbH_{\xi}$ on an infinite sum of the spiral induction from various spirals
\begin{eq}
	\bfI = \bigoplus_{\frakp_*} \bigoplus_k\pH^k\Ind_{\frakl_{d}\subset\frakp_{d}}^{\frakg_{\ubar d}}\scrC
\end{eq}
which is an ind-object in the category of $G_{\ubar 0}$-equivariant perverse sheaves on $\frakg^{\nil}_{\ubar d}$.
They have conjectured that for each simple constituent $\calS$ of $\bfI$, the $\bbH_{\xi}$-module $\Hom\left( \calS, \bfI \right)$ is irreducible. \autoref{theo:classification} confirms this conjecture. The case where $M$ is a maximal torus and $\scrC$ is trivial has earlier been studied by Vasserot in~\cite{vasserot05}. \par

Our strategy is very close to that of \cite{vasserot05}. We use the technique of convolution algebras developed in Chriss--Ginzburg~\cite{CG}, Lusztig \cite{lusztig88},\cite{lusztig95} and Evens--Mirkovi\'c~\cite{EM}.\par

Firstly, we can attach to the datum $(G, G_{\ubar 0}, \frakg, \frakg_*)$ a loop group $\rmL G$ and a loop algebra $\rmL \frakg$. There is an one-dimension torus $\bfC^{\times}$ acting on $\rmL G$ and $\rmL \frakg$, compatible with the adjoint action of $\rmL G$ on $\rmL\frakg$, such that the fixed points are given by $G_{\ubar 0}$ and $\frakg_{\ubar d}$. The quadruple $(M, M_0, \frakm, \frakm_*)$ defines a Levi subgroup $M$ of the loop group $\rmL G$. The subgroup $M$ and the $\bfZ$-graded Lie algebra $\frakm$ is stable under the $\bfC^{\times}$-action and the fixed points are given by $M_0$ and $\frakm_{d}$. Similarly, any spiral $\frakp_*$ define a $\bfC^{\times}$-stable parahoric subgroup $P\subseteq \rmL G$. \par

One would then like to imitate the proof of Lusztig~\cite{lusztig88} in the case of graded affine Hecke algebras, with $G$ replaced by $\rmL G$, parabolics replaced by parahorics and graded affine Hecke algebra's replaced by dDAHAs. One immediate obstacle is the infinite dimensionality of the geometric objects such as $\rmL G$, which prevents us from making free use of six operations for $\ell$-adic sheaves and perverse sheaves on related geometric objects, such as $\rmL\frakg$, the affine Springer resolution $\rmL\dot\frakg = \rmL G \times^P\frakp\to \frakg$ and the affine Steinberg variety $\rmL\ddot\frakg = \rmL\dot\frakg\times_{\rmL\frakg}\rmL\dot\frakg$. \par

As shown in~\cite{vasserot05}, the upshot is to consider directly the $\bfC^{\times}$-fixed points of those varieties. A suitable assumption on the $\bfC^{\times}$-actions makes the fixed points some infinite disjoint union of algebraic varieties of finite type. Moreover, the fixed point components can be described in terms of flag varieties, Springer resolutions and Steinberg varieties of some Levi subgroups. The fixed point components can be parametrised with combinatorial data, such as Coxeter complexes. The $\bbH_{\xi}$-action is then constructed by reduction to its parabolic subalgebras. Each proper parabolic subalgebra of $\bbH_{\xi}$ is a graded affine Hecke algebra, which is treated by Lusztig in~\cite{lusztig88} \cite{lusztig95}. It turns out that it suffices to define the action on proper parabolic subalgebras of $\bbH_{\xi}$, since the defining relations of $\bbH_{\xi}$ involve at most two simple reflections in the Weyl group. We will construct a specialised convolution algebra $\ha\calH$, which governs the infinite sum of perverse sheaves $\bfI$, and a homomorphism $\Phi:\bbH_{\xi}\to \ha\calH$. We will also show that the image of $\Phi$ is dense in a suitable product topology of $\ha\calH$. The conjecture of multiplicity-one then results from the density of the image of $\Phi$. \par

In~\autoref{sec:affineCox}, we recall the relative affine Weyl group and Coxeter complexes introduced in \cite{LYIII}.\par

In~\autoref{sec:steinberg}, we describe our principal geometric objects: varieties $\calX^{\nu, \nu'}$, $\calT^{\nu}$ and $\calZ^{\nu,\nu'}$, which play the role of fixed point components of $\left(\rmL G / P\right)\times \left(\rmL G / P\right)$, $\rmL \dot\frakg$ and $\rmL\ddot\frakg$, respectively.\par

In~\autoref{sec:convolution}, we define and study the specialised convolution algebra $\ha\calH$, which is equipped with a product topology. It can be viewed as the full convolution algebra specialised at an infinitesimal character $a$.\par

In~\autoref{sec:action}, we define a degenerate double affine Hecke algebra (dDAHA) $\bbH = \bbH_{\xi}$ and we construct a homomorphism $\Phi: \bbH\to \ha\calH$. We prove that the image of $\Phi$ is dense in~\autoref{subsec:density}.\par

In~\autoref{sec:simple}, we compare our construction with that of Lusztig--Yun~\cite{LYIII} and prove the multiplicity-one conjecture of {\it loc. cit.} We classify irreducible $\bbH$-modules with prescribed eigenvalues in~\autoref{theo:classification}. We also relate irreducible modules with torus fixed points of affine Springer fibres in~\autoref{theo:standard}.\par

\section{$\bfZ/m$-grading, spirals and splittings} \label{sec:grad}
We recall in this section the notion of spirals and splittings as defined in~\cite{LYI}.

\subsection{$\bfZ/m$-grading on $G$} \label{subsec:grad}
Let $G$ \index{$G$} be a connected simply connected simple algebraic group over $\bfC$. The Lie algebra is denoted $\frakg = \Lie G$. \index{$\frakg$}\par
 We fix a positive integer $m\in \bfN^*$. For any integer $k\in \bfZ$, we will denote $\ubar k = k \mod m\in \bfZ / m$. Let
\begin{eq}
	\frakg = \bigoplus_{\ubar i\in \bfZ / m}\frakg_{\ubar i}
\end{eq}
\index{$\frakg_{\ubar i}$}
be a $\bfZ/m$-grading on $\frakg$ such that $\left[ \frakg_{\ubar i}, \frakg_{\ubar j} \right]\subseteq \frakg_{\ubar i+\ubar j}$ for all $\ubar i, \ubar j\in \bfZ/m$. We define a homomorphism $\theta: \mu_m\to \Aut(\frakg)$ by setting
\begin{eq}
	\theta(\zeta)\mid_{\frakg_{\ubar j}} = \zeta^j, \quad \forall j\in \bfZ.
\end{eq}
The simplicity of $G$ implies that $\Aut(G)\cong \Aut(\frakg)$, so we can write $\theta: \mu_m\to \Aut(G)$. \par

We will assume throughout that $\theta$ is inner, meaning that the image of $\theta$ lands on the subgroup $\Ad(G)\subset \Aut(G)$.
\par

We fix once and for all a non-zero integer $d\in \bfZ_{\neq 0}$. The adjoint action of $G$ on $\frakg$ restricts to an action of the fix-point subgroup $G_{\ubar 0} = G^{\theta}\subset G$ on the graded piece $\frakg_{\ubar d}$. Let $\frakg_{\ubar d}^{\nil} = \frakg_{\ubar d}\cap \frakg^{\nil}$ be the closed subvariety of nilpotent elements. The geometric objects of interest are the $G_{\ubar 0}$ orbits in $\frakg_{\ubar d}^{\nil}$ and the $G_{\ubar 0}$-equivariant derived category $\rmD_{G_{\ubar 0}}\left( \frakg^{\nil}_{\ubar d} \right)$. \par

Let $\Pi\left( \frakg_{\ubar d} \right)$ denote the set of pairs $\left( \rmO, \scrL \right)$ where $\rmO\subset \frakg^{\nil}_{\ubar d}$ is a nilpotent $\Gz$-orbit and $\scrL\in \Loc_{G_{\ubar 0}}(\rmO)$ is an (isomorphism class of) irreducible $\Gz$-equivariant local system on $\rmO$.  \par

\subsection{Jacobson--Morosov theorem}\label{subsec:JM}
Recall the theorem of Jacobson--Morosov in the $\bfZ/m$-graded setting. Let $e\in \frakg_{\ubar d}^{\nil}$. According to~\cite[2.3]{LYI}, we can complete $e$ into an $\fraksl_2$-triple $\phi = \left( e, h, f \right)$ with $h\in \frakg_{\ubar 0}$ and $f\in \frakg_{-\ubar d}$. Consequently, there is a cocharacter $\lambda\in \bfX_*(G_{\ubar 0})$, the exponentiation of $h$, which we will denote by $\lambda: t\mapsto t^h$. Moreover, the set of such triple $\phi$ with a given $e$ forms a principal homogeneous space under the action of the unipotent part of the stabiliser of $e$ in $G_{\ubar 0}$. \par
Using the Jacobson--Morosov theorem, one can show that the number of $G_{\ubar 0}$-orbits in $\frakg^{\nil}_{\ubar d}$ is finite and moreover, each orbit is invariant by dilatation. We will also consier the action of the one-dimensional torus $\bfC^{\times}_q = \bfC^{\times}$ on $\frakg_{\ubar d}$, given by $(q, x)\mapsto q^{-2} x$.

\subsection{Spirals, nilpotent radical and splittings}\label{subsec:spirals}
Let $\lambda\in \bfX_*\left(G_{\ubar 0}\right)_{\bfQ}$ be a co-character and let $\epsilon = \pm 1$. We associate to it a $\bfZ$-graded Lie algebra $\prescript{\epsilon}{}\frakp^{\lambda}_* =\bigoplus_{n\in \bfZ}  \prescript{\epsilon}{}\frakp^\lambda_n$ where
\begin{eq}
	\prescript{\epsilon}{}\frakp^{\lambda}_n = \bigoplus_{\substack{r\in \bfQ \\r \ge \epsilon n}} \prescript{\lambda}{r}\frakg_{\ubar n}.
\end{eq}
We also define a $\bfZ$-graded Lie subalgebra of $\frakg$
\begin{eq}
	\prescript{\epsilon}{}\frakl^{\lambda}_* = \bigoplus_{n\in \bfZ} \prescript{\epsilon}{}\frakl^{\lambda}_n, \quad \prescript{\epsilon}{}\frakl^{\lambda}_n = \prescript{\lambda}{\epsilon n}\frakg_{\ubar n}.
\end{eq}
Such a graded sub-space $\prescript{\epsilon}{}\frakp^{\lambda}_*$ \index{$\frakp^{\lambda}_*$} is called an  {\bf $\epsilon$-spiral} of $\frakg$ and such an $\prescript{\epsilon}{}\frakl^{\lambda}_*$ \index{$\prescript{\epsilon}{}\frakl^{\lambda}_*$} is called a {\bf splitting} of the spiral $\prescript{\epsilon}{}\frakp^{\lambda}_*$. We let $\prescript{\epsilon}{}\frakl^{\lambda}$ be the same Lie algebra as $\frakl^{\lambda}_*$ with grading forgotten. \par

We define the $\bfZ$-graded Lie algebra $\prescript{\epsilon}{}\fraku^{\lambda}_*$ with
\begin{eq}
	\prescript{\epsilon}{}\fraku^{\lambda}_* = \bigoplus_{n\in \bfZ} \prescript{\epsilon}{}\fraku^{\lambda}_n ,\quad \prescript{\epsilon}{}\fraku^{\lambda}_n = \bigoplus_{r > \epsilon n} \prescript{\lambda}{r}\frakg_{\ubar n}
\end{eq}
\index{$\fraku^{\lambda}_*$}
to be the {\bf nilpotent radical} of the spiral $\prescript{\epsilon}{}\frakp^{\lambda}_*$. \par

Then $\prescript{\epsilon}{}\fraku^{\lambda}_*$ forms a homogeneous ideal of the graded Lie algebra $\prescript{\epsilon}{}\frakp^{\lambda}_*$ and that for each $n\in \bfZ$ the subspace $\prescript{\epsilon}{}\frakl^{\lambda}_n\subseteq \prescript{\epsilon}{}\frakp^{\lambda}_n$ is mapped isomorphically onto the quotient $\prescript{\epsilon}{}\frakp^{\lambda}_n / \prescript{\epsilon}{}\fraku^{\lambda}_n$ via the obvious projection.  The Lie algebra $\frakl^{\lambda}_*$ is the Lie algebra of a pseudo-Levi sub-group of $G$, ~\cite[2.2.5]{LYIII}.

We will write $\frakp^{\lambda}_*$, $\frakl^{\lambda}_*$ and $\fraku^{\lambda}_*$ instead of $\prescript{\epsilon}{}\frakp^{\lambda}_*$, $\prescript{\epsilon}{}\frakl^{\lambda}_*$ and $\prescript{\epsilon}{}\fraku^{\lambda}_*$ in a context where the signature $\epsilon$ is clear.

\subsection{Spiral induction and restriction}
With the datum $\left( \frakp_*, \frakl_*, \fraku_* \right) =\left( \prescript{\epsilon}{}\frakp^{\lambda}_*, \prescript{\epsilon}{}\frakl^{\lambda}_*, \prescript{\epsilon}{}\fraku^{\lambda}_* \right)$ of a spiral together with a splitting, we can define the functor of induction. Let $P_0 = \exp(\frakp_0)$ and $L_0 = \exp(\frakl_0)$.  \par

Fix an integer $d\in \bfZ\setminus\left\{ 0 \right\}$\index{$d$} and $\epsilon \in \left\{ 1, -1 \right\}$. We consider the following diagram
\begin{eq}
	\frakg_{\ubar d} \xleftarrow{\alpha} G_{\ubar 0}\times^{P_0}\frakp_{d}\xleftarrow{\beta} G_{\ubar 0}\times \frakp_{d}\xrightarrow{\gamma}  \frakl_{d},
\end{eq}
where
\begin{eq}
				\quad \alpha(g, x) = \Ad(g) x; \quad \beta(g, x) = (g, x); \quad \gamma(g, x) = x\mod{\fraku_{d}}.
\end{eq}
Then, the morphism $\alpha$ is proper whereas $\gamma$ is a smooth fibration and $\beta$ is a $P_0$-torsor.\par

Note that they depend on the sign $\epsilon\in \left\{ \pm 1\right\}$. Any such $\bfZ$-graded Lie algebra $\frakp_*^{ \lambda }$ is called a ($\epsilon$-){\bf spiral} of $\frakg$, the Lie subalgebra $\fraku^{\lambda}_*\subset \frakp^{\lambda}_*$ is called the {\bf unipotent radical}. There is a spiral version of the Levi decomposition $\frakp^\lambda_* = \frakl^\lambda_* \oplus \fraku^\lambda_*$. If $\lambda, \lambda'\in \bfX_*(G_{\ubar 0})_{\bfQ}$ are such that $\frakp^\lambda_* = \frakp^{\lambda'}_*$, then $\fraku^\lambda_* = \fraku^{\lambda'}_{*}$ but $\frakl^{\lambda}_*$ and $\frakl^{\lambda'}_*$ can be different. Any such subalgebra $\frakl^{\lambda}_*\subset\frakp^{\lambda}_*$ is called a {\bf splitting} of $\frakp^\lambda_*$.

We have the following sequences
\begin{eq}\label{equa:suite-indspirale}
	\frakl_{d}\xleftarrow{\alpha} \Gz\times^{U_{0}}\frakp_{d}\xrightarrow{\beta} \Gz\times^{P_{0}}\frakp_{d}\xrightarrow{\gamma} \frakg_{\ubar d}, \qquad
	\frakg_{\ubar d}\xleftarrow{\delta} \frakp_{d}\xrightarrow{\varepsilon} \frakl_{d}.
\end{eq}
The spiral induction and restriction are defined as
\begin{eq}
	\Ind^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} &= \gamma_*(\beta^!)^{-1}\alpha^!: \rmD_{L_{0}}\left( \frakl_{d} \right)\to \rmD_{\Gz}\left( \frakg_{\ubar d} \right) \\
	\Res^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} &= \varepsilon_!\delta^*: \rmD_{\Gz}\left( \frakg_{\ubar d} \right)\to \rmD_{L_{0}}\left( \frakl_{d} \right).
\end{eq}
In fact, for the sign $\epsilon = d / |d|$ the functors induce
\begin{eq}
	\Ind^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} \rmD_{L_{0}}\left( \frakl^{\nil}_{d} \right)\to \rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right), \quad \Res^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} : \rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right)\to \rmD_{L_{0}}\left( \frakl^{\nil}_{d} \right).
\end{eq}
Since every variety in the sequence~\eqref{equa:suite-indspirale} can be equipped with a $\Cq$-action (by weight -2 on the lie-algebraic part), there is also a $G_{\ubar 0}\times\Cq$-equivariant version of the induction and restriction
\begin{eq}
	\Ind^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} &: \rmD_{L_{0}\times\Cq}\left( \frakl_{d} \right)\to \rmD_{\Gz\times\Cq}\left( \frakg_{\ubar d} \right) \\
	\Res^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} &: \rmD_{\Gz\times \Cq}\left( \frakg_{\ubar d} \right)\to \rmD_{L_{0}\times\Cq}\left( \frakl_{d} \right).
\end{eq}

\subsection{Adjunction}\label{subsec:adjonction}
We can rephrase the definition of spiral induction and restriction as follows: there is a diagram of algebraic stacks
\begin{eq}
	\begin{tikzcd}
		\frakl_{d} \arrow{d}{\breve\beta} & \frakp_{d} \arrow{l}{\varepsilon} \arrow{d}{\bar\beta} & G_{\ubar 0}\times \frakp_{d} \arrow{l}{\alpha} \arrow{d}{\beta} \\
		\left[ \frakl_{d} / P_{0} \right]  & \arrow{l}{\bar\varepsilon} \left[ \frakp_{d} / P_{0} \right] \arrow{d}{\bar\gamma} & G_{\ubar 0}\times^{P_{0}} \frakp_{d} \arrow{d}{\gamma} \arrow{l}{\bar \alpha}  \\
		\left[ \frakl_{d} / L_{0} \right] \arrow{u}{q} & \left[ \frakg_{\ubar d} / G_{\ubar 0} \right] & \frakg_{\ubar d} \arrow{l}{\breve \alpha}
	\end{tikzcd}
\end{eq}
Then $\Ind^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} = \bar\gamma_*\bar\varepsilon^! q_* $ and $\Res^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}}=  q^* \bar\varepsilon_!\bar\gamma^*$. \par
It is clear that $\left( \Res^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}}, \Ind^{\frakg_{\ubar d}}_{\frakl_{d}\subset \frakp_{d}} \right)$ is a pair of adjoint functors.

\subsection{Admissible systems and block decomposition}\label{subsec:blockdecomp}
Recall that an admissible system on $\frakg_{\ubar d}$, as defined in~\cite{LYI}, is a datum $\left(M, \frakm_*, \rmO, \scrC  \right)$, where $M\subset G$ is a subgroup whose Lie algebra is equipped with a $\bfZ$-grading $\frakm_* = \bigoplus_{n\in \bfZ} \frakm_n$ arising as $\frakl^{\lambda}_*$ as splitting of a spiral, $\rmO\subset \frakm^{\nil}_d$ is a $M_0$-orbit and $\scrC\in \Loc_{M_0}(\rmO)$ is a cuspidal local system in the sense of~\cite{lusztig95b}. \par

An isomorphism of admissible systems $(M, \frakm_*, \rmO, \scrC)\cong (M', \frakm'_*, \rmO', \scrC')$ is a pair $(g, \eta)$ of element $g\in \Gz$ such that $gMg^{-1} = M'$, $\Ad_g \frakm_n = \frakm'_n$, $\Ad_g\rmO = \rmO'$ and an isomorphism $\eta : g^* \scrC'\cong \scrC$.  Let $\frakT\left(\frakg_{\ubar d}\right)$ denote the groupoid of admissible systems on $\frakg_{\ubar d}$ and let $\ubar\frakT\left(\frakg_{\ubar d}\right)$ denote the set of isomorphism classes its objects. \par

Let $\zeta = (M, \frakm_*, \rmO, \scrC)\in \ubar\frakT\left( \frakg_{\ubar d} \right)$ be an admissible system on $\frakg_{\ubar d}$. We define $\Perv_{\Gz}\left( \frakg_{\ubar d}^{\nil} \right)_{\zeta}$ to be the Serre subcategory of $\Perv_{\Gz}\left( \frakg_{\ubar d}^{\nil} \right)$ generated by the constituents of $\Ind^{\frakg_{\ubar d}}_{\frakm_d\subset \frakp_d}\IC\left( \scrC \right)$ for various spiral $\frakp_*$ which has $\frakm_*$ as splitting. We define $\rmD_{\Gz}\left( \frakg_{\ubar d}^{\nil} \right)_{\zeta}$ to be the full subcategory of $\rmD_{\Gz}\left( \frakg_{\ubar d}^{\nil} \right)$ of objects $\scrK$ such that $\prescript{p}{}\scrH^k\scrK\in \Perv_{\Gz}\left( \frakg_{\ubar d}^{\nil} \right)_{\zeta}$ for all $k\in \bfZ$. We also let $\Pi\left( \frakg_{\ubar d} \right)_\zeta\subset \Pi\left( \frakg_{\ubar d} \right)$ be the subsets of pairs $\left( \rmO, \scrL \right)$ such that $\IC\left( \scrL \right)\in \Perv\left( \frakg^{\nil}_{\ubar d} \right)_\zeta$. We will call these subcategories and subsets the {\bf blocks} of $\zeta$. \par

According to~\cite[0.6]{LYI}, there are decompositions into subcategories
\begin{eq*}\label{eq:dcpoth}
	\Perv_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right) = \bigoplus_{\zeta\in \ubar\frakT\left( \frakg_{\ubar d} \right)} \Perv_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right)_{\zeta}, \qquad \rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right) = \bigoplus_{\zeta\in \ubar\frakT\left( \frakg_{\ubar d} \right)} \rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right)_{\zeta}.
\end{eq*}

\subsection{}

The following is an immediate consequence of the decompositions~\eqref{eq:dcpoth}.
\begin{lemm}
	Let $\frakp_* = \frakp^{\lambda}_*$ and $\frakl_* = \frakl^{\lambda}_*$ be spiral and splitting as above. Let $\zeta = (M, \frakm_*,\scrC)$ be an admissible system on $\frakl_d$. Then the spiral induction and restriction preserve series:
	\begin{eq}
		\Ind^{\frakg_{\ubar d}}_{\frakl_d\subset \frakp_d}: \rmD_{L_0}\left( \frakl^{\nil}_d \right)_{\zeta}\to \rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right)_{\zeta} \\
		\Res^{\frakg_{\ubar d}}_{\frakl_d\subset \frakp_d}: \rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right)_{\zeta}\to \rmD_{L_0}\left( \frakl^{\nil}_d \right)_{\zeta}.
	\end{eq}
	On the other hand, if $\zeta' = \left( M', \frakm'_*, \scrC' \right)$ is an admissible sysmtem on $\frakg_{\ubar d}$ such that no $\Gz$-conjugate of $(M', \frakm'_*)$ is a $\bfZ$-graded Levi of $(L, \frakl_*)$, then $\Res^{\frakg_{\ubar d}}_{\frakl_d\subset \frakp_d}\rmD_{\Gz}\left( \frakg^{\nil}_{\ubar d} \right)_{\zeta'} = 0$.
\end{lemm}
\qed

\subsection{Spriral attached to a nilpotent element}
Let $\phi =\left( e, h, f \right)$ be an $\fraksl_2$-triplet with $e\in \frakg_{\ubar d}$, $h\in \frakg_{\ubar 0}$ and $f\in \frakg_{-\ubar d}$. We can attach to $\phi$ a spiral $\frakp^{\phi}_*$ and a splitting $\frakl^{\phi}_*$ as follows: $\phi$ can be integrated to a homomorphism of algebraic groups $\phi: \SL_2 \to G$. The cocharacter $\lambda\in \bfX_*(\Gz)$ defined by $\lambda(t) = \phi\begin{pmatrix}t & 0 \\ 0 & t^{-1}\end{pmatrix}$ gives rise to a spiral $\frakp^{(d/2)\lambda}_*$, denoted by $\frakp^{\phi}_*$, and a splitting $\frakl^{(d/2)\lambda}_*$, denoted by $\frakl^{\phi}_*$. The algebraic subgroups $P^{\phi}_0 = \exp\left( \frakp^{\phi}_0 \right), L^{\phi} = \exp\left( \frakl^{\phi}_* \right)$ and $L^{\phi}_0 = \exp\left( \frakl^{\phi}_0 \right)$ of $G$ are well-defined. They are called {\bf spiral}, and respectively {\bf splitting}, {\bf attached to $\phi$}. \par
It is clear that $e\in \frakl^{\phi}_d$, $h\in \frakl^{\phi}_0$ and $f\in \frakl^{\phi}_{-d}$. We will need the following results, proven in~\cite[2.9]{LYI}:
\begin{lemm}
	\begin{enumerate}
		\item
			The image of $\gamma$ is $\ba\rmC$, the Zariski closure of $\rmC$. Moreover, the morphism $\gamma$ restricts to an isomorphism $\gamma^{-1}(\rmC)\cong \rmC$.
		\item
			There is an isomorphism $\pi_0\left( Z_{L_0}(e) \right)\cong \pi_0\left( Z_{\Gz}(e) \right)$.
	\end{enumerate}
\end{lemm}
\qed \par

\section{Affine root system attached to $\frakg$}\label{sec:affineCox}
This section is a recall of the definition of the affine alcove complex defined in \cite{LYIII}, which turns out to be isomorphic to the (geometric realisation of the) Coxeter complex for the affine Weyl group. \par

\subsection{$\bfZ/m$-graded Lie algebras} \label{subsec:liegraduee}
We retain the setting of~\autoref{sec:grad}. \par

We assume the $\bfZ/m$-grading is induced by some co-character $\til\theta\in \bfX_*\left( G^{\ad} \right)$ \index{$\til\theta$} in the sense that
\begin{eq}
	\frakg_{\ubar i} = \bigoplus_{\substack{k\in \bfZ \\ \ubar i = \ubar k}}\prescript{\til\theta}{k}{\frakg}.
\end{eq}
We say that $\til\theta$ is a {\bf lifting} of the $\bfZ/m$-grading on $\frakg$ or that $\til\theta$ lifts the $\bfZ/m$-grading $\theta$ on $\frakg$. Let $G_{\ubar 0} = \exp\left( \frakg_{\ubar 0} \right)\subseteq G$\index{$G_{\ubar 0}$}, which is a reductive subgroup of the same rank as $G$. This assumption makes the affine root system, defined in~\autoref{subsec:affroot}, untwisted. We fix once and for all a lifting $\til\theta$. \par

The torus $\Cq$ acts linearly on the Lie algebra $\frakg$ by weight $-2$.  Let $u\in \bfX^*\left( \Cq \right)$ denote the defining character.  \index{$u$}\par

We will use a subscript $q$ to indicate a direct product with $\Cq$, such as $G_{\ubar 0, q} = G_{\ubar 0}\rtimes \Cq$\index{$G_{\ubar 0, q}$}. \par

Besides, we fix a maximal torus $T\subseteq G$ \index{$T$} centralised by $\til\theta$ so that $\til\theta\in \bfX_*\left( T / Z_G \right)$. In particular, $T$ is in $G_{\ubar 0}$.

\subsection{Affine root hyperplane arrangement}\label{subsec:affroot}
Let $\bbA$ denote the vector space $\bfX_*(T)\otimes \bfQ$. Since $T$ and the adjoint torus $T^{\ad} = T / Z_G$\index{$T^{\ad}$} are isogenous, we may regard $\til\theta/m$ as a point of $\bbA$. We will denote $\bfx = \til\theta/m$.\index{$\bfx$}
 The root system of $G$ is a subset $R\subset \bfX^*(T)\subset \bbA^*$, viewed as linear functions on $\bbA$. We define $S = R\times\bfZ$ to be the affine root system of $G$.\index{$S$} Each affine root $a = (\alpha, n)\in S$ is a non-constant affine function on $\bbA$. For $a\in S$, the zero of this linear function is affine hyperplane of $\bbA$, denoted by $H_{a}$. Let $\frakH = \left\{ H_{a}\;;\; a \in S \right\}$\index{$\frakH$} be the collection of such affine hyperplanes. The affine hyperplanes yield a stratification of $\bbA$ into {\bf facets} (i.e. subsets of $\bbA$ determined by a fintie number of equations $a = 0$ or $a > 0$ with $a\in S$). Let $\frakF$\index{$\frakF$} denote the collection of facets. The facets of maximal dimension are called {\bf alcoves}. The set of alcoves is denoted by $\frakA$\index{$\frakA$}. \par

The affine Weyl group (with respect to $T$) is given by $W_S = W(G, T)\ltimes \bfX_*(T)$\index{$W$}, which acts on $\bbA$ by reflections and translations. This action preserves the collection $\frakH$, inducing thus an action on $\frakF$. The restriction of the action to $\frakA\subseteq \frakF$ is simply transitive. We will abbreviate $W = W_S$. \par

Let $\frakE$ be the collection of affine subspaces of $\bbA$ which are non-empty intersection of a finite subset of $\frakH$. Elements of $\frakE$\index{$\frakE$} are called {\bf relevant subspaces}. \par

Let $W_{\bfx} = \Stab_{W}\left( \bfx \right)$\index{$W_{\bfx}$} be the stabiliser of $\bfx\in \bbA$. The rational co-character $\bfx$ gives rise to an identification $W_{\bfx} \cong W\left(G_{\ubar 0}, T\right)$ by identifying $\bfx$ as origin.

\subsection{Simple reflections and subsets}\label{subsec:srefl}
Given a alcove $\kappa\in \frakA$, let $\Delta^{\kappa}\subseteq S$ be the corresponding set of affine simple roots. If we consider the injective map $\Delta^{\kappa}\hookrightarrow W$,\quad $a\mapsto s_a$, then, the pair $\left(W, \Delta^{\kappa}\right)$ forms a Coxeter system. \par
Simple $W$ acts simply transitively on the set of alcoves $\frakA$, given any two alcoves $\kappa,\kappa'\in \frakA$, there is a unique $w\in W$ such that $\kappa' = w\kappa$. Then, $w$ yields isomorphisms of Coxeter systems
\begin{eq}
	(W, \Delta^{\kappa}) &\cong \left( W, \Delta^{\kappa'} \right) \\
	y\mapsto wyw^{-1},&\quad  s_a \mapsto w s_a w^{-1} = s_{w a}.
\end{eq}
Thus we can define $\left(\calW^{\can}, \Delta\right)$ to be the {\bf canonical Weyl group}: for any $\kappa\in \frakA$, there is a canonical isomorphism $\left(\calW^{\can}, \Delta^{\can}\right) \cong \left(W, \Delta^{\kappa}\right)$\index{$\calW^{\can}$}\index{$\Delta^{\can}$}. \par

For any subset $J^{\can}\subseteq \Delta^{\can}$, we will denote $\left(W_{J^{\can}}, J^{\can}\right)$ \index{$W_{J^{\can}}$} the Coxeter sub-system of $\left(\calW^{\can}, \Delta^{\can}\right)$ generated by $J^{\can}$. For any alcove $\kappa\in \frakA$, we denote $\left(W_{J^{\kappa}}, J^{\kappa}\right)\subseteq \left(W, \Delta^{\kappa}\right)$ the Coxeter sub-system which is the image of $\left( W_{J^{\can}}, J^{\can} \right)$ in $(W, \Delta^{\kappa})$ under the canonical isomorphism $\left(\calW^{\can}, \Delta^{\can}\right)\cong \left(W, \Delta^{\kappa}\right)$.  \par

Given any $\kappa\in \frakA$, there is a natural identification of $\left( \frakF, \le \right)$ with the Coxeter complex of $(W, \Delta^{\kappa})$. The cell $yW_{J^{\kappa}}$ in the Coxeter complex is identified with the facet $y\left(\partial_{J^{\can}}\kappa\right)\subseteq \bbA$, where
\begin{eq}
	\partial_{J^{\can}}\kappa = \left\{y\in \bbA\;;\;\begin{split}& a(y) = 0, & \forall a\in J^{\kappa} \\ & a(y)  > 0, & \forall s_{a}\in \Delta^{\kappa}\setminus J^{\kappa}\end{split}\right\}.
\end{eq}
\index{$\partial_{J^{\can}}$}
Here $s_{a}$ means the reflection with respect to the hyperplane $H = \left\{y\in \bbA\;;\; a(y) = 0\right\}$.
\par

Given two alcoves $\kappa, \kappa'\in \frakA$. Let $w\in W$ be such that $w\kappa = \kappa'$. Then for any $J^{\can}\subsetneq \Delta^{\can}$ we have $y\left(\partial_{J^{\can}}\kappa'\right) = yw\left(\partial_{J^{\can}}\kappa\right)$. Therefore, to each facet $y\left(\partial_{J^{\can}}\kappa\right)\in \frakF$, there is a proper subset $J^{\can}\subsetneq \Delta^{\can}$ attached, independent of the alcove $\kappa\in \frakA$. The subset $J^{\can}\subseteq \Delta^{\can}$ is called the {\bf type} of the facet $y\left(\partial_{J^{\can}}\kappa'\right)\in \frakF$. Let $\frakF_{J^{\can}}\subseteq \frakF$ denote the set of facets of type $J^{\can}$.
\par

For any pair of proper subsets $J^{\can}\subseteq K^{\can}\subsetneq \Delta^{\can}$, there is a boundary map $\partial_{K^{\can}}: \frakF_{J^{\can}}\to \frakF_{K^{\can}}$ which sends any facet $\nu$ of type $J^{\can}$ to its sub-facet of type $K^{\can}$. Then $W_{J^{\kappa}}$ \index{$W_{J^{\kappa}}$} is equal to the stabiliser of $\partial_{J^{\can}}\kappa$ in $W$.
\par

\subsection{Correspondence between spirals and facets} \label{subsec:spiralfacet}
We fix the sign $\epsilon = d / |d|$ which is required in the definition of spirals. Let $\frakP_T$ be the set of spirals $\frakp_*$ such that $\frakp_* = \pre{\epsilon}\frakp_*^{\lambda}$ for some $\lambda\in \bfX_*\left( T \right)_{\bfQ}$, \cfauto{subsec:spirals}. There is a bijection
\begin{eq}
	\frakF \cong \frakP_T.
\end{eq}
Given a facet $\nu\in \frakF$, we choose a point $y\in \nu$ and set $\lambda_y = m\epsilon\left( \bfx - y \right) = \epsilon\left( \til\theta - my \right)\in \bfX_*\left( T \right)_{\bfQ}$. It gives rise to the spiral $\pre{\epsilon}\frakp^{\lambda_y}_*$, which does not depend on the choice of $y\in \nu$, see \cite[3.4.4]{LYIII}. \par

We will denote $\frakp^{\nu}_* = \pre{\epsilon}\frakp^{\lambda_y}_*$ as well as $\fraku^{\nu}_* = \pre{\epsilon}\fraku^{\lambda_y}_*$ and $\frakl^{\nu}_* = \pre{\epsilon}\frakl^{\lambda_y}_*$ for any $y\in \nu$.  We will also denote $P^{\nu}_0 = \exp\left( \frakp^{\nu}_0 \right)$, $L^{\nu} = \exp\left( \frakl^{\nu} \right)$, $L^{\nu}_0 = \exp\left( \frakl^{\nu}_0 \right)$ and $U^{\nu}_0 = \exp\left( \fraku^{\nu}_0 \right)$. Those are subgroups of $G$.

\subsection{Graded pseudo-Levi attached to relevant affine subspaces} \label{subsec:subsp-pslevi}
Let $\frakM^{\bfZ-\gr}_T$ be the set of quadruplets $(M, M_0, \frakm, \frakm_*)$ with $M$ a pseudo-Levi subgroup of $G$ containing the maximal torus $T$ and $\frakm_*$ a $\bfZ$-grading on $\frakm = \Lie M$ which makes $\frakm$ a graded Lie algebra such that $T \subseteq M_0$, where $M_0 = \exp\left( \frakm_0 \right)$. There is an injection
\begin{eq}
	 \frakE\hookrightarrow \frakM^{\bfZ-\gr}_T
\end{eq}
defined as follows: Given any relevant subspace $\bbE\in\frakE$, choosing any facet $\nu\in \frakF$ which spans $\bbE$ as affine subspace, if we set
\begin{eq}
	\left(M^\bbE, M^\bbE_0,  \frakm^\bbE, \frakm^\bbE_*\right) = \left(L^{\nu} , L^{\nu}_0,\frakl^{\nu}, \frakl^{\nu}_* \right),
\end{eq}
then the splitting $\left(M^{\bbE}, M^{\bbE}_0, \frakm^{\bbE}, \frakm^{\bbE}_*\right)$ does not depends on the choice of $\nu$, \cite[3.4.7]{LYIII}.

\subsection{Cuspidal local system and relative Weyl group} \label{subsec:cusp}
Let $(M, M_0, \frakm,\frakm_*)\in \frakM^{\bfZ-\gr}_T$\index{$M,M_0, \frakm, \frakm_*$}. Assume that there is an $M$-equivariant cuspidal local system $\scrC$\index{$\scrC$} on a nilpotent orbit $\rmO \subseteq \frakm$ such that $\rmO_{d} = \rmO\cap \frakm_{d}\neq \emptyset$. Suppose that the sextuple $\xi = \left(M, M_0, \frakm,\frakm_*, \rmO, \scrC  \right)$\index{$(M, M_0, \frakm,\frakm_*, \rmO, \scrC)$} is an admissible system~\cite[3.1]{LYI}. In particular, $\scrC\mid_{\frakm_{d}}$ is an irreducible $M_0$-equivariant local system on $\rmO_{d}$~\cite[4.2(c),4.4(a)]{lusztig95b}. Let $(e, h, f)$ be an $\fraksl_2$-triple with $e\in \rmO_{d}$, $h\in \frakm_0$ and $f\in \frakm_{-d}$ and let $\varphi\in \bfX_*\left( G_{\ubar 0} \right)$ be given by the Jacobson--Morosov theorem so that $\varphi$ acts on $e$ with weight $2$. Consider the obvious inclusion $Z_{M}(e)\hookrightarrow Z_{M_q}(e)$, which induces $\pi_0\left(Z_{M}(e)\right)\to \pi_0\left(Z_{M_q}(e)\right)$. The last map is an isomorphism, since
\begin{eq}
	Z_{M}(e)\times \bfC^{\times} &\cong Z_{M\times \Cq}(e) \\
	\left( g, q \right)&\mapsto \left( g \til\theta(q), q \right)
\end{eq}
and since $\pi_0\left(Z_{M}(e)\times \bfC^{\times}\right)\cong\pi_0\left(Z_{M}(e)\right)$. It follows that the $M$-equivariant local system $\scrC$ is naturally equipped with an $M_q$-equivariant structure.  Likewise, $\scrC\mid_{\frakm_{d}}$ is equipped with an $M_q$-equivariant structure.
We fix such a system for the rest of the article. \par

According to~\cite[3.1(d)]{LYI}, there exists $\nu\in \frakF$ such that $\frakm_* = \frakl^\nu_*$. Let $\bbE^M\in \frakE$\index{$\bbE^M$} be the relevant affine subspace spanned by $\nu$ and let $W_{\bbE^M}\subset W$\index{$W_{\bbE^M}$} be the subgroup which fixes $\bbE^M$ pointwise. \par

The affine root hyperplanes in $\frakH$ induce, by intersection with $\bbE^M$, a collection of {\bf $\bbE^M$-affine root hyperplanes}
\begin{eq}
	\frakH(\bbE^M) = \left\{ H\cap \bbE^M ; H\in \frakH,\; H\cap \bbE^M\;\text{is a hyperplane in $\bbE^M$} \right\}\subseteq \frakE.
\end{eq}
One defines similarly the collection of {\bf $\bbE^M$-facets} $\frakF(\bbE^M)\subseteq \frakF$\index{$\frakF(\bbE^M)$}, as well as {\bf $\bbE^M$-alcoves} $\frakA\left( \bbE^M \right)\subseteq\frakF$\index{$\frakA(\bbE^M)$} and {\bf relevant $\bbE^M$-subspaces} $\frakE\left( \bbE^M \right)\subseteq\frakE$\index{$\frakE(\bbE^M)$}. \par

Let $I^{\can}\subsetneq \Delta$ be the type of $\nu$ so that $\nu\in \frakF_{I^{\can}}$. Let $\kappa\in \frakA$ be a alcove such that $\nu = \partial_{I^{\can}}\kappa$.

Let $J^{\can}\subsetneq \Delta$ be a proper subset containing $I^{\can}$. We denote $\sigma = \partial_{J^{\can}}\kappa\in\frakF\left( \bbE^M \right)$, which is a sub-facet of $\nu$. Consider the datum of graded pseudo-Levi $\left(L^{\sigma}, L^{\sigma}_0, \frakl^{\sigma}, \frakl^{\sigma}_*\right)$ such that $W\left( L^{\sigma}, T \right) \cong W_{J^{\kappa}} = \Stab_{W}(\sigma)$. Moreover, since $\left( M, M_0, \frakm, \frakm_* \right) = \left( L^{\nu}, L^{\nu}_0, \frakl^{\nu}, \frakl^{\nu}_* \right)$, $M$ is a Levi subgroup of $L^{\sigma}$ and there is a nilpotent $M$-orbit $\rmO\subseteq \frakm$ which supports a cuspidal local system $\scrC$ and an isomorphism.
Let $w_0^{J^{\kappa}} \in W_{J^{\kappa}}$ be the longest element. A theorem of Lusztig~\cite[9.2]{lusztig84},~\cite[2.5]{lusztig88} asserts that $w^{J^{\kappa}}_0$ stabilises the Levi subgroup $M$, the nilpotent orbit $\rmO\subseteq \frakm$ as well as the cuspidal local system on $\scrC$. In particular, $w^{J^{\kappa}}_0\in N_{W}\left( W_{\bbE^M} \right)$.\par
Under the canonical isomorphism $\left( W, \Delta^{\kappa} \right)\cong\left( \calW^{\can}, \Delta^{\can} \right)$, $w^{J^{\kappa}}_0$ is sent to $w^{J^{\can}}_0$. We see that $w^{J^{\can}}_0\in N_{\calW^{\can}}\left( \calW_{I^{\can}} \right)$. Therefore, the results of~\autoref{sec:relcox} is applicable to $(\calW^{\can}, \Delta^{\can})$ and $I^{\can}$. In particular, \autoref{theo:strrelcox} yields the following results.

\begin{theo}\label{theo:coxeterite}
	\begin{enumerate}[label=(\roman*)]
		\item
			In the Coxeter system $\left(\calW^{\can}, \Delta^{\can}\right)$, the shortest coset representatives for the elements of the quotient $N_{\calW^{\can}}\left( \calW_{I^{\can}} \right) /\calW_{I^{\can}}$ form a subgroup $\til \calW^{\can}$ of $N_{\calW^{\can}}\left( \calW_{I^{\can}} \right)$, which maps isomorphically onto $N_{\calW^{\can}}\left( \calW_{I^{\can}} \right) / \calW_{I^{\can}} $ under the quotient map. Moreover, if we set
			\begin{eq}
				\til \Delta^{\can} = \begin{cases}\emptyset &\text{if }\#\left( \Delta^{\can}\setminus I^{\can}\right) = 1, \\ \left\{ w^{I^{\can}\cup \{s\}}_0w^{I^{\can}}_0\in \til \calW^{\can}\;;\; s\in \Delta^{\can}\setminus I^{\can} \right\}& \text{if } \#\left( \Delta^{\can}\setminus I^{\can}\right)> 1,  \end{cases}
			\end{eq}
			then $\left( \til \calW^{\can}, \til \Delta^{\can} \right)$ forms a Coxeter system.
		\item
			If $\til\ell: \til \calW^{\can}\to \bfN$ denotes the length function on $\left( \til \calW^{\can}, \til \Delta^{\can} \right)$, then for any elements $w, y\in \til \calW^{\can}$, we have
			\begin{eq}
				\ell(w) + \ell(y) = \ell(wy)\quad \text{if and only if} \quad \til\ell(w) + \til\ell(y) = \til\ell(wy).
			\end{eq}
	\end{enumerate} \hfill\qedsymbol
\end{theo}
The {\bf canonical relative affine Weyl group} is defined to be the Coxeter system $\left(\til \calW^{\can}, \til \Delta^{\can}\right)$ of the above theorem.\index{$\til \calW^{\can}$}\index{$\til \Delta^{\can}$} \par

Then~\autoref{theo:fixed} in the present situation can be restated as

\begin{theo}\label{theo:sametype}
	The $\bbE^M$-alcoves in $\frakA\left( \bbE^M \right)\subseteq \frakF$ are of the same type $I^{\can}$. \hfill\qedsymbol
\end{theo}
This theorem justifies the notation $I^{\can}$ without $\kappa$ involved.

\subsection{Conjugates of $\xi$}\label{subsec:conjxi}

We keep the admissible system $\xi$ as in~\autoref{subsec:cusp} and the type $I^{\can}$. \par

By the identification $W\left( G_{\ubar 0}, T \right)\cong W_{\bfx}$, we can choose for each $w\in W_{\bfx}$ a lifting $\dot w\in N_{G_{\ubar 0}}\left( T \right)$. The adjoint action $\Ad_{\dot w}: \frakg\to \frakg$ sends $\rmO\subseteq \frakm$ to a nilpotent $\dot wM\dot w^{-1}$-orbit $\Ad_{\dot w}\rmO\subseteq \Ad_{\dot w} \frakm$. Since these subsets do not depend on the choice of the lifting, we denote them by $\prescript{w}{}M$, $\prescript{w}{}\frakm$ and $\prescript{w}{}\rmO$. The induced isomorphism of stacks $\Ad_{\dot w}: \left[ M\backslash \frakm \right]\cong \left[  \prescript{w}{}M\backslash \prescript{w}{}\frakm  \right]$ does not depend on the lifting $\dot w$ either. \par

The $M$-equivariant cuspidal local system $\scrC$ on $\rmO\subseteq \frakm$ can be viewed as a sheaf on the quotient stack $\left[ M\backslash\rmO \right]$. For $w\in W_{\bfx}\cap N_{W}(W_{\bbE^M})$, the isomorphism $\Ad_{\dot w}: \left[ M\backslash \frakm \right]\cong \left[  M\backslash \frakm  \right]$ is the identity morphism of $\left[ M\backslash \frakm \right]$. The result of Lusztig~\cite[9.2]{lusztig84},~\cite[2.5]{lusztig88} show that there is a canonical isomorphism $\Ad_{\dot w}^*\scrC\cong \scrC$ and it yields an action of $W_{\bfx}\cap N_{W}\left( W_{\bbE^M} \right)$ on $\scrC$, considered as a sheaf on $\left[M\backslash\rmO\right]$. \par

Let $\Xi$\index{$\Xi$} be the subset $W_{\bfx}\cdot \frakA(\bbE^M)$ of $\frakF$. By \autoref{theo:sametype}, all facets in $\Xi$ are of type $I^{\can}$, so that $\Xi\subseteq \frakF_{I^{\can}}$. \par

Let $\nu\in \Xi$. There exists $w\in W_{\bfx}$ such that $L^{\nu} = \prescript{w}{} M$. We define $\rmO^{\nu} = \prescript{w}{}\rmO\subseteq \frakl^{\nu}$ and $\scrC^{\nu} = \Ad_{\dot w}^*\scrC$\index{$\scrC^{\nu}$}\index{$\rmO^{\nu}$}, which is a $\ba\bfQ_{\ell}$-local system on $\left[ L^{\nu}\backslash \rmO^{\nu} \right]$. Clearly, for different choices of $w$, the resulting $\scrC^{\nu}$'s are canonically isomorphic. The $L^{\nu}$-equivariant local system $\scrC^{\nu}$ can be endowed with an $L^{\nu}_{q}$-equivariant structure in a unique way. We will sometimes suppress the index $\nu$ from the notation $\scrC^{\nu}$ if it is clear in the context.  \par

Denote $W_{\nu} = \Stab_{W}(\nu)$\index{$W_{\nu}$}. The non-canonical {\bf relative Weyl group} is defined to be $\til W^{\nu} = N_{W}\left( W_{\nu} \right) / W_{\nu}$\index{$\til W^{\nu}$}. We choose a $\kappa\in \frakA$ such that $\partial_{I^{\can}}\kappa = \nu$. The canonical isomorphism $\left(\calW^{\can}, \Delta^{\can}  \right)\cong\left(W, \Delta^{\kappa}  \right)$ induces $\til \calW^{\can}\cong \til W^{\nu}$. It is easy to see that this isomorphism does not depend on the choice of $\kappa$. We denote by $\til \Delta^{\nu}\subseteq \til W^{\nu}$ \index{$\til \Delta^{\nu}$} the image of $\til \Delta^{\can}$ under this isomorphism. \par

For each $\nu\in \frakF_{I^{\can}}$, we define $\bbE^{\nu}$\index{$\bbE^{\nu}$} to be the relevant affine subspace of $\bbA$ spanned by $\nu$. If $\bbE^{\nu} = \bbE^{\nu'}$ for $\nu,\nu'\in \Xi$, then $W_{\nu} = W_{\nu'}$ and hence there is an identification $\til W^{\nu} = \til W^{\nu'}$. As before, we can define a {\bf canonical} $\bbE^{\can}$\index{$\bbE^{\can}$} with a left action of $\til \calW^{\can}$ such that for each $\nu\in \Xi$, there is a canonical isomorphism $\bbE^{\can}\cong \bbE^{\nu}$ through which the actions of $\til \calW^{\can}$ and $\til W^{\nu}$ are intertwined. The relative Weyl group $\til W^{\nu}$ acts faithfully on $\bbE^{\nu}$. The induced $\til W^{\nu}$-action on the set of $\bbE^{\nu}$ alcoves $\frakA\left( \bbE^{\nu} \right)$ is simply transitive, thanks to~\autoref{theo:sametype}.

\subsection{Relative position}\label{subsec:relpos}

\begin{prop}\label{prop:doublequo}
	\begin{enumerate}[label=(\roman*)]
		\item
			There is a canonical left $\calW^{\can}$-action on $\frakF_{I^{\can}}$, which commutes with the left $W$-action and induces a left $\til \calW^{\can}$-action on $\frakA(\bbE^{\nu})$ for each $\nu\in \Xi$.
		\item
			There is a canonical inclusion $\til \calW^{\can}\hookrightarrow \calW_{I^{\can}}\backslash \calW^{\can} /\calW_{I^{\can}}$.
		\item
			There is a canonical bijection
			\begin{eq}
				W\backslash \left(\frakF_{I^{\can}} \times \frakF_{I^{\can}}\right)&\cong \calW_{I^{\can}}\backslash \calW^{\can} / \calW_{I^{\can}} \\
			\end{eq}
			which induces for each $\nu\in \Xi$ a bijection
			\begin{eq}
				\til W^{\nu}\backslash \left(\frakA\left( \bbE^{\nu} \right) \times \frakA\left( \bbE^{\nu} \right)\right)&\cong \til \calW^{\can}.\\
			\end{eq}
	\end{enumerate}
\end{prop}
\begin{proof}
	For each $w\in \calW^{\can}$ and each $\kappa\in \frakA$, we denote by $w^\kappa\in W$ the image of $w$ under the canonical isomorphism $\left( \calW^{\can}, \Delta^{\can} \right)\cong\left( W, \Delta^{\kappa} \right)$ \cfauto{subsec:srefl}. We define the left action of $\calW^{\can}$ on the set of $\bbA$-alcoves $\frakA$ by
	\begin{eq}
		\calW^{\can}\times \frakA  \to \frakA \\
		(w, \kappa)\mapsto (w^{\kappa})^{-1} \kappa.
	\end{eq}
	This action descends in a unique way via the boundary map $\partial_{I^{\can}}:\frakA\to  \frakF_{I^{\can}}$, giving a right $\calW^{\can}$-action on $\frakF_{I^{\can}}$. Obviously, for each $\nu\in \frakF_{I^{\can}}$, the subgroup $N_{\calW^{\can}}\left( \calW_{I^{\can}} \right)$ stabilises the subset $\frakA\left( \bbE^{\nu} \right)\subseteq \frakF_{I^{\can}}$ and induces a right action of $\til \calW^{\can}\cong N_{\calW^{\can}}\left( \calW_{I^{\can}} \right) / \calW_{I^{\can}} $ on $\frakA\left( \bbE^{\nu} \right)$. This proves (i).\par

	Choosing any $\kappa\in \frakA$, we define
	\begin{eq*}\label{equa:relpos}
		\calW_{I^{\can}}\backslash \calW^{\can} / \calW_{I^{\can}}&\to W\backslash \left(\frakF_{I^{\can}} \times \frakF_{I^{\can}}\right) \\
		\calW_{I^{\can}}w\calW_{I^{\can}} &\mapsto W\left(\partial_{I^{\can}}\kappa , w^{\kappa}\partial_{I^{\can}}\kappa\right).
	\end{eq*}
	Clearly, it is a bijection and does not depend on the choice of $\kappa$. Now, we write $\nu = \partial_{I^{\can}}\kappa$. By definition, if $w\in \til \calW^{\can}$, then $w$ normalises $\calW_{I^{\can}}$, so $w^{\kappa}\in \til W^{\kappa}$ normalises $W_{\nu}$. From the fact that
	\begin{eq}
		\frakA\left( \bbE^{\nu} \right) = \left\{\mu\in \frakF_{I^{\can}}\;;\; W_{\mu} = W_{\nu}  \right\}
	\end{eq}
	and that $W_{w^{\kappa}\nu} = W_{\nu}$, it follows that $w^{\kappa}\nu\in \frakA\left( \bbE^{\nu} \right)$. Therefore the bijection~\eqref{equa:relpos} restricts to an injection
	\begin{eq}
		\til \calW^{\can}\to \til W^{\nu}\backslash \left(\frakA\left( \bbE^{\nu} \right)\times\frakA\left( \bbE^{\nu} \right)\right)
	\end{eq}
	It is also surjective since $\til \calW^{\can}\cong \til W^{\nu}$ and that the $\til W^{\nu}$-action on $\frakA\left(\bbE^{\nu}\right)$ is transitive~\cfauto{subsec:conjxi}.

	\par

	This proves (ii) and (iii).

\end{proof}

\begin{defi}
	For any two elements $\nu, \nu'\in \frakF_{I^{\can}}$, the image of $(\nu, \nu')\in \frakF_{I^{\can}}\times \frakF_{I^{\can}}$ under
	\begin{eq}
		\frakF_{I^{\can}}\times \frakF_{I^{\can}} \to W\backslash\left(\frakF_{I^{\can}}\times \frakF_{I^{\can}} \right) \cong \calW_{I^{\can}} \backslash \calW^{\can} / \calW_{I^{\can}}
	\end{eq}
	is called the {\bf relative position} of $\nu$ and $\nu'$. The relative position is called {\bf good} if it is in the image of the inclusion $\til \calW^{\can}\hookrightarrow \calW_{I^{\can}} \backslash \calW^{\can} / \calW_{I^{\can}}$, {\bf bad} otherwise.
\end{defi}
The following criterion is immediate.
\begin{lemm}\label{lemm:posrel}
	Two elements $\nu,\nu'\in \frakF_{I^{\can}}$ are in good relative position if and only if $\bbE^{\nu} = \bbE^{\nu'}$ as subspaces of $\bbA$. \hfill\qedsymbol
\end{lemm}

\section{Steinberg type varieties}\label{sec:steinberg}
We keep the assumptions of the previous sections. In particular, there is a sextuple $\left( M, M_0, \frakm, \frakm_*, \rmO, \scrC \right)$ \cfauto{subsec:cusp}, a co-character $\til\theta\in \bfX_*\left( T^{\ad} \right)$ which lifts the $\bfZ/m$-grading on $\frakg$ and $\bfx = \til\theta / m\in \bbA$.\par
In addition, we fix henceforth a non-zero integer $d\in \bfZ\setminus\left\{ 0 \right\}$ and we set $\epsilon = d / |d|\in \left\{ 1, -1 \right\}$\index{$\epsilon$}.\par

\subsection{Varieties of pairs of flags}
We denote $\ubar{\Xi} = W_{\bfx}\backslash \Xi$\index{$\ubar\Xi$}. We will write $\ubar{\nu} = W_{\bfx}\nu$\index{$\ubar{\nu}$} in $\ubar{\Xi}$ for each $\nu\in \Xi$. Since the left $\calW^{\can}$-action of~\autoref{prop:doublequo}(i) on $\Xi$ commutes with the left $W_{\bfx}$-action, we get a canonical left $\calW^{\can}$-action on $\ubar\Xi$.\par

For $\nu, \nu'\in \Xi$, we set
\begin{eq}
	\calX^{\nu, \nu'} = \left(G_{\ubar 0} / P_0^{\nu}\right) \times \left(G_{\ubar 0} / P_0^{\nu'}\right).
\end{eq}
\index{$\calX^{\nu, \nu'}$}
Then, $G_{\ubar 0}$ acts on $\calX^{\nu, \nu'}$ by diagonal left translation.
\begin{prop}\label{prop:canX}
	The $G_{\ubar 0}$-variety $\calX^{\nu, \nu'}$ depends only on the classes $\ubar\nu, \ubar\nu'\in \ubar{\Xi}$. More precisely, there is a $G_{\ubar 0}$-variety $\calX^{\ubar\nu, \ubar\nu'}$, depending only on the $W_{\bfx}$-orbits $\ubar\nu,\ubar\nu'\in \ubar\Xi$, and a canonical $G_{\ubar 0}$-equivariant isomorphism
	\begin{eq}
		\calX^{\nu, \nu'}\cong\calX^{\ubar\nu, \ubar\nu'}.
	\end{eq}
\end{prop}
\begin{proof}
	Indeed, if $y,y'\in W_{\bfx}$, choosing a lifting $\dot y, \dot y'\in N_{G_{\ubar 0}}(T)$ of $y$ and $y'$, we have a $G_{\ubar 0}$-equivariant isomorphism
	\begin{eq}
		\calX^{\nu, \nu'}&\cong\calX^{y\nu, y'\nu'} \\
		\left( gP_0^{\nu},g'P_0^{\nu'} \right)&\mapsto \left( g\dot y^{-1}P_0^{\nu},g'\dot y'^{-1}P_0^{\nu'} \right),
	\end{eq}
	which is canonical in the sense that it allows us to define a canonical object $\calX^{\ubar\nu, \ubar\nu'}$.
\end{proof}

\subsection{Orbits and double cosets}
Let $\nu, \nu'\in \Xi$. The orbits of the $G_{\ubar 0}$-action on $\calX^{\nu, \nu'}$ are identified with the orbits of diagonal left translation of $W_{\bfx}$ on $\left(W_{\bfx} / W_{\bfx, \nu}\right) \times \left(W_{\bfx} / W_{\bfx,\nu'}\right)$, where $W_{\bfx, \nu} = W_{\bfx}\cap W_{\nu}$. There is a canonical inclusion
\begin{eq}
	\left(W_{\bfx} / W_{\bfx, \nu}\right) \times \left(W_{\bfx} / W_{\bfx, \nu'}\right) &\hookrightarrow \Xi \times \Xi \\
				(u, v) &\mapsto (u \nu, v\nu'),
\end{eq}
which induces a map
\begin{eq}
	G_{\ubar 0}\backslash\calX^{\nu, \nu'}\cong W_{\bfx}\backslash \left(\left(W_{\bfx} / W_{\bfx, \nu}\right) \times \left(W_{\bfx} / W_{\bfx, \nu'}\right)\right) &\rightarrow W\backslash \left(\frakF_{I^{\can}} \times \frakF_{I^{\can}}\right) \cong \calW_{I^{\can}}\backslash \calW^{\can} /\calW_{I^{\can}}  \\
\end{eq}
which we denote by $\Pi_{\nu, \nu'} : G_{\ubar 0}\backslash\calX^{\nu, \nu'}\to \calW_{I^{\can}}\backslash \calW^{\can}/\calW_{I^{\can}}$.\index{$\Pi_{\nu, \nu'}$} The map $\Pi_{\nu, \nu'}$ depends only on the $W_{\bfx}$-orbits $\ubar\nu, \ubar\nu'\in \ubar\Xi$ in the sense of proposition~\autoref{prop:canX}. \par
For $w\in \calW^{\can}$ we will denote $[w] = \calW_{I^{\can}}w\calW_{I^{\can}} \in \calW_{I^{\can}}\backslash \calW^{\can} / \calW_{I^{\can}}$\index{$[w]$} the double coset containing $w$.
\begin{lemm}\label{lemm:uniqueorbit}
				Let $[w]\in \calW_{I^{\can}}\backslash \calW^{\can} / \calW_{I^{\can}}$.
				\begin{enumerate}[label=(\roman*)]
								\item
									The fibre $\Pi_{\nu, \nu'}^{-1}\left[ w\right]$ is finite and the union of $G_{\ubar 0}$-orbits $\bigcup_{O\in \Pi^{-1}_{\nu, \nu'}[w]} O$ is locally closed.
								\item
									If $w\in \til \calW^{\can}$, then $\#\left(\Pi_{\nu, \nu'}^{-1}\left[ w \right] \right)\le 1 $. Moreover, $\#\left(\Pi_{\nu, \nu'}^{-1}\left[ w \right] \right)= 1$ if and only if $\ubar{\nu} = w\ubar{\nu'}$.
				\end{enumerate}
\end{lemm}
\begin{proof}
	The assertion (i) is standard and follows from the Bruhat decomposition.\par
	Suppose now that $w\in \til \calW^{\can}$ and $\nu,\nu'\in \Xi$ such that $\ubar\nu = w\ubar\nu'$. We may assume that $\nu =  w\nu'$ since $\calX^{\nu, \nu'}$ and $\Pi_{\nu, \nu'}$ depends only on the $W_{\bfx}$-orbits $\ubar\nu$ and $\ubar\nu'$. Denote $w^{\nu}\in \til W^{\nu}$ the image of $w$ under the isomorphism $\left( \til \calW^{\can}, \til \Delta \right)\cong\left( \til W^{\nu}, \til \Delta^{\nu} \right)$ so that $\nu' = w^{\nu}\nu$ \cfauto{subsec:relpos}. It suffices to show that the map
	\begin{eq}
		\varphi: W_{\bfx}\backslash \left(\left(W_{\bfx} / W_{\bfx, \nu}\right) \times \left(W_{\bfx} / W_{\bfx, \nu'}\right)\right) &\to W\backslash \left(\frakF_{I^{\can}} \times \frakF_{I^{\can}}\right)\\
		W_{\bfx}\cdot (u, u') &\mapsto W\cdot (u \nu, u'w^{\nu}\nu)
	\end{eq}
	is injective over $W\cdot\left(\nu, w^{\nu}\nu  \right)\in W\backslash \left(\Xi \times \Xi\right)$.
	Assume we have $u, u', v, v'\in W_{\bfx}$ such that $\left( uW_{\bfx, \nu}, u'W_{\bfx, \nu'} \right)$ and $\left( vW_{\bfx, \nu}, v'W_{\bfx, \nu'} \right)$ are sent to $W\cdot\left(\nu, w^{\nu}\nu  \right)$, in other words
	\begin{eq}
		W\cdot\left( u\nu, u'w^{\nu}\nu \right) =  W\cdot\left( \nu, w^{\nu} \nu \right) = W\cdot\left( v\nu, v'w^{\nu}\nu \right).
	\end{eq}
	We shall prove that $W_{\bfx}\cdot\left( u\nu, u'w^{\nu}\nu \right)= W_{\bfx}\cdot\left( v\nu, v'w^{\nu}\nu \right)$. \par

	Indeed, as
	\begin{eq}
		W\cdot\left( \nu, w^{\nu} \nu \right)= W\cdot\left( u\nu, u'w^{\nu} \nu \right)= W\cdot\left( \nu, u^{-1}u'w^{\nu} \nu \right)
	\end{eq}
	there exists $q\in W_{\nu} = W_{\nu}$ such that $qw^{\nu}\nu = u^{-1}u'w^{\nu}\nu$, or equivalently
	\begin{eq}
		\left(w^{\nu}\right)^{-1}q^{-1}u^{-1}u'w^{\nu}\in W_{\nu}.
	\end{eq}
	Since $w^{\nu}$ normalises $W_{\nu}$, we obtain $q^{-1}u^{-1}u'\in W_{\nu}$ and hence $u^{-1}u'\in W_{\nu}\cap W_{\bfx}$. Similarly, we have $v^{-1}v'\in W_{\nu}\cap W_{\bfx}$. Therefore $v'^{-1}vu^{-1}u'\in W_{\nu}\cap W_{\bfx}$. Finally, we conclude that
	\begin{eq}
		W_{\bfx}\cdot\left(u\nu, u' \nu'  \right)&= W_{\bfx}v^{-1}v'u'^{-1}u\left(\nu, u^{-1}u' \nu'  \right) = W_{\bfx}\cdot\left(v^{-1}v'u'^{-1}u\nu, v^{-1}v' \nu'  \right)\\
		&= W_{\bfx}\cdot\left(\nu, v^{-1}v' \nu'  \right) = W_{\bfx}\cdot\left(v\nu, v' \nu'  \right).
	\end{eq}
\end{proof}
Recall that there is a canonical bijection $P^{\nu}_0\backslash G_{\ubar 0}/P^{\nu'}_0\cong G_{\ubar 0}\backslash \calX^{\nu, \nu'}$.
\begin{prop}\label{prop:gooddoublecosets}
	Let $\Omega\in P^{\nu}_0\backslash G_{\ubar 0}/P^{\nu'}_0$ be a double coset and let $O_{\Omega}\in G_{\ubar 0}\backslash \calX^{\nu, \nu'}$ be the corresponding orbit. Then $\Pi_{\nu, \nu'}\left(O_{\Omega}\right) \in \til \calW^{\can}$ if and only if for any $g\in \Omega$, the following natural inclusion
	\begin{eq}
		\left(\frakp^{\nu}_{N}\cap \Ad_{g}\frakp^{\nu'}_N\right) / \left(\fraku^{\nu}_{N}\cap \Ad_{g}\fraku^{\nu'}_N\right) \to \Ad_{g}\frakp^{\nu'}_N / \Ad_{g}\fraku^{\nu'}_N
	\end{eq}
	is an isomorphism for each $N\in \bfZ$.
\end{prop}
\begin{proof}
	Let $g\in \Omega$. According to~\cite[5.1]{LYI}, there is a splitting $\frakl_*$ of $\frakp^{\nu}_*$ and a splitting $\frakl'_*$ of $\Ad_g\frakp^{\nu'}_*$ such that $L^{\nu}_0 = \exp\left( \frakl_0 \right)$ and $L^{\nu'}_0 = \exp\left( \frakl'_0 \right)$ contain a common maximal torus $T'$ of $G_{\ubar 0}$. \par
	Let $g'\in G_{\ubar 0}$ be such that $g'T'g'^{-1} = T$. Then $\Ad_{g'}\frakp^{\nu}_*$ and $\Ad_{g'g}\frakp^{\nu'}_*$ are in $\frakP_T$ and there are unique $y, w\in \calW^{\can}$ such that $\Ad_{g'}\frakp^{\nu}_* = \frakp^{\nu y}_*$ and $\Ad_{g'g}\frakp^{\nu'}_* = \frakp^{\nu yw}_*$. From the definition of $\Pi_{\nu, \nu'}$, we see that $\Pi_{\nu, \nu'}\left( O_{\Omega} \right) = [w]$. By~\autoref{lemm:posrel}, $[w]\in \til \calW^{\can}$ if and only if $\bbE^{\nu y} = \bbE^{\nu yw}$. Looking at the Levi decomposition $\frakp^{\nu y}_* = \frakl^{\nu y}_*\oplus\fraku^{\nu y}_*$ and $\frakp^{\nu yw}_* =  \frakl^{\nu yw}_*\oplus\fraku^{\nu yw}_*$, we see that $\bbE^{\nu y} = \bbE^{\nu yw}$ if and only if
	\begin{eq*}\label{equa:condition1}
		\left(\frakp^{\nu y}_{N}\cap \frakp^{\nu yw}_N\right) / \left(\fraku^{\nu y}_{N}\cap \fraku^{\nu yw}_N\right) \to \frakp^{\nu yw}_N / \fraku^{\nu yw}_N
	\end{eq*}
	is an isomorphism for all $N\in \bfZ$. Taking into account the definitions $\Ad_{g'}\frakp^{\nu}_* = \frakp^{\nu y}_*$ and $\Ad_{g'g}\frakp^{\nu'}_* = \frakp^{\nu yw}_*$, the condition that the map~\eqref{equa:condition1} being an isomorphism is equivalent to that the map
	\begin{eq*}\label{equa:condition2}
		\left(\Ad_{g'}\frakp^{\nu}_{N}\cap \Ad_{g'g}\frakp^{\nu'}_N\right) / \left(\Ad_{g'}\fraku^{\nu}_{N}\cap \Ad_{g'g}\fraku^{\nu'}_N\right) \to \Ad_{g'g}\frakp^{\nu'}_N / \Ad_{g'g}\fraku^{\nu'}_N
	\end{eq*}
	being an isomorphism for all $N\in \bfZ$. We obtain the desired equivalent condition by applying $\Ad_{g'^{-1}}$ to the map~\eqref{equa:condition2}.
\end{proof}
\begin{rema}\label{rema:goodbad}
	In the terminologies of~\cite[5.2]{LYI}, if a double coset $\Omega\in P_0^{\nu}\backslash G_{\ubar 0} / P_0^{\nu'}$ satisfies the condition of~\autoref{prop:gooddoublecosets}, it is called {\bf good}. It is called {\bf bad} if otherwise.
\end{rema}

\subsection{Stratification of $\calX^{\nu, \nu'}$}
We define for each $w\in \calW^{\can}$ and each $\nu, \nu'\in \Xi$
\begin{eq}
				\calX^{\nu, \nu'}_{w} = \bigcup_{O\in\Pi^{-1}_{\nu, \nu'}[w]}O\subseteq \calX^{\nu, \nu'}
\end{eq}
\index{$\calX^{\nu, \nu'}_w$}
to be a locally closed subset of $\calX^{\nu,\nu'}$ equipped with the reduced subscheme structure. Obviously, $\calX^{\nu, \nu'}_w = \calX^{\nu, \nu'}_{w'}$ if $[w] = [w']$, so the variety $\calX^{\nu, \nu'}_w$ does not depend on the representative $w$ for the class $[w]$.
\begin{prop} \label{prop:strX}
	For each $\nu, \nu'\in \Xi$ and each $w\in \til \calW^{\can}$ such that $w\ubar{\nu'} = \ubar{\nu}$, there is an isomorphism of $G_{\ubar 0}$-schemes
				\begin{eq}
								\begin{tikzcd}[row sep=.5em]
									G_{\ubar 0} / P_0^{\nu}\cap P_0^{w^{-1}\nu} \arrow{r}{\cong} &  \calX_w^{\nu, w^{-1}\nu}\arrow{r}{\cong} & \calX_w^{\nu, \nu'}  \\
									g\cdot\left(P_0^{\nu}\cap P_0^{ w^{-1}\nu}\right) \arrow[mapsto]{r}& \left(gP_0^{\nu}, gP_0^{w^{-1}\nu}\right) &
								\end{tikzcd}.
				\end{eq}
\end{prop}
\begin{proof}
	By~\autoref{lemm:uniqueorbit}, $\calX_w^{\nu, \nu'}$ is a $G_{\ubar 0}$-orbit. The proposition results from the fact that
	\begin{eq}
		\Stab_{G_{\ubar 0}} \left( e P_0^{\nu}, e P_0^{\nu w} \right) = P_0^{\nu}\cap P_0^{w^{-1}\nu }.
	\end{eq}
\end{proof}

\subsection{Induction of cuspidal local system} \label{subsec:indcusp}

Recall that we have for each $\nu\in \Xi$ a Levi decomposition of spiral $\frakp^{\nu}_* = \frakl^{\nu}_* \oplus \fraku^{\nu}_*$, a nilpotent $L^{\nu}$-orbit $\rmO^{\nu} \subseteq \frakl^{\nu}$ and an $L^{\nu}$-equivariant cuspidal local system $\scrC^\nu$ on $\rmO^{\nu}$, \cfauto{subsec:conjxi}. We denote $\rmO_{d}^{\nu} = \rmO^{\nu}\cap \frakl_{d}$. By restriction, the local system $\scrC^{\nu}\mid_{\rmO^{\nu}_{d}}$ is $L^{\nu}_{0}$-equivariant. \par

For each $\nu\in \Xi$, we introduce a variety
\begin{eq}
	\calT^{\nu} = G_{\ubar 0}\times^{P_0^{\nu}} \left( \rmO^{\nu}_{d}\times \fraku^{\nu}_{d} \right).
\end{eq}
It is a smooth variety equipped with an action of $G_{\ubar 0, q}$.
By induction of groups, there is a $G_{\ubar 0}$-equivariant local system
\begin{eq}
	\dot\scrC^{\nu} = G_{\ubar 0}\times^{P_0^{\nu}}\left( \scrC^{\nu}\mid_{\rmO_{d}^\nu}\boxtimes \ba\bfQ_\ell \right)
\end{eq}
on $\calT^{\nu}$.
Let $\alpha^{\nu}: \calT^{\nu} \to \frakg_{\ubar d}$ be the natural morphism given by the adjoint action of $G_{\ubar 0}$. \par

If $\nu, \nu'\in \Xi$ are such that $\ubar\nu = \ubar\nu'$, then there is a $G_{\ubar 0}$-equivariant canonical isomorphism $\calT^{\nu}\cong \calT^{\nu'}$ which identifies $\alpha^{\nu}$ with $\alpha^{\nu'}$ and $\dot\scrC^{\nu}$ with $\dot\scrC^{\nu'}$. We will write $\dot\scrC$ for $\dot\scrC^{\nu}$ when the index $\nu$ is clear in the context. \par

We define the {\bf Lusztig sheaf} for the relative alcove $\nu\in \Xi$ to be $\bfI^{\nu} = \alpha^{\nu}_! \IC(\dot\scrC)\in \rmD_{G_{\ubar 0}}\left( \frakg_{\ubar d} \right)$. For the same reason as in the last paragraph, $\bfI^{\nu}$ depends only on the $W_{\bfx}$-orbit $\ubar \nu$.
\index{$\calT^{\nu}$}\index{$\dot\scrC^{\nu}, \dot\scrC$}
\index{$\bfI^{\nu}$}

\begin{prop}\label{prop:BBDG}
				For $\ubar\nu\in \ubar\Xi$, the following statements hold
				\begin{enumerate}[label=(\roman*)]
								\item
									The complex $\bfI^{\nu}$ is supported on the nilpotent cone, i.e. $\bfI^{\nu} = \alpha^{\nu}_! \IC(\dot\scrC)\in \rmD_{G_{\ubar 0}}\left( \frakg_{\ubar d}^{\nil} \right)$
								\item
									We have $\bfI^{\nu} \cong \bigoplus_{k\in \bfZ} \prescript{p}{}\rmH^k\bfI^{\nu}[-k]$ and each factor $\prescript{p}{}\rmH^k\bfI^{\nu}$ is a semi-simple perverse sheaves.
				\end{enumerate}
\end{prop}
\begin{proof}
	The (i) follows from to the assumption that $\epsilon = d / |d|$ and~\cite[7.1(a)]{LYI}.
	The (ii) follows from the Be\u \i linson--Bernstein--Deligne--Gabber decomposition theorem and that the purity of $\IC(\dot\scrC)$~\cite[1.4]{lusztig95b}. \par
\end{proof}

\subsection{Steinberg type varieties}\label{subsec:steinberg}

For $\nu, \nu'\in \ubar\Xi$, we set
\begin{eq}
	\calZ^{\nu, \nu'} = \calT^{\nu} \times_{\frakg_{\ubar d}} \calT^{\nu'}
\end{eq}
\index{$\calZ^{\nu, \nu'}$}
and we denote $q_1:\calZ^{\nu, \nu'} \to \calT^{\nu}$ and $q_2:\calZ^{\nu, \nu'} \to \calT^{\nu'}$ the canonical projections. \par

For the same reason as in~\autoref{subsec:indcusp}, this variety is independent of the choice of $\nu$ and $\nu'$ in their $W_{\bfx}$-conjugacy classes. \par

We have a $G_{\ubar 0,q}$-equivariant local system on $\calZ^{\nu, \nu'}$
\begin{eq}
	\scrK^{\nu, \nu'} = \calHom\left(q_2^*\dot\scrC^{\nu'}, q_1^!\dot\scrC^{\nu}\right) \cong s^!\left(\dot\scrC^{\nu} \boxtimes \bfD \dot\scrC^{\nu'} \right),
\end{eq}
where $s = (q_1, q_2): \calZ^{\nu, \nu'}\to \calT^{\nu}\times\calT^{\nu'}$. We shall simply write $\scrK = \scrK^{\nu, \nu'}$. \par

There is a canonical $G_{\ubar 0,q}$-equivariant map which forgets the Lie algebra components
\begin{eq}
				\calZ^{\nu, \nu'} \to \calX^{\nu, \nu'},
\end{eq}
which depends only on the $W_{\bfx}$-orbits $\ubar\nu$ and $\ubar\nu'$.
For $w\in \calW^{\can}$, we denote $\calZ^{\nu, \nu'}_w$\index{$\calZ^{\nu, \nu'}_w$} the pre-image of $\calX^{\nu, \nu'}_w$ under the above map.
\begin{prop} \label{prop:strZ}
	For $\nu, \nu'\in \ubar\Xi$ and $w\in \til \calW^{\can}$ such that $w\ubar{\ubar\nu'} = \ubar{\ubar\nu}$, there is an equivariant isomorphism of $G_{\ubar 0}$-schemes
				\begin{eq}
								\begin{tikzcd}[row sep=.5em]
									G_{\ubar 0} \times^{P_0^{\nu}\cap P_0^{w^{-1}\nu}}\left(\rmO_{d}^{\nu}\times \left(\fraku_{d}^{\nu}  \cap \fraku_{d}^{w^{-1}\nu}\right)\right) \arrow{r}{\cong} &  \calZ_w^{\nu, w^{-1}\nu}\arrow{r}{\cong} & \calZ_w^{\nu, \nu'}  \\
												(g, x) \arrow[mapsto]{r}& \left( \left( g , x\right), \left(g, x  \right) \right) &
								\end{tikzcd}.
				\end{eq}
\end{prop}

\section{Convolution algebra}\label{sec:convolution}
We keep the assumptions of~\autoref{sec:steinberg}. In particular, there is an admissible system $\left( M, M_0, \frakm, \frakm_*, \rmO, \scrC \right)$ and the cuspidal local system $\scrC$ induces a sheaf $\dot\scrC$ on $\calT^{\nu}$ for each $\ubar\nu\in \Xi$.

\subsection{Convolution product}\label{subsec:convolution}
Let $\nu, \nu'\in \Xi$. We have two projections $q_1:\calZ^{\nu, \nu'}\to \calT^{\nu}$ and $q_2:\calZ^{\nu, \nu'}\to \calT^{\nu'}$~\cfauto{subsec:steinberg}.
We define
\begin{eq}
				\calH^{\nu, \nu'} = \Ext^*_{G_{\ubar 0}}\left(q_2^* \dot\scrC, q_1^! \dot\scrC  \right).
\end{eq}
\index{$\calH^{\nu, \nu'}$} It is isomorphic to $\rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu,\nu'}, \scrK  \right)$ in a natural way. For the same reason as in~\autoref{subsec:indcusp}, the definition of $\calH^{\nu, \nu'}$ is independent of $\nu$ and $\nu'$ in their $W_{\bfx}$-conjugacy class. \par
By the formalism of~\cite{lusztig95} and~\cite{EM}, for $\nu, \nu', \nu''\in \Xi$, there is a convolution morphism
\begin{eq}
				\calH^{\nu, \nu'}\otimes \calH^{\nu', \nu''} \to \calH^{\nu, \nu''}.
\end{eq}
The $\rmH^*_{G_{\ubar 0}}$-module $\rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}, \scrK  \right)$ is isomorphic by Verdier duality to the extension space $\Ext^*_{G_{\ubar 0}}\left( \bfI^{\nu'}, \bfI^{\nu} \right)$ and the convolution product is identified with the Yoneda product.
\par

We recall the construction of {\it loc. cit}. Let $\ha\rmO^{\nu}\subseteq \frakl^{\nu}$\index{$\ha\rmO^{\nu}$} be the closure of $\rmO^{\nu}$ in $\frakl^{\nu}$ and let $\ha\rmO^{\nu}_{d} = \ha\rmO^{\nu}\cap \frakl^{\nu}_{d}$. Since the orbit $\rmO^{\nu}$ is distinguished, $\rmO^{\nu}_{d}$ must be open dense in $\frakl^{\nu}_{d}$ {\it cf.}~\cite[4.4a]{lusztig95b}. This implies that $\ha\rmO_{d}^{\nu}$ is either empty or equal to $\frakl_{d}^{\nu}$. We set $\ha\calT^{\nu} = G_{\ubar 0}\times^{P_0^{\nu}}\left( \ha \rmO^{\nu}_{d}\times \fraku^{\nu}_{d} \right)$ \index{$\ha\calT^{\nu}$} and $\ha\calZ^{\nu, \nu'} = \ha\calT^{\nu}\times_{\frakg_{\ubar d}} \ha\calT^{\nu}$\index{$\ha\calZ^{\nu, \nu'}$}. We denote $u^{\nu}:\calT^{\nu}\hookrightarrow\ha\calT^{\nu}$ and $u^{\nu,\nu'}:\calZ^{\nu,\nu'}\hookrightarrow\ha\calZ^{\nu,\nu'}$ the open embeddings. We remark that $\ha\calT^{\nu}$ is smooth and is proper over $\frakg_{\ubar d}$. We denote $q_1: \calZ^{\nu, \nu'}\to \calT^{\nu}$, $q_2: \calZ^{\nu, \nu'}\to \calT^{\nu'}$, $\ha q_1: \ha\calZ^{\nu, \nu'}\to \ha\calT^{\nu}$ and $\ha q_2: \ha\calZ^{\nu, \nu'}\to \ha\calT^{\nu'}$ the canonical projections. \par
By the cleaness of $\scrC$, we have $u^{\nu}_!\dot\scrC = u^{\nu}_*\dot\scrC$. We will thus identify $\dot\scrC$ with its direct image on $\ha\calT^{\nu}$ without any confusion. Define $\scrK = \calHom\left( q_2^* \dot\scrC, q_1^! \dot\scrC \right)$\index{$\scrK$} on $\calZ^{\nu,\nu'}$ and $\ha\scrK = \calHom\left( \ha q_2^* \dot\scrC, \ha q_1^! \dot\scrC \right)$\index{$\ha\scrK$} on $\ha\calZ^{\nu,\nu'}$. We have
\begin{eq}
				\ha\scrK &\cong \calHom\left( \ha q_2^* u^{\nu}_!\dot{\scrC}, \ha q_1^! \dot{\scrC} \right) \cong\calHom\left( u^{\nu,\nu'}_!q_2^*\dot{\scrC}, \ha q_1^! \dot{\scrC} \right)\\
				&\cong u^{\nu,\nu'}_*\calHom\left( q_2^*\dot{\scrC}, u^{\nu,\nu'!}\ha q_1^! \dot{\scrC} \right) \cong u^{\nu,\nu'}_*\calHom\left( q_2^*\dot{\scrC}, q_1^! u^{\nu'*}\dot{\scrC} \right)\cong u^{\nu,\nu'}_*\scrK.
\end{eq}
Therefore
\begin{eq}
				\calH^{\nu, \nu'} = \rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu, \nu'}, \scrK \right)\cong \rmH^*_{G_{\ubar 0}}\left( \ha\calZ^{\nu, \nu'}, \ha\scrK \right).
\end{eq}
The cleaness assures that we can safely work on the non-proper version $\calT^{\nu}$, $\calZ^{\nu,\nu'}$ and $\scrK$.
\par

Now we describe the convolution product. Given $\nu, \nu', \nu''\in \Xi$, we consider the following diagram
\begin{eq}
				\begin{tikzcd}[column sep=5em]
					\calZ^{\nu, \nu'} \times \calZ^{\nu', \nu''}\arrow{d}{t} \arrow[hookleftarrow]{r}&  \calZ^{\nu, \nu'} \times_{\calT^{\nu'}} \calZ^{\nu', \nu''} \arrow{r}{\mu} \arrow{d}{r} & \calZ^{\nu, \nu''}\arrow{d}{s} \\
					\left(\calT^{\nu}\times \calT^{\nu'}\right)\times\left(\calT^{\nu'}\times \calT^{\nu''}\right) & \calT^{\nu}\times \calT^{\nu'}\times \calT^{\nu''} \arrow[swap]{l}{\gamma} \arrow{r}{u} &  \calT^{\nu} \times \calT^{\nu''}
				\end{tikzcd}
\end{eq}
There is a sequence of maps
\begin{eq}
	&\rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu, \nu'}, \scrK \right)\otimes \rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu', \nu''}, \scrK \right)\to\rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}\times\calZ^{\nu', \nu''}, t^!\left(\dot{\scrC}\boxtimes\bfD\dot{\scrC}\boxtimes\dot{\scrC}\boxtimes\bfD\dot{\scrC}\right) \right) \\
	& \xrightarrow{t_*t^!\left(\id\to \gamma_*\gamma^*\right)}
	\rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}\times\calZ^{\nu', \nu''}, t^!\gamma_*\left(\dot{\scrC}\boxtimes\left(\bfD\dot{\scrC}\otimes\dot{\scrC}\right)\boxtimes\bfD\dot{\scrC}\right) \right) \\
	&\cong \rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}\times_{\calT^{\nu'}}\calZ^{\nu', \nu''}, r^!\left(\dot{\scrC}\boxtimes\left(\bfD\dot{\scrC}\otimes\dot{\scrC}\right)\boxtimes\bfD\dot{\scrC}\right) \right) \\
	&\to \rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}\times_{\calT^{\nu'}}\calZ^{\nu', \nu''}, r^!\left(\dot{\scrC}\boxtimes\bfD\ba\bfQ_{\ell}\boxtimes\bfD\dot{\scrC}\right) \right)\cong \rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}\times_{\calT^{\nu'}}\calZ^{\nu', \nu''}, r^!u^!\left(\dot{\scrC}\boxtimes\bfD\dot{\scrC}\right) \right) \\
	&\cong \rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}\times_{\calT^{\nu'}}\calZ^{\nu', \nu''}, \mu^!\scrK\right)\to \rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu, \nu''}, \scrK \right).
\end{eq}
The convolution product is then defined to be the composite $\calH^{\nu, \nu'}\otimes\calH^{\nu', \nu''} \to\calH^{\nu, \nu''}$. \par

For any $G_{\ubar 0}$-stable closed subschemes $V\subset \calZ^{\nu, \nu'}$, $V'\subset \calZ^{\nu', \nu''}$ and $V''\subset \calZ^{\nu, \nu''}$, we define similarly the convolution product so that the folloing diagram commutes
\begin{eq}
	\begin{tikzcd}
		\rmH^*_{G_{\ubar 0}}\left( V, i_V^!\scrK \right) \otimes \rmH^*_{G_{\ubar 0}}\left( V', i_{V'}^!\scrK \right) \arrow{r}\arrow{d}&  \rmH^*_{G_{\ubar 0}}\left( V'', i_{V''}^!\scrK \right) \arrow{d}\\
		\calH^{\nu, \nu'}\otimes\calH^{\nu', \nu''}\arrow{r} & \calH^{\nu, \nu''} \\
	\end{tikzcd},
\end{eq}
where the vertical arrows are given by adjunction co-units $i_{V!}i_V^!\to \id$, etc. Henceforth, we will simply denote $\scrK$ for $i^!_V\scrK$, for any such closed immersion $i_V$.

\subsection{Polynomial action}\label{subsec:poly}

For each $\nu\in \Xi$, let $A^{\nu} = \rmZ^0\left(L^{\nu}\right)$\index{$A^{\nu}$} be the neutral component of the centre of $L^{\nu}$. We have $\bbE^{\nu}\cong \bfX_{*}\left(A^{\nu}\right)_{\bfQ}$. We define $\bfS^{\nu}$ to be the algebra of $\ba\bfQ_{\ell}$-valued polynomial functions on $\bbE^{\nu}$, so that $\bfS^{\nu} \cong \rmH^*_{A^{\nu}}$. \index{$\bfS^{\nu}$}  If $\nu, \nu'\in \Xi$, then the canonical isomorphism $\bbE^{\nu}\cong \bbE^{\nu'}$ induces a canonical isomorphism $\bfS^{\nu}\cong \bfS^{\nu'}$. We define equally the canonical object $\bfS^{\can}$ such that $\bfS^{\can}\cong\bfS^{\nu}$ for all $\nu\in \Xi$ \index{$\bfS^{\can}$}. \par

\begin{lemm}\label{lemm:bimod-SS}
	For any $\nu,\nu'\in \Xi$, the equivariant cohomology group $\calH^{\nu,\nu'}$ has a $(\bfS^{\nu}, \bfS^{\nu'})$-bimodule structure, which makes the convolution product $\calH^{\nu,\nu'}\times \calH^{\nu',\nu''}\to \calH^{\nu,\nu''}$ a $\bfS^{\nu'}$-bilinear map of $(\bfS^{\nu}, \bfS^{\nu'})$-bimodules. \par
	More generally, given any $G_{\ubar 0,q}$-invariant subvariety $i_V:V\hookrightarrow \calZ$, the cohomology $\rmH^*_{G_{\ubar 0}}(V, i_V^!\scrK)$ has a structure of $(\bfS^{\nu}, \bfS^{\nu'})$-bimodules.
\end{lemm}
\begin{proof}
	We will only prove the statement for $\calH^{\nu,\nu'}$. By the induction principle of equivariant cohomology, there are isomorphisms
	\begin{eq}
		\rmH^*_{G_{\ubar 0,q}}\left( \calT^{\nu}, \ba\bfQ_{\ell} \right)\cong \rmH^*_{L^{\nu}_{0,q}}\left( \rmO_{d}^{\nu}, \ba\bfQ_{\ell} \right)\cong \rmH^*_{\Stab_{L^{\nu}_{0,q}}(\mathbf{o})}.
	\end{eq}
	where $\mathbf{o}$ is any point in the $L^{\nu}_0$-orbit $\rmO^{\nu}_{d}$. By the Jacobson--Morosov theorem, there exists a cocharacter $\varphi\in X_*(L^{\nu}_0)$ which acts on $\mathbf{o}$ by weight $2$. Hence there is a morphism
	\begin{eq*}\label{equa:Stabo}
		A^{\nu}\times\bfC^{\times}_q&\to G_{\ubar 0} \times \Cq \\
		\left( g, u \right)&\mapsto \left( g\varphi(u), u \right).
	\end{eq*}
	whose image is a maximal reductive subgroup of $\Stab_{L^{\nu}_{0,q}}\left( \mathbf{o} \right)$, \cite[2.3]{lusztig88}. Thus we obtain an isomorphisms of rings
	\begin{eq}
		\bfS^{\nu}\otimes \ba\bfQ_{\ell}[u] \cong \rmH^*_{A^{\nu}_q}\cong \rmH^*_{G_{\ubar 0,q}}(\calT^{\nu}, \ba\bfQ_{\ell}),
	\end{eq}
	Since the orbit $\rmO^{\nu}_d$ is distinguished, the cocharacter $\varphi$ acts on $\frakl^{\nu}_d$ by weight $2$. Hence by the formula~\eqref{equa:Stabo}, the torus $\Cq$ of left hand side acts trivially on $\frakl^{\nu}_d$. Substituting $u\mapsto -d/m$ into the left-hand side, we obtain $\bfS^{\nu} \cong \rmH^*_{G_{\ubar 0}}(\calT^{\nu}, \ba\bfQ_{\ell})$, which acts on $\calH^{\nu,\nu'}$ by cup product via the pull-back:
	\begin{eq}
		q_1^*:\rmH^*_{G_{\ubar 0}}(\calT^{\nu}, \ba\bfQ_{\ell})\to \rmH^*_{G_{\ubar 0}}(\calZ^{\nu, \nu'}, \ba\bfQ_{\ell})
	\end{eq}
	with the map $q_1$ as defined in~\autoref{subsec:steinberg}. Similarly, we have $\bfS^{\nu'}\cong \rmH^*_{G_{\ubar 0}}\left( \calT^{\nu'}, \ba\bfQ_{\ell} \right)$ which acts via $q^*_2$. Thus, the tensor product $\bfS^{\nu}\otimes_{\rmH_{G_{\ubar 0}}} \bfS^{\nu'}$ acts on $\rmH^*_{G_{\ubar 0}}\left( \calZ,\scrK \right)$ via the cup product.

\end{proof}
\begin{rema}
	The delicate point of this $\bfS^{\nu}$-action is that in the isomorphism~\ref{equa:Stabo}, the $\Cq$-part is not the identity map. In other words, the quotient map $\rmH^*_{G_{\ubar 0,q}}(\calT^{\nu}, \ba\bfQ_{\ell})\to \rmH^*_{G_{\ubar 0}}(\calT^{\nu}, \ba\bfQ_{\ell})$ implicit in the proof is not simply forgetting the $\Cq$-part of the equivariance.
\end{rema}

\subsection{Cohomology of components $\calZ^{\nu, \nu'}_w$}\label{subsec:cohom}
We define for $\nu, \nu'\in \Xi$ and $[w]\in  \calW_{I^{\can}} \backslash \calW^{\can}/ \calW_{I^{\can}}$
\begin{eq}
				\calH^{\nu, \nu'}_w = \rmH^*_{G_{\ubar 0}}\left(\calZ^{\nu, \nu'}_w, i_w^!\scrK \right),
\end{eq}
\index{$\calH^{\nu, \nu'}_w$}
where $i_w: \calZ_w^{\nu, \nu'}\to \calZ^{\nu, \nu'}$ is the inclusion.
By the construction of~\autoref{lemm:bimod-SS}, $\calH^{\nu, \nu'}_w$ comes equipped with an $(\bfS^{\nu},\bfS^{\nu'})$-bimodule structure. \par

\begin{prop}\label{prop:SS-bimod}
	Let $\nu, \nu'\in \Xi$ and $[w]\in \calW_{I^{\can}}\backslash \calW^{\can} / \calW_{I^{\can}}$.
	\begin{enumerate}[resume,label=(\roman*)]
		\item
			If $w\ubar\nu' \neq \ubar\nu$, then $\calH^{\nu, \nu'}_w = 0$.
		\item
			If the relative position $[w]$ is bad, i.e. $[w]\notin \til \calW^{\can}$, then $\calH^{\nu, \nu'}_w = 0$.
		\item
			If $[w]$ is good, i.e. $[w]\in \til \calW^{\can}$, and if $w\ubar{\nu'} = \ubar{\nu}$, then $\calH^{\nu, \nu'}_w$ is a free of rank $1$ as left graded $\bfS^{\nu}$-module and as right graded $\bfS^{\nu'}$-module, vanishing in odd degrees.
	\end{enumerate}
\end{prop}
\begin{proof}
	If $w\ubar\nu' \neq \ubar\nu$, then $\calX^{\nu, \nu'}_w = \emptyset$ and $\calZ^{\nu, \nu'}_w = \emptyset$ so $\calH^{\nu, \nu'}_w = 0$. \par
	We turn to (ii). Suppose that $[w]\notin \til \calW^{\can}$. Recall that there is a left $G_{\ubar 0}$-action on $\calX^{\nu, \nu'}_w$ with a finite number of orbits. Let
	\begin{eq}
		\calX^{\nu, \nu'}_w = \bigsqcup_{\Omega} \calX_{\Omega}
	\end{eq}
	be a decomposition into $G_{\ubar 0}$ orbits, where $\Omega$ is taken over a finite subset of $P^{\nu}_0\backslash G_{\ubar 0}/P^{\nu'}_{0}$. We define $\calZ_\Omega\subseteq \calZ^{\nu, \nu'}$ to be the pre-image of $\calX_\Omega$ under the projection $\calZ^{\nu, \nu'}\to \calX^{\nu, \nu'}$. Notice that all double cosets $\Omega$ that appear in the decomposition are {\it bad} in the sense of~\cite[5.2]{LYI} since $[w]$ is not in $\til \calW^{\can}$.\par
	Let
	\begin{eq}
		\dot\frakp^{\nu}_{d} = \left(\rmO^{\nu}_{d}\times \fraku^{\nu}_{d}\right)\times_{\frakg_{\ubar d}} \calT^{\nu'}
	\end{eq}
	so that $\calZ^{\nu, \nu'} \cong G_{\ubar 0}\times^{P_0^{\nu''}}\dot\frakp^{\nu'}_{d}$. Moreover, for each $\Omega$ if we denote
	\begin{eq}
		\dot\frakp^{\nu}_{d,\Omega} = \left(\rmO^{\nu}_{d}\times \fraku^{\nu}_{d}\right)\times_{\frakg_{\ubar d}}\left(\Omega\times^{P^{\nu'}_0}\left(\rmO^{\nu'}_{d}\times \fraku^{\nu'}_{d}\right)\right)\subseteq\dot\frakp^{\nu}_{d}
	\end{eq}
	then there is a diagram of cartesian squares
	\begin{eq}
		\begin{tikzcd}
			\dot\frakp^{\nu}_{d,\Omega}\arrow{r}{r'}\arrow{d}{q_1'} & \calZ_{\Omega} \arrow{r}{q_2}\arrow{d}{q_1}& \calT^{\nu'} \arrow{d}\\
			\frakp^{\nu}_{d} \arrow{r}{r} & \calT^{\nu} \arrow{r}& \frakg_{\ubar d}
		\end{tikzcd}
	\end{eq}
	Let $i:\calZ_{\Omega}\hookrightarrow \calZ^{\nu, \nu'}$ be the inclusion. Now
	\begin{eq}
		\rmH^*_{G_{\ubar 0}}\left( \calZ_{\Omega}, i^!\scrK \right) \cong \Ext^*_{G_{\ubar 0}}\left( q_2^*\dot\scrC^{\nu'}, q_1^!\dot\scrC^{\nu}\right)\cong\Ext^*_{G_{\ubar 0}}\left( q_{1!}q_{2}^*\dot\scrC^{\nu'}, \dot\scrC^{\nu}\right) \\
		\cong\Ext^*_{P_{0}}\left( r^*q_{1!}q_{2}^*\dot\scrC^{\nu'}, r^*\dot\scrC^{\nu}\right)\cong\Ext^*_{P_{0}}\left( q'_{1!}r'^*q_{2}^*\dot\scrC^{\nu'}, r^*\dot\scrC^{\nu}\right).
	\end{eq}
	Taking into account the cuspidality of $\scrC^{\nu'}$ and the fact that $\Omega$ is bad,~\cfauto{prop:gooddoublecosets} and~\autoref{rema:goodbad}, it is shown in~\cite[5.3]{LYI} that $q'_{1!}r'^*q_{2}^*\dot\scrC^{\nu'} = 0$. Thus $\rmH^*_{G_{\ubar 0}}\left( \calZ_{\Omega}, i^!\scrK \right)  = 0$ for each $\Omega$. By an argument of long exact sequence of cohomoloy, we conclude that
	\begin{eq}
		\calH^{\nu, \nu'}_w = \rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu, \nu'}_{w}, \scrK \right) = 0.
	\end{eq}
	This proves (ii).
	\par

	We turn to (iii). Suppose that $w\nu' = \nu$ with $w\in \til \calW^{\can}$. Using the description~\autoref{prop:strZ}, by the induction property of equivariant cohomology, we see that
	\begin{eq}
		\calH^{\nu, \nu'}_w &= \rmH_{G_{\ubar 0}}^*\left(G_{\ubar 0}\times^{P_0^{\nu}\cap P_0^{\nu'}}\left( \rmO^{\nu}_{d} \times \fraku^{\nu}_{d}\cap \fraku^{\nu'}_{d} \right), \scrK\right) \cong \rmH_{P_{0}^{\nu}\cap P_{0}^{\nu'}}^*\left( \rmO^{\nu}_{d} \times \fraku^{\nu}_{d}\cap \fraku^{\nu'}_{d} , \scrK\mid_{\rmO^{\nu}_{d}\times \fraku^{\nu}_{d}\cap\fraku^{\nu'}_{d}}\right) \\
		&\cong \rmH_{L^{\nu}_{0}}^*\left( \rmO^{\nu}_{d} , \scrK\mid_{\rmO^{\nu}_{d}}\right).
	\end{eq}
	From the definition of $\scrK$, we see that
	\begin{eq}
		\scrK\mid_{\rmO^{\nu}_{d}} = \calHom\left( \scrC^{\nu}, \scrC^{\nu} \right)\left[ 2\dim G_{\ubar 0} / P^{\nu}_{0}\cap P^{\nu'}_{0} - 2\dim \fraku^{\nu}_{d} \cap \fraku^{\nu'}_{d} \right]
	\end{eq}
	According to a result of Lusztig~\cite[4.4(a)]{lusztig95b}, the subscheme $\rmO^{\nu}_{d}\subseteq \frakl^{\nu}_{d}$ is the unique open $L_0^{\nu}$-orbit whenever it is non-empty, whence up to even shift of cohomological degree, there are isomorphisms
	\begin{eq}
		\rmH^*_{L^{\nu}_{0}}\left( \rmO^{\nu}_{d}, \scrK \right) \cong \rmH^*_{L^{\nu}_{0}}\left( \rmO^{\nu}_{d}, \calHom\left( \scrC^{\nu}, \scrC^{\nu} \right) \right)\cong\rmH^*_{\Stab_{L^{\nu}_{0}}\left( \mathbf{o} \right)}\left( \left\{ \mathbf{o}  \right\},  \scrC^*_{ \mathbf{o}}\otimes \scrC_{ \mathbf{o}} \right)\cong \bfS^{\nu}.
	\end{eq}
	The last isomorphism is due to the fact that the fibre $\scrC_{\mathbf{o}}$ in $\mathbf{o}$ is a irreducible representation of $\pi_0\left( \Stab_{L^{\nu}_{0}}(\mathbf{o}) \right)$. Similarly, $\calH^{\nu, \nu'}_w\cong \bfS^{\nu'}$, \cfauto{subsec:cusp}. See also~\cite[4.2]{lusztig88}.
	\par
	By construction, the $\bfS^{\nu}$-action is the same as the cup product on $\rmH_{L^{\nu}_{0,t}\times\Cq}^*\left( \rmO^{\nu}_{d} , \scrK\right)$. Therefore $\calH^{\nu, \nu'}_w$ is free left graded $\bfS^{\nu}$-module of rank $1$. Applying~\autoref{prop:strZ} to $\nu = w\nu'$ and reasoning similarly with the transposed $\calZ^{\nu', \nu}_w\cong\calZ^{\nu, \nu'}_{w^{-1}}$, we see that $\calH^{\nu, \nu'}_w$ is free right graded $\bfS^{\nu'}$-module of rank $1$. It clearly vanishes in odd degrees. This complete the proof. \par
\end{proof}
\begin{rema}
	The special feature of convolution algebra with a cuspidal local system as coefficient is that not every Bruhat cell on the partial flag variety contributes to the cohomology, but only those corresponding to the subset $\til W\subset \calW_I \backslash \calW / \calW_I$ do.
\end{rema}
\subsection{Filtration by length}\label{subsec:filtration}

Recall that for each $\nu,\nu'\in \Xi$, the cohomology $\calH^{\nu, \nu'}_w = \rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{w}, \scrK \right)$ is equipped with an $(\bfS^{\nu},\bfS^{\nu'})$-bimodule structure~\cfauto{subsec:poly}.

Let $\pre{I}\calW^I \subseteq \calW^{\can}$\index{$\pre{I}\calW^I$} be the set of representatives of minimal length of the double quotient $[w]\in  \calW_{I^{\can}} \backslash \calW^{\can}/ \calW_{I^{\can}}$. Note that $\til \calW^{\can}\subseteq \pre{I}\calW^I$. We introduce a partial order $\le_{\ell}$\index{$\le_{\ell}$} on $\calW_{I^{\can}} \backslash \calW^{\can}/ \calW_{I^{\can}}$ in the following way: for $y, y'\in \pre{I}\calW^I$, the relation $[y] \le_{\ell} [y']$ holds if and only if $\ell(y) < \ell(y')$ or $y=y'$. Similarly, the relation $[y] \le_{\ell} [y']$ means $\ell(y) < \ell(y')$. \par

For each $\nu, \nu'\in \Xi$, we filter the schemes $\calX^{\nu, \nu'}, \calZ^{\nu, \nu'}$ and the convolution algebras accordingly. Note that by Bruhat decomposition, for each pair of elements $w,y\in \til \calW^{\can}$, if the variety $\calX^{\nu,\nu'}_{w}$ lies in the closure of $\calX^{\nu,\nu'}_{y}$, then $w\le_{\ell} y$. For $[w]\in \calW_{I^{\can}} \backslash \calW^{\can}/ \calW_{I^{\can}}$. We set
\begin{eq}
	\calX^{\nu, \nu'}_{\le w} = \bigcup_{\substack{y\in \pre{I}\calW^{I} \\ y \le_{\ell} w}}\calX^{\nu, \nu'}_y ,\quad
				\calZ^{\nu, \nu'}_{\le w} = \bigcup_{\substack{y\in \pre{I}\calW^{I} \\ y \le_{\ell} w}}\calZ^{\nu, \nu'}_y,\quad
				\calX^{\nu, \nu'}_{< w} = \calX^{\nu,\nu'}_{\le w} \setminus \calX^{\nu,\nu'}_{w},\quad
				\calZ^{\nu, \nu'}_{< w} = \calZ^{\nu, \nu'}_{\le w}  \setminus \calZ^{\nu,\nu'}_{w}.
\end{eq}
\index{$\calX^{\nu, \nu'}_{\le w}, \calZ^{\nu, \nu'}_{\le w}$}
Those are reduced closed subschemes of $\calX^{\nu, \nu'}$ and $\calZ^{\nu, \nu'}$. \par

For each $w\in \calW^{\can}$, corresponding to the triplet $\calZ^{\nu, \nu'}_{< w}\subseteq \calZ^{\nu, \nu'}_{\le w} \supseteq \calZ^{\nu, \nu'}_{w}$, there is a long exact sequence of graded $\bfS^{\nu}$-$\bfS^{\nu'}$-bimodules
\begin{eq}
				\rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{< w}, \scrK \right) \to
				\rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{\le w}, \scrK  \right) \to
				\rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{w}, \scrK  \right) \xrightarrow{\partial}
				\rmH_{G_{\ubar 0}}^{*+1}\left( \calZ^{\nu, \nu'}_{<w}, \scrK  \right).
\end{eq}
By~\autoref{prop:SS-bimod}, the cohomology group $\rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{w}, \scrK \right)$ vanishes in odd degrees. Therefore, by induction on $w$, we have $\partial = 0$ in the above sequence. This yields the following short exact sequence
				\begin{eq}
								0\to \rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{< w}, \scrK  \right) \to
								\rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{\le w}, \scrK  \right) \to
								\rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{w}, \scrK  \right)  \to 0.
				\end{eq}
We define then
\begin{eq}
	\calH^{\nu, \nu'}_{\le w} = \rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{\le w}, \scrK \right), \quad
	\calH^{\nu, \nu'}_{< w} = \rmH_{G_{\ubar 0}}^*\left( \calZ^{\nu, \nu'}_{< w}, \scrK \right).
\end{eq}
\index{$\calH^{\nu, \nu'}_{\le w}, \calH^{\nu, \nu'}_{< w}$}
By induction on $w$, we see that the natural morphism $\calH^{\nu, \nu'}_{\le w}\to \calH^{\nu, \nu'}$ is injective. Hence $\calH^{\nu, \nu'}_{\le w}$ is an $(\bfS^{\nu},\bfS^{\nu'})$ sub-bimodule of $\calH^{\nu, \nu'}$. The sub-quotient $\calH^{\nu, \nu'}_{\le w} / \calH^{\nu, \nu'}_{< w}$ is isomorphic to the bimodule $\calH^{\nu, \nu'}_{w}$. We denote by $\calH^{\nu,\nu'}_* = \bigoplus_{w\in \pre{I}\calW^I}\calH^{\nu,\nu'}_w$ the associated graded $(\bfS^{\nu}, \bfS^{\nu'})$-bimodule.

\subsection{Convolution product on graded pieces}\label{subsec:grconv}

For $y\in \til \calW^{\can}$, recall that there are two lengths attached to $y$: $\ell(y)$ and $\til\ell(y)$. The former is the length of the element $y$ in the full affine Coxeter system $\left( \calW^{\can}, S^{\can} \right)$, whereas the latter is the length of $y$ in the relative Coxeter system $\left(\til \calW^{\can}, \til S^{\can}  \right)$. The~\autoref{theo:coxeterite} implies that $\ell(yw) = \ell(y) + \ell(w)$ if and only if $\til\ell(yw)= \til\ell(y) + \til \ell(w)$ for any $y,w\in \til \calW^{\can}$. \par

\begin{lemm}
	The convolution product on $\calH^{\nu,\nu'}$ induces a product on the associated graded $(\bfS^{\nu},\bfS^{\nu'})$-bimodules $\calH^{\nu,\nu'}_*$.
\end{lemm}
\begin{proof}
	Let $w,w'\in \til \calW^{\can}$. We suppose that $\til \ell(ww') = \til\ell(w) + \til\ell(w')$, so that $\ell(ww') = \ell(w) + \ell(w')$. For any element $y\in Y$, let
	\begin{eq}
		\overline{C_y} = \bigsqcup_{y' \le_{\ell} y}\calW_{I^{\can}}y'\calW_{I^{\can}}\subseteq \calW^{\can}
	\end{eq}
	We then have $\overline{C_w}\cdot \overline{C_{w'}} \subseteq \overline{C_{ww'}}$. \par
	Let $\nu\in \Xi$ and denote $\nu' =  w^{-1}\nu,\;\nu'' = w'^{-1}\nu'$. The image of the convolution
	\begin{eq}
		\calZ^{\nu, \nu'}_{\le w} \times_{\calT^{\nu'}}\calZ^{\nu', \nu''}_{\le w'}\to \calZ^{\nu, \nu''}
	\end{eq}
	lies in the closed subscheme $\calZ^{\nu, \nu''}_{\le ww'}$.
	It follows that the restriction of the convolution product of cohomology~\cfauto{subsec:convolution}
	\begin{eq}
		\calH^{\nu, \nu'}_{\le w} \otimes \calH^{\nu', \nu''}_{\le w'} \to \calH^{\nu, \nu''}
	\end{eq}
	factorises through the sub-bimodule $\calH^{\nu, \nu''}_{\le ww'}\subseteq \calH^{\nu, \nu''}$. It follows that the convolution product respects the filtration by length, so it induces a product on the graded pieces.
\end{proof}
With , the following lemma of transversality follows easily from the Bruhat decomposition.
\begin{lemm}\label{lemm:isostr}
	Given $\nu\in \Xi$, $w,w'\in \til \calW^{\can}$ such that $\til\ell(ww') = \til\ell(w) + \til\ell(w')$, let $\nu' =  w^{-1}\nu$ and $\nu'' = w'^{-1}\nu'$.
	\begin{enumerate}[label=(\roman*)]
		\item\label{lemm:isostr-i}
			The projection $\calZ^{\nu, \nu'}\times_{\calT^{\nu'}} \calZ^{\nu', \nu''}\to \calZ^{\nu,\nu''}$ induces an isomorphism $\ba\mu:\calZ^{\nu, \nu'}_{w}\times_{\calT^{\nu'}} \calZ^{\nu', \nu''}_{w'}\cong  \calZ^{\nu, \nu''}_{ww'}$.
		\item\label{lemm:isostr-ii}
			The convolution product on graded pieces $\calH^{\nu, \nu'}_{w} \otimes \calH^{\nu', \nu''}_{w'} \to \calH^{\nu, \nu''}_{ww'}$ agrees with the Gysin map of the closed embedding
			\begin{eq}
				\calZ^{\nu, \nu''}_{ww'}\cong \calZ^{\nu, \nu'}_{w}\times_{\calT^{\nu'}} \calZ^{\nu', \nu''}_{w'} \hookrightarrow \calZ^{\nu, \nu'}_{w}\times \calZ^{\nu', \nu''}_{w'}.
			\end{eq}
		\item\label{lemm:isostr-iii}
			The convolution product
			\begin{eq}
				\calH^{\nu, \nu'}_{w}\otimes \calH^{\nu', \nu''}_{w'} \to \calH^{\nu, \nu''}_{ww'}
			\end{eq} is surjective.
			\label{prop:surj}
		\item \label{lemm:isostr-iv}\label{coro:convo-poly}
			The polynomial actions~\cfauto{subsec:poly} yields an isomorphism of algebras $\bfS^{\nu}\cong\calH^{\nu, \nu}_e$.
	\end{enumerate}
\end{lemm}
\begin{proof}
	The statement~\ref{lemm:isostr-i} follows easily from the equivalence $l(yw) = l(y) + l(w)\,\Leftrightarrow \til l(yw) = \til l(y) + \til l(w)$ and the Bruhat decomposition. The statement~\ref{lemm:isostr-ii} and~\ref{lemm:isostr-iii} are consequences of the ``transversality'' result~\ref{lemm:isostr-i}, {\it cf. \cite[7.6.12]{CG}}. The statement~\ref{lemm:isostr-ii} follows from~\ref{lemm:isostr-iii} and~\autorefitem{prop:SS-bimod}{iii}.
\end{proof}

\subsection{Completion}\label{subsec:special}

Let $\frakm_0\subset \rmH^{*}_{G_{\ubar 0}}$ denote the graded maximal ideal (positive-degree part) and let $\rmH^{\wedge}$ be the $\frakm_0$-adic completion, equipped with the $\frakm_0$-adic topology. Define the following topological vector spaces
\begin{eq}
	\ha\calH^{\nu, \nu'} = \calH^{\nu, \nu'}\otimes_{\rmH^*_{G_{\ubar 0}}} \rmH^{\wedge}, \quad &\ha\calH^{\nu, \nu'}_{\le w} = \calH^{\nu, \nu'}_{\le w}\otimes_{\rmH^*_{G_{\ubar 0}}}\rmH^{\wedge},\\
	\ha\calH = \prod_{\ubar\nu'\in \ubar\Xi}\bigoplus_{\ubar\nu\in \ubar\Xi}\ha\calH^{\nu,\nu'},\quad &\ha\calH_{\le w} = \prod_{\ubar\nu'\in \ubar\Xi}\bigoplus_{\ubar\nu\in \ubar\Xi}\ha\calH^{\nu,\nu'}_{\le w}
\end{eq}
\index{$\calH^{\nu, \nu'}_a, \calH^{\nu, \nu'}_{\le w, a}, \calH_a, \calH_{\le w, a}$}
(Recall that $\calH^{\nu, \nu'}$ depends only on the orbits $\ubar\nu, \ubar\nu'\in \ubar \Xi$ up to canonical isomorphism, \cfauto{subsec:convolution}.
In the category of topological abelian groups, the topology on the direct sum is the weak topology of the union of the finite sums, whereas the topology on the finite sum coincides with the product topology since the category is $\bfZ$-linear.) \par
Since the convolution product $\calH^{\nu,\nu'}\times \calH^{\nu', \nu''}\to\calH^{\nu,\nu''}$ (\autoref{subsec:convolution}) is $\rmH_{G_{\ubar 0}}$-linear, the completion at $\frakm_0$ yields a continuous linear map
\begin{eq}
	\ha\calH^{\nu,\nu'}\times \ha\calH^{\nu',\nu''}\to \ha\calH^{\nu, \nu''}.
\end{eq}
Letting vary $\ubar\nu, \ubar\nu', \ubar\nu''\in \ubar\Xi$, it induces on $\ha\calH$ a structure of topological ring. \par
(recall that $\bfI^{\nu}$ depends only on the class $\ubar\nu\in \ubar \Xi$) with kernel being topologically nilpotent and closed.
\begin{defi}
	A module $M$ of $\ha\calH$ is called {\bf smooth} if the action of $\ha\calH$ on $M$ is continuous when $M$ is equipped with the discrete topology. Equivalently, $M$ is smooth if for each $m\in M$, the annihilator
	\begin{eq}
		\ann(m) = \left\{h\in \calH_a\;;\; hm = 0  \right\}
	\end{eq}
	is open in $\ha\calH$.
\end{defi}
Recall the isomorphism $\calH^{\nu, \nu'} \cong \Ext^*_{G_{\ubar 0}}(\bfI^{\nu'}, \bfI^{\nu})$. The following fact is standard about extension algebras.
\begin{lemm}\label{lemm:simples}
	There is a canonical bijection between the set of isomorphic classes of smooth simple modules of $\ha\calH$ and the set of simple constituents of $\bigoplus_{\ubar\nu\in \ubar\Xi}\bigoplus_{k\in \bfZ}\prescript{p}{}\rmH^k\bfI^{\nu}$.
\end{lemm}
	See~\cite[6.1]{vasserot05} for a proof. \hfill\qedsymbol

\subsection{Spectra of the polynomial actions}\label{subsec:spectra}
Let $\nu\in \Xi$. We consider the folloing ring map
\begin{eq}
	\Phi^{\nu,\nu}:\bfS\cong\bfS^{\nu}\cong \calH^{\nu,\nu}_e\hookrightarrow \ha\calH^{\nu,\nu}_e,
\end{eq}
The first map is the canonical isomorphism~\cfauto{subsec:poly}, followed by the isomorphism~\autorefitem{lemm:isostr}{iv}, the obvious inclusion, the $\frakm_0$-adic completion.\par

Let $\bfx^{\nu}\in \bbE^{\nu}$\index{$\bfx^{\nu}$} be the image of the orthogonal projection (with respect to the Killing form) of $\bfx\in \bbA$ onto $\bbE^{\nu}$ and let $\bfx_{\nu}^{\can}$\index{$\bfx^{\can}_{\nu}$} be the image of $\bfx^{\nu}$ under the canonical isomorphism $\bbE^{\nu}\cong \bbE^{\can}$ \cfauto{subsec:conjxi}.\par
The following results describe the spectrum of the $\bfS$-actions.

\begin{lemm}[{\cite[4.3.4]{LYIII}}]\label{coro:support}
	For any $\nu\in \Xi$, the map $\Phi^{\nu,\nu'}$ identify $\ha\calH^{\nu, \nu'}_e$ with the completion of $\bfS$ at $\bfx_{\nu}\in \bbE$.
\end{lemm}
\qed \par

\section{Morphism from the degenerate double affine Hecke algebra}\label{sec:action}

Following the strategy of \cite{vasserot05} and \cite{LYIII}, we will construct a morphism $\bbH \to \ha\calH$ from the degenerate double affine Hecke algebra (dDAHA) to the convolution algebra.
\subsection{Degenerate double affine Hecke algebra}\label{subsec:daha}
In order to describe the degenerate double affine Hecke algebra, we need to define an affine root system on the affine subspace $\bbE\subset\bbA$. The restriction of the affine root system $S$ to the subspace $\bbE$ yield an affine root system $\til S'$ on $\bbE$. However, we define $\til S$ to be an affine root system on $\bbE$ isogeneous to $\til S'$ such that $\til S$ is isomorphic to the affinisation of some finite root system $R$.
For each $a\in \til\Delta$, there is a positive integer $c_a\ge 2$ introduced in~\cite[2.4.8]{LYIII}. This number is described in~\cite[\S 7]{lusztig95c}. The constants $c_a$ need to be renormalised according to the isogeny that we have chosen for $\til S$. In the case of principal block (i.e. $M = T$ is a maximal torus), we have $c_a = 2$ for each $a\in \Delta$. In the case where $R$ is of type $B/C$, we assume $R$ to be of type $BC$, so that $c_a = c_b$ whenever $a,b\in \til\Delta$ are conjugate by the extended Weyl group. \par
Define the degenerate double affine Hecke algebra attached to the admissible system $\xi$ as the one defined in~\autoref{subsec:dDAHA} with the affine root system $(\bbE, \til S')$ and with parameters $\left\{(d/2m)c_a\right\}_{a\in \til S}$. \par
More explicitely, the dDAHA is defined to be $\bbH = \bfC \til\calW\otimes\bfS$ with the following commutation laws:
\begin{eq}
	\til s_a\; f - \til s_a(f)\; \til s_a = (d/2m)c_a \frac{f - \til s_a(f)}{a}, \quad a\in \til \Delta
\end{eq}
where $\til s_a\in \til \calW$ is the orthogonal reflection on $\bbE$ with respect to the relative affine hyperplane $\{a = 0\}$ on $\bbE$.

\begin{rema}
	The class of degenerate DAHAs which can be constructed in the present setting is limited for non-simply laced root systems. Since the constants $c_a$ are certain integers determined by the cuspidal pair $(\rmO, \scrC)$, only certain integral proportions between parameters can appear.
\end{rema}

\subsection{Graded affine Hecke algebras}\label{subsec:affinehecke}
Let $J^{\can}\subsetneq \Delta^{\can}$ be a proper subset containing $I^{\can}$ \cfauto{subsec:cusp}. We have a parabolic Coxeter subsystem $\left( \til \calW_{J^{\can}}, \til J^{\can} \right)$ of the canonical relative Weyl group $\left( \til \calW^{\can}, \til \Delta^{\can} \right)$, where $\til J^{\can} = \left\{ \til s_{a}\;;\; a\in \Delta \setminus J \right\}$ \cfauto{subsec:relCox}. We define $\bbH_J$ to the subalgebra of $\bbH$ generated by $\bfS$ and $s_a$ for $a\in \til\Delta$. In the case where $J^{\can} = I^{\can}$, we recover the polynomial algebra
\begin{eq}
	\bbH_{I^{\can}} = \bfS.
\end{eq}
\par

Define $\Xi_{J^{\can}} = \partial_{J^{\can}}\Xi\subseteq \frakF_{J^{\can}}$\index{$\Xi_{J^{\can}}$}. Let $\sigma\in \Xi_{J^{\can}}$. The tuple $\left(H, H_0, \frakh, \frakh_*\right) = \left(L^{\sigma}, L^{\sigma}_0, \frakl^{\sigma}, \frakl^{\sigma}_*\right)$\index{$H, H_0, \frakh, \frakh_*$} is a graded pseudo-Levi subgroup of $G$ \cfauto{subsec:spiralfacet}. We define $\frakF^{\sigma}_{I^{\can}} =  \left\{ \nu\in \frakF_{I^{\can}}; \partial_{J^{\can}}\nu = \sigma \right\}$. Then the stabiliser $\Stab_{W}(\sigma)$ acts transitively on $\frakF_{I^{\can}}^{\sigma}$. \par

Each $\nu\in \frakF_{I^{\can}}^{\sigma}$ gives rise to a parabolic subgroup $P^{\sigma\le\nu} \subseteq H$\index{$P^{\sigma\le \nu},\frakp^{\sigma\le\nu}_*, \fraku^{\sigma\le\nu}_*$} whose Lie algebra $\frakp^{\sigma\le\nu}_*$ is $\bfZ$-graded and that $\frakp^{\sigma\le\nu}_n = \frakp^{\nu}_n\cap \frakh$. We define also $\fraku^{\sigma\le\nu}_n = \fraku^{\nu}_n\cap \frakh$ so that $U^{\sigma\le\nu} = \exp\left( \fraku^{\sigma\le\nu} \right)$ is the unipotent radical of $P^{\sigma\le \nu}$. Notice that we have a Levi decomposition $\frakp^{\sigma\le \nu}_* = \frakl^{\nu}_*\oplus \fraku^{\sigma\le \nu}_*$ which respects the gradings.\par

Since there exists $\nu_0\in \frakF^{\sigma}_{I^{\can}}\cap \Xi$, the Levi $\frakl^{\nu_0}$ carries a $L^{\nu_0}$-equivariant cuspidal local system $\scrC^{\nu_0}$,~\cfauto{subsec:conjxi}. Since for each $\nu\in \frakF^{\sigma}_{I^{\can}}$, the quotient stacks $\left[ L^{\nu}\backslash \frakl^{\nu} \right]$ and $\left[ L^{\nu_0}\backslash \frakl^{\nu_0} \right]$ are canonically isomorphic, there is for each $\nu\in \frakF^{\sigma}_{I^{\can}}$ a $L^{\nu}$-equivariant cuspidal local system $\scrC^\nu$ on a nilpotent orbit in $\frakl^{\nu}$.

Choose any $\nu\in \frakF^{\sigma}_{I^{\can}}$. Let $\dot\frakh_{\rmO} = H\times^{P^{\sigma\le\nu}}\left(\rmO^{\nu}\times \fraku^{\sigma\le\nu} \right)$\index{$\dot\frakh_{\rmO}$} and let $\alpha:\dot\frakh_{\rmO}\to \frakh$ be the natural morphism defined by the adjoint action. As before, we denote $H_{q} = H\times \Cq$\index{$H_{q}$}. The group $H_q$ acts on $\dot\frakh$ in a natural way. The $L^{\nu}_q$-equivariant cuspidal local system $\scrC^{\nu}$ on $\rmO^{\nu}\subseteq \frakp^{\sigma\le\nu}$ induces a $H_{q}$-equivariant local system $\dot\scrC$ on $\dot\frakh_{\rmO}$. Put $\ddot\frakh_{\rmO} = \dot\frakh_{\rmO}\times_{\frakh_{\rmO}}\dot\frakh_{\rmO}$\index{$\ddot\frakh_{\rmO}$}. There is a complex $\scrK = \calHom\left(q_2^*\dot{\scrC},q_1^!\dot{\scrC}\right)$\index{$\scrK$} on $\ddot\frakh_{\rmO}$, where $q_1, q_2: \ddot\frakh_{\rmO} \to\dot\frakh_{\rmO}$ are the canonical projections. The cohomology $\rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO}, \scrK \right)$, being isomorphic to $\Ext^*_{H_{q}}\left(\alpha_* \dot{\scrC}, \alpha_* \dot{\scrC}  \right)$, becomes a ring with the Yoneda product. \par

\begin{theo}[8.11,\cite{lusztig95}; 4.4.5,\cite{LYIII}]\label{theo:isoft}
				There is a canonical isomorphism of algebras
				\begin{eq}
					\bbH_{J^{\can}}\cong \rmH^*_{H_q}\left( \ddot\frakh_{\rmO}, \scrK \right)\mid_{u=d/2m}.
				\end{eq}
\end{theo}
\qed \par

\subsection{Localisation of graded affine Hecke algebras}\label{subsec:fixedpoint}
The pseudo-Levi subalgebra $\frakh$ arising from $\sigma\in \Xi$ is $\bfZ$-graded. Let $\bfC^{\times}_{\deg} = \bfC^{\times}$ be the torus which acts on $\frakh_n$ by weight $2n$ for each $n\in \bfZ$. Define a cocharacter
\begin{eq}
	\iota:\bfC^{\times} \to \bfC^{\times}_{\deg}\times \Cq, \quad t\mapsto \left( t, t^{-d} \right).
\end{eq}
Then $\iota$ acts on the variety $\frakh$ and all related geometric objects.
Fixed points of $\iota$ are given by
\begin{eq}
	\left(\frakh\right)^{\iota} = \frakh_{d}, \quad  H^{\iota} = H_{0}.
\end{eq}
We apply a version of the equivariant localisation with respect to $\iota$ on the variety $\ddot\frakh_{\rmO}$. \par

We will denote $\Xi_{\sigma} = \Xi\cap \frakF^{\sigma}_{I^{\can}}$ and $\ubar\Xi_{\sigma}$ the image of $\Xi_{\sigma}$ in $\ubar\Xi$.\index{$\Xi_{\sigma}, \ubar\Xi_{\sigma}$}
For each $\nu, \nu'\in \Xi_{\sigma}$, we set
\begin{eq}
	\calZ_H^{\nu, \nu'} = \left(H_0\times^{P_0^{\sigma\le\nu}}\left( \rmO^{\nu}_{d}\times \fraku^{\sigma\le\nu}_{d}\right)\right)\times_{\frakh_{d}}\left(H_0\times^{P_0^{\sigma\le\nu'}}\left( \rmO^{\nu'}_{d}\times \fraku^{\sigma\le \nu'}_{d} \right)\right).
\end{eq}
\index{$\ha\calZ_H^{\nu, \nu'}$}
It is a variant of the Steinberg-type varieties $\ha\calZ^{\nu, \nu'}$. Following the same procedure as in~\autoref{subsec:convolution}, we define a $H_{0}\times\Cq$-equivariant complex $\scrK_H$\index{$\scrK_H$} on $\calZ_H^{\nu, \nu'}$ as well as a convolution product on the cohomology $\rmH^*_{H_{0,q}}\left( \calZ_{H}^{\nu, \nu'}, \scrK_H \right)$.
\begin{lemm}
	\label{lemm:dec}
	If $i:\ddot\frakh_{\rmO}^{\iota}\to \ddot\frakh_{\rmO}$ denote the inclusion of fixed points, then there is a canonical isomorphism of algebras
	\begin{eq}
		\bigoplus_{\ubar\nu, \ubar\nu'\in \ubar\Xi_{\sigma}}\rmH^*_{H_{0}}\left( \calZ_{H}^{\nu, \nu'}, \scrK_H \right)\cong\rmH^*_{H_{0}}\left( \ddot\frakh_{\rmO}^{\iota}, i^*\scrK\right)
	\end{eq}
\end{lemm}
\begin{proof}
	The $\iota$-fixed points can be described as
	\begin{eq}
		\ddot\frakh_{\rmO}^{\iota} = \bigsqcup_{\ubar\nu, \ubar\nu'\in W_{\bfx}\backslash\frakF_{I^{\can}}^{\sigma}} \left(H_0\times^{P_0^{\sigma\le\nu}}\left( \rmO^{\nu}_{d}\times \fraku^{\sigma\le\nu}_{d}\right)\right)\times_{\frakh_{d}}\left(H_0\times^{P_0^{\sigma\le\nu'}}\left( \rmO^{\nu'}_{d}\times \fraku^{\sigma\le \nu'}_{d} \right)\right).
	\end{eq}
	It is shown in the proof of \cite[4.5.4]{LYIII} that $\rmO^{\nu}_{d} = \emptyset$ if $\nu\notin \Xi_{\sigma}$. Similarly $\rmO^{\nu'}_{d} = \emptyset$ if $\nu'\notin \Xi_{\sigma}$.
	By the cleaness of $\scrC$ (\autoref{subsec:convolution}), the restriction of cohomlogy yields an isomorphism
	\begin{eq}
		\bigoplus_{\ubar\nu, \ubar\nu'\in \ubar\Xi_{\sigma}}\rmH^*_{H_{0}}\left(\calZ_{H}^{\nu, \nu'}, \scrK_H \right)\cong \rmH^*_{H_{0}}\left( \ddot\frakh_{\rmO}^{\iota}, i^*\scrK \right).
	\end{eq}

\end{proof}

The cocharacter $\iota:\bfC^{\times}\to H_{0,q}$ yields a closed point $\iota\in \Spec(\rmH^*_{H_{0,q}})$.  The localisation theorem ({\it cf.} Evens--Mirkovi\'c~\cite[4.10]{EM}) states that we have an isomorphism of algebras after the completion of equivariant cohomology groups at the maximal ideal $\frakm_{\iota / 2m}\subset \rmH^*_{H_{0,q}}$:
\begin{eq*}\label{equa:EMloc}
	\rmH_{H_{q}}^{*}\left( \ddot\frakh_{\rmO}, \scrK \right)^{\wedge}_{\frakm_{\iota / 2m}}\cong\rmH_{H_{0,q}}^{*}\left( \ddot\frakh_{\rmO}^{\Cx}, i^*\scrK \right)^{\wedge}_{\frakm_{\iota / 2m}}.
\end{eq*}
Combining~\autoref{theo:isoft},~\autoref{lemm:dec} and~\eqref{equa:EMloc}, we obtain
\begin{eq*}\label{equa:phi-tf}
	\bbH_{J^{\can}} &\cong \rmH_{H_q}^{*}\left( \ddot\frakh_{\rmO}, \scrK \right)\mid_{u=d/2m} \to \rmH_{H_q}^{*}\left( \ddot\frakh_{\rmO}, \scrK \right)^{\wedge}_{\frakm_{\iota / 2m}}\mid_{u=d/2m} \\
	&\cong \rmH_{H_{0,q}}^{*}\left( \ddot\frakh_{\rmO}^{\iota}, i^*\scrK \right)^{\wedge}_{\frakm_{\iota / 2m}}\mid_{u=d/2m}\cong \bigoplus_{\ubar{\nu}', \ubar{\nu}''\in \ubar\Xi_{\sigma}} \rmH_{H_{0}}^{*}\left( \calZ_H^{\nu', \nu''}, \scrK_H \right)^{\wedge}_{\frakm_{0}}.
\end{eq*}
The composition is injective and the image is dense in the $\frakm_0$-adic topology on the last term.

\subsection{Map from $\bbH_J$ to $\ha\calH$}\label{subsec:reduction} We introduce intermediate morphisms which relate $\calZ^{\nu, \nu'}_{H}$ \cfauto{subsec:fixedpoint} and $\calZ^{\nu, \nu'}$ \cfauto{subsec:convolution} and deduce a ring homomorphism on the convolution algebras. \par
We fix $\sigma\in \Xi_{J^{\can}}$. For $\nu,\nu'\in \Xi_{\sigma}$, we put
\begin{eq}
				\til\calZ_H^{\nu, \nu'} = \left(P^{\sigma}_0\times^{P_0^{\nu}}\left( \rmO^{\nu}_{d}\times  \fraku^{\nu}_{d} \right)\right)\times_{\frakg_{\ubar d}}\left(P^{\sigma}_0\times^{P_0^{\nu'}}\left( \rmO^{\nu}_{d}\times  \fraku^{\nu'}_{d}  \right)\right).
\end{eq}
\index{$\til\calZ_H^{\nu, \nu'}$}

\begin{lemm}
\label{lemm:reduction-map}
	For any $\nu,\nu'\in \Xi_{\sigma}$, there is a diagram of canonical morphisms
				\begin{eq}
								\begin{tikzcd}
									\calZ^{\nu, \nu'}_H  &\arrow[two heads]{l}{p}\til \calZ^{\nu, \nu'}_H \arrow[hookrightarrow]{r}{i} & \calZ^{\nu, \nu'}
								\end{tikzcd}
				\end{eq}
				in which
				\begin{enumerate}[label=(\roman*)]
								\item
												$p$ is an affine fibration with fibers isomorphic to $\fraku^{\sigma}_{d}$, whereas
								\item
												$i$ is a closed immersion, which factorises as
												\begin{eq}
																\begin{tikzcd}
																	\til \calZ_H^{\nu, \nu'}\arrow{dr}[swap]{(e, \id)}\arrow{r}{i} & \bigcup_{[w]\in \calW_{I^{\can}}\backslash \calW_{J^{\can}} /  \calW_{I^{\can}} }\calZ^{\nu, \nu'}_w \arrow{r}{\subseteq}  & \calZ^{\nu, \nu'}\\
																	&G_{\ubar 0}\times^{P_0^{\sigma}}\til \calZ_H^{\nu, \nu'}\arrow{u}{\cong}
																\end{tikzcd}.
												\end{eq}
				\end{enumerate}
\end{lemm}
\begin{proof}
	Since the adjoint action of $P_0^{\nu}$ on $\rmO_{d}\times \left( \fraku^{\nu}_{d} / \fraku^{\sigma}_{d} \right)$ factorises through the quotient $P_0^{\nu} \to P_0^{\nu} / U_0^{\sigma}\cong P_0^{\sigma\le\nu}$ \cfauto{subsec:affinehecke}, we have the isomorphisms
				\begin{eq}
								P^{\sigma}_0\times^{P_0^{\nu}}\left( \rmO_{d}\times \left( \fraku^{\nu}_{d} / \fraku^{\sigma}_{d} \right) \right) \cong \left(P^{\sigma}_0 / U^{\sigma}_0\right)\times^{P_0^{\nu} / U_0^{\sigma}}\left( \rmO_{d}\times \left( \fraku^{\nu}_{d} / \fraku^{\sigma}_{d} \right) \right) \cong H_0\times^{P_0^{\sigma\le\nu}}\left( \rmO_{d}\times  \fraku^{\sigma\le\nu}_{d} \right)
				\end{eq}
				which fits into the following sequence
				\begin{eq}
								P^{\sigma}_0\times^{P_0^{\nu}}\left( \rmO_{d}\times  \fraku^{\nu}_{d} \right) \xrightarrow{\text{projection}}
								P^{\sigma}_0\times^{P_0^{\nu}}\left( \rmO_{d}\times \left( \fraku^{\nu}_{d} / \fraku^{\sigma}_{d} \right) \right) \cong
								H_0\times^{P_0^{\sigma\le\nu}}\left( \ha\rmO_{d}\times  \fraku^{\sigma\le\nu}_{d} \right)
				\end{eq}
				in which the first morphism is a locally trivial fibration with fibers $\fraku_{d}^{\sigma}$.
				Substituting these morphisms into the definition of $\til\calZ^{\nu,\nu'}_H$ yields morphism $p$. The morphism $i$ comes from the inclusion $P^{\sigma}_0\hookrightarrow G_{\ubar 0}$. This proves (i).\par
				The statement (ii) is due to the fact that the image of the inclusion of $G_{\ubar 0}$-schemes
				\begin{eq*}\label{equa:indG}
					G_{\ubar 0}\times^{P^{\sigma}_0}\left(\left(P^{\sigma}_0/ P_0^{\nu}\right) \times \left( P^{\sigma}_0/ P_0^{\nu'}\right)\right) \to \left(G_{\ubar 0}/ P_0^{\nu}\right) \times \left( G_{\ubar 0}/ P_0^{\nu'}\right).
				\end{eq*}
				identified with $\bigcup_{[w]\in \calW_{I^{\can}}\backslash \calW_{J^{\can}} /  \calW_{I^{\can}}  }\calX^{\nu, \nu'}_w$.
			\end{proof}

Convolution products can be defined similarly on $Z^{\nu,\nu'}_H$ and on $\til Z^{\nu,\nu'}_H$.

\begin{prop}\label{prop:reduction-convolution}
	The morphisms $i$ and $p$ considered in~\autoref{lemm:reduction-map} induce maps
				\begin{eq}
					\rmH^*_{H_{0}}\left( \calZ^{\nu, \nu'}_{H}, \scrK_H \right) &\cong \rmH^*_{P^{\sigma}_{0}}\left( \til\calZ^{\nu, \nu'}_{H}, \til\scrK_{H} \right)\hookrightarrow \rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu, \nu''}, \scrK \right),
				\end{eq}
				which commute with the convolution products.
\end{prop}
\begin{proof}
				We have a commutative diagram
				\begin{eq}
								\begin{tikzcd}
												\left(\calT^{\nu}\times\calT^{\nu'}\right) \times \left(\calT^{\nu'}\times\calT^{\nu''}\right) & \calT^{\nu}\times\calT^{\nu'}\times\calT^{\nu''}\arrow[swap]{l}{d}\\
												\calZ^{\nu, \nu'} \times \calZ^{\nu', \nu''} \arrow{u}& \calZ^{\nu, \nu'} \times_{ \calT^{\nu'}} \calZ^{\nu', \nu''} \arrow{u}\arrow[swap]{l}{\gamma} \arrow{r}{\mu}\arrow[start anchor=center, end anchor=center, phantom]{ul}{\square} & \calZ^{\nu, \nu''} \\
												\til\calZ^{\nu, \nu'}_H \times \til\calZ^{\nu', \nu''}_H \arrow[hookrightarrow]{u}{i\times i} \arrow[two heads]{d}{p\times p} & \til\calZ^{\nu, \nu'}_H \times_{ \calT^{\nu'}} \til\calZ^{\nu', \nu''}_H \arrow[swap]{l}{\til\gamma_H} \arrow{r}{\til\mu_H}\arrow[hookrightarrow]{u}{i\times i} \arrow[two heads]{d}{p\times p}\arrow[start anchor=center, end anchor=center, phantom]{ul}{\square}\arrow[start anchor=center, end anchor=center, phantom]{dr}{\square}& \til\calZ^{\nu, \nu''}_H \arrow[hookrightarrow]{u}{i}\arrow[two heads]{d}{p}\arrow{r}\arrow[start anchor=center, end anchor=center, phantom]{dr}{\square} & \frakp^{\sigma}_{d}\arrow[two heads]{d}\\
												\calZ^{\nu, \nu'}_H \times \calZ^{\nu', \nu''}_H & \calZ^{\nu, \nu'}_H \times_{\calT^{\nu'}_H} \calZ^{\nu', \nu''}_H \arrow[swap]{l}{\gamma_H} \arrow{r}{\mu_H} & \calZ^{\nu, \nu''}_H \arrow{r} & \frakh_{d}\\
								\end{tikzcd},
				\end{eq}
				containing four cartesian squares as indicated by a box ``$\square$''. The second line of it defines the convolution product on $\rmH^*_{G_{\ubar 0}}\left( \calZ^{\nu,\nu'}, \scrK \right)$, the third line defines the convolution product on $\rmH^*_{P^{\sigma}_{0}}\left( \til\calZ^{\nu,\nu'}_H, \til\scrK_H \right)$, whereas the fourth line defines the convolution product on $\rmH^*_{H_{0}}\left( \calZ^{\nu,\nu'}_H, \scrK_H \right)$. The diagram induces a corresponding commutative diagram of cohomology, thanks to base change and composability of units and co-units of adjuntion and the fact that there is a canonical isomorphism $i^*\scrK \cong \til \scrK_H$.
\end{proof}

Let $\ubar \Xi_{J^{\can}} = W_{\bfx} \backslash\Xi_{J^{\can}}$, so that there are partitions of sets
\begin{eq}
	\Xi = \bigsqcup_{\sigma\in \Xi_{J^{\can}}}\Xi_{\sigma},\quad \ubar\Xi = \bigsqcup_{\sigma\in \ubar\Xi_{J^{\can}}}\ubar\Xi_{\sigma}.
\end{eq}

We obtain a sequence of injective maps of algebras
\begin{eq}
	\bbH_{J^{\can}} &\xrightarrow{\text{\eqref{equa:phi-tf}}} \prod_{\ubar\sigma\in \ubar\Xi_{J^{\can}}}\bigoplus_{\ubar{\nu}', \ubar{\nu}''\in \ubar\Xi_{\sigma}} \rmH_{H_{0}}^{*}\left( \calZ_H^{\nu', \nu''}, \scrK \right)^{\wedge}
	\xrightarrow{(\ref{prop:reduction-convolution})} \prod_{\ubar\sigma\in \ubar\Xi_{J^{\can}}}\bigoplus_{\ubar{\nu}', \ubar{\nu}''\in \ubar\Xi_{\sigma}} \rmH_{G_{\ubar 0}}^{*}\left( \calZ^{\nu', \nu''}, \scrK \right)^{\wedge}\\
	&\hookrightarrow \prod_{\ubar{\nu}''\in \ubar\Xi}\bigoplus_{\ubar{\nu}'\in \ubar\Xi} \rmH_{G_{\ubar 0}}^{*}\left( \calZ^{\nu', \nu''}, \scrK \right)^{\wedge}= \ha\calH,
\end{eq}
\cfauto{subsec:fixedpoint}. We denote the composite of it by $\Phi_{J^{\can}}: \bbH_{J^{\can}}\to \ha\calH$.\index{$\Phi_{J^{\can}}$}

\begin{prop}\label{prop:genpara}
	For $\nu, \nu'\in \Xi$ and $w\in \til \calW_{J^{\can}}$ such that $\ubar\nu =  w\ubar\nu'$, the graded piece $\ha\calH^{\nu, \nu'}_{\le w} / \ha\calH^{\nu, \nu'}_{< w}\cong \ha\calH^{\nu, \nu'}_{w}$ is a free of rank 1 as left $\ha\calH^{\nu, \nu}_{e}$-module and as right $\ha\calH^{\nu', \nu'}_{e}$-module generated by the image of $\Phi_{J^{\can}}(w)$.
\end{prop}
\begin{proof}
	By~\cite[4.9]{lusztig88}, there is a decomposition of $\rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO, e}, \scrK \right)$-bimodules
	\begin{eq}
		\rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO}, \scrK \right) \cong \bigoplus_{w\in \til \calW_{J^{\can}}}\rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO, w}, \scrK \right),
	\end{eq}
	By the construction of Lusztig's isomorphism~\autoref{theo:isoft}, the graded piece $\rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO, w}, \scrK \right) = \rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO, \le w}, \scrK \right) / \rmH^*_{H_{q}}\left( \ddot\frakh_{\rmO, < w}, \scrK \right)$ is freely generated by the image of $w\in \bbH_J$. The maps~\eqref{equa:phi-tf} and~\ref{prop:reduction-convolution} then sends it to a generator of $\ha\calH^{\nu,\nu'}_w$. \par
\end{proof}

\subsection{Map from $\bbH$ to $\ha\calH$}\label{subsec:repdaha} We prove that the maps $\Phi_{J^{\can}}$ are compatible one to another and thus define a morphism from the dDAHA to $\ha\calH$. The proof of the following lemma is straightfoward.
\begin{lemm}\label{prop:compatible}
	Given two proper subsets $J^{\can}, K^{\can}$ of $S^{\can}$ such that $I^{\can}\subseteq J^{\can}\subseteq K^{\can}\subsetneq S^{\can}$, there is a canonical inclusion of graded affine Hecke algebras $\bbH_{ J^{\can}}\subseteq \bbH_{K^{\can}}$ and the maps $\Phi_{J^{\can}}$ and $\Phi_{K^{\can}}$ are compatible in the sense that
	\begin{eq}
		\Phi_{K^{\can}}\mid_{\bbH_{J^{\can}}} = \Phi_{J^{\can}}.
	\end{eq}
\end{lemm}
\qed \par
\begin{coro}
	There is a ring homomorphism
	\begin{eq}
		\Phi : \bbH \to \ha\calH.
	\end{eq}
	\index{$\Phi$}
\end{coro}
\begin{proof}
	From the definition of $\bbH$, we clearly have $\bbH \cong \varinjlim_{\substack{I^{\can}\subseteq J^{\can}\subsetneq S^{\can}}} \bbH_{ J^{\can}}$. By \autoref{prop:compatible}, the system $\{\Phi_{J^{\can}}\}_{J^{\can}}$ is compatible under restriction. We define $\Phi = \varinjlim_{\substack{I^{\can}\subseteq J^{\can}\subsetneq S^{\can}}}\Phi_{J^{\can}}$, which is a ring homomorphism
	\begin{eq}
		\Phi : \bbH \cong \varinjlim_{\substack{I^{\can}\subseteq J^{\can}\subsetneq S^{\can}}} \bbH_{ J^{\can}} \to \ha\calH.
	\end{eq}
\end{proof}

\subsection{Density of image}\label{subsec:density}
Following the line of~\cite[4.8, 4.9]{vasserot05}, we prove that the image of $\Phi$ is dense in the product topology of the convolution algebra $\ha\calH$.\par
Recall that $\ha\calH$ defined in~\autoref{subsec:special} is equipped with a topology.  \par
\begin{lemm}\label{prop:generator}
	Let $w\in \til \calW^{\can}$. For each $\nu\in \Xi$, letting $\nu' =  w^{-1}\nu$, the graded piece $\ha\calH^{\nu, \nu'}_{w}$ is free of rank $1$ as left $\ha\calH^{\nu, \nu}_{e}$-module and as right $\ha\calH^{\nu', \nu'}_e$-module, with generator the image of $\Phi(w)$ in $\calH^{\nu, \nu'}_{w,a}$. \par
\end{lemm}
\begin{proof}
	We prove the assertion by induction on $\til\ell(w)$. When $\til\ell(w) = 0$, this is trivial. When $\til\ell(w) = 1$, $w = \til s_a\in \til \calW$ is a relative simple reflection and the result follows from~\autoref{prop:genpara}. \par
	Suppose then $\til \ell(w) \ge 2$. We choose $s\in \til \calW$ such that $\til\ell(s) = 1$ and $\til\ell(ws) = \til \ell(w) - 1$. Since the image of $\Phi(ws)$ in $\ha\calH^{\nu,  (ws)^{-1}\nu }_{w}$ and the image of $\Phi(s)$ in $\ha\calH^{(ws)^{-1}\nu , \nu'}_{s}$ are free generators by inductive hypothesis, by the surjectivity~\autorefitem{lemm:isostr}{iii}, $\Phi(w) = \Phi(ws)\Phi(s)$ in turn generates freely $\ha\calH^{\nu, \nu'}_{ws}$.
\end{proof}

\begin{lemm}\label{prop:dense}
	In the particular case where $J^{\can} = I^{\can}$, the image of the morphism
	\begin{eq}
		\Phi_{I^{\can}}: \bfS^{\can} = \bbH_{ I^{\can}} \to \ha\calH_{e}
	\end{eq}
	is dense.
\end{lemm}
\begin{proof}
	By definition $\ha\calH_{e} = \prod_{\ubar\nu\in \ubar\Xi}\ha\calH^{\nu,\nu}_{e}$. Since the composite of $\Phi_{I^{\can}}$ with each projection $\prod_{\ubar\nu\in \ubar\Xi}\ha\calH^{\nu,\nu}_{e}\to \ha\calH^{\nu,\nu}_{e}$ is equal to the map $\Phi^{\nu,\nu}$ of~\autoref{coro:support}. By~\autoref{coro:support}, $\ha\calH_{e}^{\nu,\nu}$ is a topological $\bfS^{\can}$-module supported on $\bfx^{\can}_{\nu}\in \bbE^{\can}$ and that the image of the composition
	\begin{eq}
		\bfS^{\can}\xrightarrow{\Phi_{I^{\can}}} \ha\calH_{e}\xrightarrow{\text{proj}} \ha\calH^{\nu, \nu}_{e}
	\end{eq}
	is dense.
	If $\nu,\nu'\in \Xi$ are such that $\bfx_{\nu}^{\can} = \bfx_{\nu'}^{\can}$, then $\nu$ and $\nu'$ are conjugate by $W_{\bfx}$, which implies that $\ubar\nu = \ubar \nu'\in \ubar\Xi$. As there is a canonical isomorphism $\Spec \bbH_{I}\cong \bbE^{\can}\otimes_{\bfQ}\ba\bfQ_{\ell}$, we may consider $\bfx^{\can}_{\nu}$ as closed points of $\Spec \bfS^{\can}$. \par

	Since we have seen that $\bfx^{\can}_{\nu}\neq \bfx^{\can}_{\nu'}$ if $\ubar\nu\neq \ubar \nu'$, for any finite subset $\ubar\Xi'\subseteq \ubar\Xi$, the composite
	\begin{eq}
		\bfS^{\can} \xrightarrow{\Phi_{I^{\can}}} \prod_{\nu\in \ubar \Xi}\ha\calH^{\nu,\nu}_{e}\rightarrow \prod_{\nu\in \ubar \Xi'}\ha\calH^{\nu,\nu}_{e}
	\end{eq}
	is dense as well, by the chinese remainder theorem. Hence, by the definition of product topology, the image of $\Phi_{I^{\can}}$ is dense.
\end{proof}

\begin{theo}\label{theo:density}
				The image of $\Phi:\bbH \to \ha\calH$ is dense.
\end{theo}
\begin{proof}
	Taking the subset $I^{\can} = I^{\can}_{\bbE}\subseteq S^{\can}$, we have $\Phi_{I^{\can}}: \bbH_{ I^{\can}}\to \ha\calH_{e}$ with dense image, after~\autoref{prop:dense}, whence $\image\left( \Phi \right)\cap \ha\calH_{e}$ is dense in $\ha\calH_{e}$. We can prove by induction on $\ell(w)$ that $\image\left( \Phi \right)\cap \ha\calH_{\le w}$ is dense in $\ha\calH_{\le w}$ for each $w\in \til \calW^{\can}$. Indeed, assume that $\image\left( \Phi \right)\cap \ha\calH_{\le w'}$ is dense in $\calH_{\le w',a}$ for all $w'$ such that $\ell(w') < \ell(w)$. Then
	\begin{eq}
		\sum_{\ell(w') < \ell(w)}\left(\image\left( \Phi \right) \cap \ha\calH_{\le w'}\right) \subseteq\image\left( \Phi \right) \cap \ha\calH_{< w}.
	\end{eq}
	is dense in $\ha\calH_{< w}$. Consider following diagram
	\begin{eq}
		\begin{tikzcd}
			0 \arrow{r} & \image\left( \Phi \right) \cap \ha\calH_{< w} \arrow{r}\arrow{d}{\subseteq} &\image\left( \Phi \right) \cap \ha\calH_{\le w} \arrow{r}\arrow{d}{\subseteq} & \frac{\image\left( \Phi \right) \cap \ha\calH_{\le w}}{\image\left( \Phi \right) \cap \ha\calH_{< w}}\arrow{r} \arrow{d}{\subseteq}& 0 \\
			0 \arrow{r} & \ha\calH_{< w}\arrow{r} & \ha\calH_{\le w}\arrow{r} & \ha\calH_{w} \arrow{r} & 0
		\end{tikzcd}
	\end{eq}
	The image of the right vertical arrow, denoted $V$, is an $\bbH_{ I^{\can}}$-submodule. By the density of $\image(\Phi_{I^{\can}})\subseteq \ha\calH_{e}$, the closure $\ba V$ is a $\ha\calH_{e}$-submodule of $\ha\calH_{w}$. By~\autoref{prop:generator}, since $\ba V$ contains $\Phi(w)$, we have $\ba V = \ha\calH_{w}$, so the right vertical arrow has dense image. Since the left arrow also has dense image, so is the middle arrow.\par

	Since $\bigcup_{w\in \til \calW^{\can}}\ha\calH_{\le w}$ is dense in $\ha\calH$, so is the image of $\Phi$.
\end{proof}

\section{Simple and proper standard modules}\label{sec:simple}

\subsection{Geometric parametrisation of simple modules}
Consider the block $\calO_{\bfx_{\nu}}(\bbH)$ of the category $\calO$ of the dDAHA $\bbH$ associated to the $W$-orbits $\{\bfx_{\nu}\}_{\nu\in \Xi}$ in $\bbE$, as defined in~\autoref{subsec:OH}. \par
The following full faithfulness results from~\autoref{theo:density}.

Recall the Lusztig sheaf $\bfI^{\nu}$ defined in~\autoref{subsec:indcusp}. Let $\bfI = \bigoplus_{\nu\in \ubar \Xi}\bfI^{\nu}$ be the (infinite) sum.
\begin{theo}[Springer correspondence]\label{theo:classification}
	\begin{enumerate}[label=(\roman*)]
		\item\label{theo:class-i}
			The pull-back via the homomorphism $\Phi$ of~\autoref{subsec:repdaha}, denoted by $\Phi^*$, induces an equivalence of category
			\begin{eq}
				\Phi^*: \ha\calH\mof^{\text{sm}}\cong \rmO_{\bfx_{\nu}}\left( \bbH \right).
			\end{eq}
		\item\label{theo:class-ii}
			For any $z\in \frakg_{\ubar d}^{\nil}$ and $\chi\in \Irr\Rep\left( \pi_0\left( G_{\ubar 0,z} \right)\right)$\index{$(z, \chi)$}, the representation of $\bbH$
			\begin{eq}
				\bfL_{z, \chi} = \Phi^*\left[\bfI : \mathrm{IC}\left(\chi\right)\right]
			\end{eq}
			\index{$\bfL_{z, \chi}$}
			is in $\calO_a\left( \bbH \right)$ and irreducible if it is non-zero.
		\item\label{theo:class-iii}
			The simple objects in $\calO_{\bfx_{\nu}}\left( \bbH \right)$ are given by $\{\bfL_{z, \chi}\}_{(z, \chi)}$, where $(z, \chi)$ runs over the $G_{\ubar 0}$-conjugacy classes of pairs, where $z\in \frakg_{\ubar d}^{\nil}$ and $\chi$ is a irreducible local system on the $G_{\ubar 0}$-orbit of $z$ such that $\IC\left( \chi \right)$ appears as a simple constituent of $\bfI$.
	\end{enumerate}
\end{theo}
\begin{rema}
	\begin{enumerate}[label=(\roman*)]
		\item
			The assertion \ref{theo:class-ii} confirms the multiplicity-one conjecture in~\cite{LYIII}.
		\item
			We have supposed that the points $\bfx_{\nu}$ are rational in $\bbE$. However, the non-rational case can be easily reduced to the rational case.
		\item
			With more care, one can remove the hypothesis that the grading on $\frakg$ is inner by working with twisted affine root systems.
	\end{enumerate}
\end{rema}
\begin{proof}
	The statement~\ref{theo:class-i} is standard, as the map $\Phi: \bbH\to \ha\calH$ is injective with dense image, see~\cite[7.6]{vasserot05}.
	The assertion (ii) follows from (i) together with~\autoref{lemm:simples} and~\autoref{theo:density}.\par

	Now let $\bfL\in\calO_a \left( \bbH\right)$ be irreducible.	Using (i), $\bfL$ can be equipped with a smooth $\ha\calH$-module structure, which is simple. By~\autoref{lemm:simples}, it must be isomorphic to some simple constituent of $\bfI$, thus one of the $\bfL_{z, \chi}$'s. This completes the proof.
\end{proof}

\subsection{Proper standard modules}\label{subsec:standard}
Let $z\in \frakg_{\ubar d}^{\nil}$ be a nilpotent element. Let $\calT^{\nu}_e$ be the fibre of $\calT^{\nu}\to \frakg_{\ubar d}$ at $z$ and let $i_z: \calT^{\nu}_z \to \calT^{\nu}$ be the closed inclusion. Consider the vector space
\begin{eq}
	\ba\Delta_{z} &= \bigoplus_{\ubar \nu\in \ubar\Xi}\rmH^*\left( \calT^{\nu}_z, i_z^!\dot{\scrC} \right).\\
\end{eq}
By the formalism of convolution algebras, for each $\nu, \nu'\in \Xi$ there is a natural map
\begin{eq}
	\calH^{\nu, \nu'}\to  \Hom\left(\rmH^*\left( \calT^{\nu'}_z, i^!\dot{\scrC} \right),\rmH^*\left( \calT^{\nu}_z, i^!\dot{\scrC} \right)\right).
\end{eq}
Taking the sum over $\ubar\nu\in \ubar\Xi$ and the product over $\ubar\nu'\in \ubar\Xi$, we obtain a smooth $\calH_a$-action on $\ba\Delta_{z}$.
Besides, there is a natural $\pi_0\left( G_{\ubar 0,z} \right)$-action on $\ba\Delta_{z}$, which commutes with the $\calH_{a}$-action. For any $\chi\in \Irr\Rep\left( \pi_0\left( G_{\ubar 0,z} \right) \right)$, we define
\begin{eq}
\ba\Delta_{z, \chi} &= \Hom_{\pi_0\left( G_{\ubar 0, z} \right)}\left(\chi, \ba\Delta_{z}\right)\
\end{eq}
\index{$\ba\Delta_{z, \chi}$}
to be the $\chi$-isotypic component, which is a $\calH_a$-submodule of $\ba\Delta_z$. We view $\ba\Delta_z$ as a $\bbH$-module via $\Phi:\bbH\to \calH_a$~\cfauto{subsec:repdaha}.
We call $\ba\Delta_{z, \chi}$ the {\bf proper standard module} of $\bbH$.
\begin{theo}\label{theo:standard}
			For each pair $(z, \chi)$ as above, the following holds.
	\begin{enumerate}[label=(\roman*)]
		\item\label{theo:std-i}
			$\ba\Delta_{z, \chi}\in \calO_{\bfx_{\nu}}\left( \bbH \right)$.
		\item\label{theo:std-ii}
			$\ba\Delta_{z, \chi}\neq 0$ if and only if $\bfL_{z, \chi}\neq 0$.
		\item\label{theo:std-iii}
			For any pair $(z',\chi')$ as above, the Jordan--H\"older multiplicity of $\bfL_{z', \chi'}$ in $\ba\Delta_{z, \chi}$ is given by
			\begin{eq}
				\left[\ba\Delta_{z, \chi}: \bfL_{z, \chi'} \right] = \sum_k\dim \Hom_{\pi_0\left( G_{\ubar 0, z} \right)}\left(\chi,\rmH^k\left( z^!\mathrm{IC}(\chi') \right)\right).
			\end{eq}
	\end{enumerate}
\end{theo}
\begin{proof}
	The assertion~\ref{theo:std-i} results from the smoothness of the $\ha\calH$-action and~\autorefitem{theo:classification}{i}. \par
	The assertion~\ref{theo:std-ii} can be proven with the same arguments as~\cite[8.17]{lusztig95b}, using the block decomposition of the equivariant category of $\frakg^{\nil}_{\ubar d}$, \cfauto{subsec:blockdecomp}.
	The assertion (iii) is standard, see~\cite[8.6.23]{CG}.
\end{proof}

\appendix

\section{Relative Coxeter groups} \label{sec:relcox}
This section is independent of the rest of the article. We recollect certain results of Lusztig~\cite{lusztig76},~\cite{lusztig95c} and~\cite{lusztig88} concerning the structure of relative Weyl groups. We generalise these results to a more general setting.
\subsection{Set of reflections}
Let $\left( W, S \right)$ be a Coxeter system with $S\subseteq W$ the set of simple reflections. We follow Bourbaki~\cite[ch 4]{Bourbaki} and define $T = \left\{wsw^{-1}\in W\;;\; s\in S, w\in W  \right\}$ to be the subset of elements that are conjugate to some simple reflection. Elements of $T$ are called {\bf reflections}. Given any $w\in W$, let $w = w_1 w_2\ldots w_q$ be a reduced decomposition with $w_i\in S$ for all $i$. For each $t\in T$, we let
\begin{eq}
				\eta\left( w, t \right) = \begin{cases}-1 & \text{if $t = w_1 w_{2} \ldots w_{j-1} w_j w_{j-1} \ldots w_2 w_1$ for some $1\le j \le q$} \\ 1 & \text{otherwise}\end{cases}.
\end{eq}
It turns out that $\eta\left( w, t \right)$ is independent of the choice of reduced decomposition. Let $T(w) = \left\{ t\in T; \eta(w, t) = -1 \right\}$ and $D(w) = T(w) \cap S$. The following properties about $T(w)$ is standard.
\begin{prop}\label{prop:propT}
				In the above notations,
				\begin{enumerate}[label=(\roman*)]
								\item
												There is a characterisation $T(w) = \left\{ t\in T; \ell(tw) < \ell(w) \right\}\subseteq T$.
								\item
												For each $w\in W$, we have $\ell(w) = \# T(w)$.
								\item
									For each $w, y\in W$, we have $T(wy)\subseteq T(w)\cup wT(y)w^{-1}$.
				\end{enumerate}\hfill\qedsymbol
\end{prop}
\par

\subsection{Conjugation of parabolic subgroups}
For any subset $\Sigma\subseteq S$, let $\left( W_{\Sigma}, \Sigma \right)$ be the Coxeter subsystem of $\left( W, S \right)$ generated by $\Sigma$. Accordingly, let $T_{\Sigma}$ denote the set of reflections in the Coxeter system $(W_{\Sigma}, \Sigma)$.\par

We define subsets $\prescript{\Sigma}{ }W = \left\{ w\in W\;;\; T(w)\cap T_{\Sigma} = \emptyset \right\}$, $W^{\Sigma} = \left\{ w\in W\; ;\; T(w^{-1})\cap T_{\Sigma} = \emptyset \right\} = \left( \prescript{\Sigma}{ }W\right)^{-1}$. Then, $W^{\Sigma}$ and $\prescript{\Sigma}{ }W$ form a full subset of representatives for the elements of $W / W_{\Sigma}$ and $W_{\Sigma}\backslash W$, respectively. Moreover, the elements of $W^{\Sigma}$ and $\prescript{\Sigma}{ }W$ are characterised by the property that they are of minimal length in their own classes.  \par

For any two subsets $\Sigma, \Sigma'\subseteq S$, we define a subset
\begin{eq}
				\rmN\left( \Sigma, \Sigma' \right) = \left\{y\in W^{\Sigma}\cap \prescript{\Sigma'}{ }W\; ; \;  y W_{\Sigma}y^{-1} = W_{\Sigma'} \right\}
\end{eq}
\begin{prop}
				\begin{enumerate}[label=(\roman*)]
								\item
												For any subsets $\Sigma, \Sigma'\subseteq S$ and for any element $y\in \rmN\left( \Sigma, \Sigma' \right)$, the isomorphism defined by conjugation
												\begin{eq}
																\Int_y:W_{\Sigma}&\to W_{\Sigma'} \\
																w&\mapsto ywy^{-1}
												\end{eq}
												induces an isomorphism of Coxeter systems $\left( W_{\Sigma}, \Sigma \right) \cong \left( W_{\Sigma'}, \Sigma' \right)$ and in particular a bijection on the reflections $T_{\Sigma} \cong T_{\Sigma'}$.
								\item
												For any subsets $\Sigma, \Sigma', \Sigma''\subseteq S$, the multiplication on $W$ restricts to a map
												\begin{eq}
																\rmN\left( \Sigma', \Sigma'' \right)\times \rmN\left( \Sigma, \Sigma' \right) &\to \rmN\left( \Sigma, \Sigma'' \right).
												\end{eq}
				\end{enumerate}
\end{prop}
\begin{proof}
				For (i), it suffices to show that $\Int_y(\Sigma) = \Sigma'$, or equivalently that the isomorphism $\Int_y$ preserves the length. Indeed, we have for any $w\in W_{\Sigma}$
				\begin{eq}
								\ell(y) + \ell(w) = \ell\left( yw \right) = \ell\left( ywy^{-1} y \right) = \ell\left( ywy^{-1} \right) + \ell(y),
				\end{eq}
				where first equality is due to the fact that $y\in W^{\Sigma}$ and the third is due to the fact that $y\in \prescript{\Sigma'}{ } W$. This proves (i). \par
				For (ii), since $y'yW_{\Sigma}y^{-1}y'^{-1} = y'W_{\Sigma'}y'^{-1} = W_{\Sigma''}$, it remains to show that $y'y\in W^{\Sigma}\cap \prescript{\Sigma''}{ } W$. By symmetry, we need only to show that $y'y\in \prescript{\Sigma''}{ }W$. The assumption that $y\in \prescript{\Sigma'}{}W$ and $y'\in \prescript{\Sigma''}{}W$, implies $T(y)\cap T_{\Sigma''} = \empty$ and $y'T(y)y'^{-1}\cap T_{\Sigma''} = y'\left(T(y)\cap T_{\Sigma'}\right)y'^{-1} = \emptyset$. Therefore, by~\autoref{prop:propT}(iii), we deduce that
				\begin{eq}
					T(yy')\cap T_{\Sigma''} \subseteq \left(T(y)\cap T_{\Sigma''}\right) \cup\left(y'T(y)y'^{-1}T_{\Sigma''}\right) = \emptyset,
				\end{eq}
				whence $yy'\in \prescript{\Sigma''}{}W$.

\end{proof}
Now we study the decompsition of elements in $\rmN\left(\Sigma, \Sigma'  \right)$.

\begin{prop}\label{prop:dichotomy0}
				Let $\Sigma, \Sigma'\subseteq S$ be subsets and let $w\in \rmN\left( \Sigma, \Sigma' \right)$. Suppose that $W_\Sigma$ and $W_{\Sigma'}$ are finite. For each $s\in S \setminus \Sigma'$, the following conditions are equivalent:
				\begin{enumerate}[label=(\roman*)]
								\item
												$\ell(s w) = \ell(w) - 1$,
								\item
												$s\in D\left(w_0^{\Sigma'}w\right) \setminus \Sigma' = D(w)$,
								\item
												$W_{\Sigma'\cup \left\{ s \right\}}$ is finite and
												$\ell\left(w_0^{\Sigma'\cup \left\{ s \right\}}w_0^{\Sigma'}w \right) = \ell(w) - \ell\left(w_0^{\Sigma'\cup \left\{ s \right\}}w_0^{\Sigma'}\right)$.
								\item
												$ T_{\Sigma'\cup \left\{ s \right\}} \setminus T_{\Sigma'} \subseteq T\left( w_0^{\Sigma'}w \right) \setminus T_{\Sigma'} = T\left( w \right) $,
				\end{enumerate}
\end{prop}
\begin{proof}
				The conditions (i) and (ii) are equivalent by definition. \par
				Suppose first the condition (i): $\ell(sw) = \ell(w) - 1$. We claim that for any element $x\in W_{\Sigma'\cup \left\{ s \right\}}$, we have
				\begin{eq*}\label{equa:ordre}
								w_0^{\Sigma'}w  \ge x w_0^{\Sigma'}w
				\end{eq*}
				in the Bruhat order.
				We prove this by induction on $\ell(x)$. When $\ell(x) = 0$, the equation~\eqref{equa:ordre} holds trivially. Suppose that $\ell(x) > 0$, so that $\emptyset\neq D(x) \subseteq \Sigma'\cup \left\{ s \right\}$. We choose any $s'\in D(x)$ so that $\ell(s'x) < \ell(x)$ and thus $w_0^{\Sigma'}w \ge s'x w_0^{\Sigma'}w$ by the inductive hypothesis. If $s'x w_0^{\Sigma'}w > x w_0^{\Sigma'}w$, then we are done. Otherwise we have $s'\notin D\left( s'xw_0^{\Sigma'}w \right)$. Since $w\in \rmN(\Sigma, \Sigma')$, we know that $w_0^{\Sigma'}w = ww_0^{\Sigma}$ so that $\Sigma'\cup \left\{ s \right\}\subseteq D(w_0^{\Sigma'})\cup D(w) = D(w_0^{\Sigma'}w)$, which implies $s'\in D (w_0^{\Sigma'}w) \setminus D(s'xw_0^{\Sigma'}w)$. In this case, the lifting property of the Bruhat order (see~\cite[2.2.7]{bjorner06}) implies that $w_0^{\Sigma'}w \ge x w_0^{\Sigma'}w$, so~\eqref{equa:ordre} is verified. In particular, the equation~\eqref{equa:ordre} immediately implies that $W_{\Sigma'\cup \left\{ s \right\}}$ is finite and thus there exists a unique longest element $w^{\Sigma'\cup\{s\}}_0$. Now applying~\eqref{equa:ordre} to elements of $T_{\Sigma'\cup \left\{ s \right\}}\subseteq W_{\Sigma'\cup\{s\}}$, we deduce that $T_{\Sigma'\cup \left\{ s \right\}}\subseteq T(w_0^{\Sigma'}w)$. Therefore
				\begin{eq}
								T_{\Sigma'\cup \left\{ s \right\}} \setminus T_{\Sigma'} \subseteq T\left( w_0^{\Sigma'}w \right) \setminus T_{\Sigma'} = T\left( w \right)
				\end{eq}
				This implies (iv).
				\par

				Assuming (iv), we see that $T_{\Sigma'\cup\left\{ s \right\}}\subseteq T(w)$ must be finite, so is $W_{\Sigma'\cup \left\{ s \right\}}$, and that
				\begin{eq}
								\ell\left( w_0^{\Sigma'\cup \left\{ s \right\}}w_0^{\Sigma'}w \right)= \#T\left( w_0^{\Sigma'\cup \left\{ s \right\}}w_0^{\Sigma'}w \right)  = \#T\left(w \right) - \# T_{\Sigma'\cup \left\{ s \right\}} + \# T_{\Sigma'} = \ell(w) - \ell\left(w_0^{\Sigma'\cup \left\{ s \right\}}w_0^{\Sigma'}\right)
				\end{eq}
				and we have thus (iii). \par
				Assuming (iii), since $w_0^{\Sigma'\cup \{s\}}w_0^{\Sigma'}\in W^{\Sigma'}$, every reduced decomposition of it ends with $s$. This implies
				\begin{eq}
								\ell\left(sw \right) &\le \ell\left(w^{\Sigma'\cup \{ s \}}_0w^{\Sigma'}_0w \right) + \ell\left(w^{\Sigma'\cup \{ s \}}_0w^{\Sigma'}_0s\right) \\
								&= \ell\left(w \right) - \ell\left( w^{\Sigma'\cup \{ s \}}_0w^{\Sigma'}_0 \right) + \ell\left(w^{\Sigma'\cup \{ s \}}_0w^{\Sigma'}_0\right) - 1  = \ell(w) - 1,
				\end{eq}
				whence (i). \par
\end{proof}

\begin{prop}\label{prop:lien}
	For any $\Sigma, \Sigma'\subseteq S$ such that $W_{\Sigma}$ and $W_{\Sigma'}$ are finite and for any $y\in \rmN(\Sigma, \Sigma')$, there exists a sequence $\left\{ \Sigma_i\right\}_{i=1}^q$ of subsets of $S$ and a sequence $\left\{ y_i\right\}_{i=1}^{q-1}$ of elements of $W$ such that
				\begin{enumerate}
								\item
												$\Sigma_1 = \Sigma$ and $\Sigma_q = \Sigma'$,
								\item
												the subsets $\Sigma_i$ and $\Sigma_{i+1}$ differ by at most one element for all $i = 1\ldots q-1$,
								\item
												$y_i\in \rmN\left( \Sigma_i, \Sigma_{i+1} \right)$ and there exists $s_i\in S \setminus\Sigma_{i}$ such that $y_i = w_0^{\Sigma_{i}\cup \{s\}}w_0^{\Sigma_{i}}$ for all $i = 1\ldots q-1$,
								\item
												$\ell(y) = \ell(y_1) + \ldots + \ell(y_q)$.
				\end{enumerate}
\end{prop}
\begin{proof}
				We prove the statement by induction on $\ell(y)$. If $\ell(y) = 0$, it is trivial. Suppose that $\ell(y) > 0$. We choose an element $s\in D(y)$. Since $y\in \prescript{\Sigma'}{ }W$, we see that $s\in S \setminus \Sigma'$. By~\autoref{prop:dichotomy0}, we have
				\begin{eq}
								\ell(y) = \ell\left( w^{\Sigma'\cup \{s\}}_0w^{\Sigma'}_0y \right) + \ell\left( w^{\Sigma'\cup \{s\}}_0w^{\Sigma'}_0 \right).
				\end{eq}
				Now, the conjugation $\Int_{w^{\Sigma'\cup \{s\}}_0w^{\Sigma'}_0}$ sends $\Sigma'$ to some subset $\Sigma''\subseteq S$. Therefore,
				\begin{eq}
								w^{\Sigma'\cup \{s\}}_0w^{\Sigma'}_0y\in \rmN\left( \Sigma, \Sigma'' \right).
				\end{eq}
				Applying the inductive hypothesis to $w^{\Sigma'\cup \{s\}}_0w^{\Sigma'}_0y$, there exists then $\Sigma_1 = \Sigma, \cdots, \Sigma_q = \Sigma''$ and $y_1, \cdots, y_{q-1}$ satisfying the desired properties. We define then $\Sigma_{q+1} = \Sigma'$, $\Sigma'\cup \{s\} =  \Sigma''\cup \{s_q\}$ and $y_q = \left(w^{\Sigma'\cup \{s\}}_0w^{\Sigma'}_0\right)^{-1} = w^{\Sigma''\cup \{s_q\}}_0w^{\Sigma''}_0$. Then the sequences $\{\Sigma_i\}_{i=1}^{q+1}$ and $\{y_i\}_{i=1}^q$ have the desired property.
\end{proof}

\subsection{Relative Coxeter groups}\label{subsec:relCox}
Let $(W, S)$ be a Coxeter system and let $\Sigma\subseteq S$ be a subset which generates a parabolic subgroup $W_{\Sigma}$. \par

We assume the following two conditions:
\begin{enumerate}
				\item
								The parabolic subgroup $W_{\Sigma}$ is finite.
				\item
								For each subset $\Sigma'\subseteq S$ containing $\Sigma$, if the parabolic subgroup $W_{\Sigma'}$ generated by $\Sigma'$ is finite, then its longest element $w^{\Sigma'}_0$ normalises $W_{\Sigma}$.
\end{enumerate}

By the first hypothesis, there is a longest element $w_0^{\Sigma}\in W_{\Sigma}$. It defines an automorphism of group
\begin{eq}
				W_{\Sigma}\to W_{\Sigma} \\
				w \mapsto w_0^{\Sigma}ww_0^{\Sigma} \\
\end{eq}
As $w_0^{\Sigma}$ is longest, for any element $w\in W_{\Sigma}$, we have $\ell\left(ww_0^{\Sigma}\right) =\ell\left(w_0^{\Sigma}w\right) = \ell\left(w^{\Sigma}_0\right) - \ell(w)$ and thus $\ell\left(w_0^{\Sigma}ww_0^{\Sigma}\right) = \ell(w)$. In particular, it induces an involution on $\Sigma$ and on $T_{\Sigma}$. \par

Consider the quotient group $N_{W}\left( W_{\Sigma} \right) /  W_{\Sigma}$. In each class $C\in N_{W}\left(   W_{\Sigma} \right) / W_{\Sigma}$, there is exactly one element $w\in C$ of minimal length $\ell(w) = \min_{y\in C}\ell (y)$. Such an element is characterised by the property that $T(w)\cap T_{\Sigma} = \emptyset$. Let $W^{\Sigma} = \left\{ w\in W; T(w)\cap T_{\Sigma} = \emptyset \right\}$ and $\til W = N_W\left( W_{\Sigma} \right)\cap W^{\Sigma}$.
\begin{prop}
				\begin{enumerate}[label=(\roman*)]
								\item
												The normaliser $N_W(W_{\Sigma})$ is isomorphic in the canonical way to the semi-direct product $\til W\ltimes W_{\Sigma}$.
								\item
												For any $w\in \til W$, the conjugation
												\begin{eq}
																W_{\Sigma}\to W_{\Sigma} \\
																y\mapsto wyw^{-1}
												\end{eq}
												preserves the length function and thus induces a permutation of $\Sigma$ and of $T_{\Sigma}$.
								\item
												The element $w_0^{\Sigma}$ centralises $\til W$.
								\item
												For any $w\in \til W$, we have $T\left( w_0^{\Sigma}w \right) = T(w)\sqcup  T_{\Sigma}$ and $D\left( w_0^{\Sigma}w \right) = D(w)\sqcup  \Sigma$.
				\end{enumerate}
\end{prop}
\begin{proof}
	For (i), since $\til W\to N_W\left(W_{\Sigma}  \right)/W_{\Sigma}$ is a bijection, it suffices to show that $\til W$ is closed under multiplication. Indeed, if $w, y\in \til W$, then $\Int_w$ preserves the set $T_{\Sigma}$. Now $T_{wy}\subseteq T(w)\cup wT(y) w^{-1}\subseteq T_{W}\setminus T_{\Sigma}$, so $wy\in N_W(W_{\Sigma})\cap \prescript{\Sigma}{}W = \til W$. This proves (i).\par
	For (ii), we notice that $\ell(w) + \ell(y) = \ell(wy) = \ell\left( wyw^{-1} w\right)$. Since $wyw^{-1}\in W_{\Sigma}$, we have $\ell\left( wyw^{-1} w\right) = \ell\left(wyw^{-1}\right) + \ell\left( w \right)$. Thus $\ell(y) = \ell\left( wyw^{-1} \right)$. This proves (ii). \par
	For each element $w\in \til W$, the conjugation $\Int_w$ preserves $T_{\Sigma}$, the length function on $W_{\Sigma}$, thus the longest element of $\left( W_{\Sigma}, \Sigma \right)$.  This proves (iii). \par
	The statement (iv) results immediately from (ii) and (iii).
\end{proof}

\autoref{prop:dichotomy0} applied to the case $\Sigma = \Sigma'$ can be restated as follows.
\begin{prop}\label{prop:dichotomy}
				Let $w\in \til W$. For each $s\in S \setminus \Sigma$, the following conditions are equivalent:
				\begin{enumerate}[label=(\roman*)]
								\item
												$\ell(s w) = \ell(w) - 1$,
								\item
												$s\in D\left(w_0^{\Sigma}w\right) \setminus \Sigma = D(w)$,
								\item
												$W_{\Sigma\cup \left\{ s \right\}}$ is finite and
												$\ell\left(w_0^{\Sigma\cup \left\{ s \right\}}w_0^{\Sigma}w \right) = \ell(w) - \ell\left(w_0^{\Sigma\cup \left\{ s \right\}}w_0^{\Sigma}\right)$.
								\item
												$ T_{\Sigma\cup \left\{ s \right\}} \setminus T_{\Sigma} \subseteq T\left( w_0^{\Sigma}w \right) \setminus T_{\Sigma} = T\left( w \right) $,
				\end{enumerate}
\end{prop}
\par

We define
\begin{eq}
				\Sigma^{\complement} = \left\{ s \in S \setminus \Sigma ; \# W_{\Sigma\cup \left\{ s \right\}}< \infty \right\}.
\end{eq}
For each $s\in \Sigma$, we denote $\til s = w_0^{\Sigma\cup \left\{ s \right\}} w^{\Sigma}_0\in W$. Clearly, $\til s \in \til W$. We set $\til S = \left\{ \til s\in \til W; s\in \Sigma^{\complement} \right\}$.
Let $\til T\subseteq \til W$ denote the subset of elements that are $\til W$-conjugate to some element of $\til S$. For each $w\in \til W$, we denote
\begin{eq}
				\til T(w) = \left\{ t\in \til T ; \ell(tw) < \ell(w) \right\},\quad \til D(w) = \til T(w)\cap \til S.
\end{eq}
\begin{prop}\label{prop:exchange}
				Let $w\in \til W$ and let $\til s\in \til D(w)$. Suppose that we are given a decomposition $w = w_1 \ldots w_r$ with $w_i\in \til S$ for all $i$ and that $\ell(w) = \ell(w_1) + \ldots + \ell(w_r)$. Then there exists an $1 \le i \le r$ such that
				\begin{eq}
								\til s w_1 \ldots w_{i-1} = w_1 \ldots w_i.
				\end{eq}
\end{prop}
\begin{proof}
				From~\autoref{prop:dichotomy} we deduce that $\ell(\til sw) + \ell(\til s) = \ell(w)$. Let $m\ge 1$ be the minimal integer such that
				\begin{eq}
								\ell\left(\til sw_1\ldots w_m\right) < \ell(\til s) + \ell\left(w_1\ldots w_m\right)
				\end{eq}
				By~\autoref{prop:dichotomy} again, we have
				\begin{eq}
								&\ell\left(w_1\ldots w_{m-1}\right)+ \ell(w_m) - \ell(\til s) = \ell\left(w_1\ldots w_m\right) - \ell(\til s)  = \ell\left(\til sw_1\ldots w_m\right) \\
								&= \ell\left(\til sw_1\ldots w_{m-1}\right) - \ell(w_m) =  \ell\left(w_1\ldots w_{m-1}\right) + \ell(\til s) - \ell(w_m).
				\end{eq}
				Therefore $\ell\left( \til s \right) = \ell\left( w_m \right)$. \par
				Now let $\til s = u_1 \ldots u_r$, and $w_m = v_1 \ldots v_r$ be reduced decompositions in $\left( W, S \right)$ so that $r = \ell\left(\til s\right) = \ell \left( w_m \right)$ and $u_i, w_i\in S$ for all $i$. We also let $w_1 \ldots w_{m-1} = x_1 \ldots x_{r'}$ be a reduced decomposition for $w_1 \ldots w_{m-1}\in W$. Now for each $j = 1 \ldots r$, we have
				\begin{eq}
								&\ell\left( w_1 \ldots w_m \right)- (r-j+1) \le \ell\left( u_j\ldots u_r w_1 \ldots w_m \right)= \ell\left(u_j\ldots u_r x_1 \ldots x_{r'}v_1 \ldots v_r\right) \\
								& = \ell\left( u_{j-1}\ldots u_1 \til s w_1 \ldots w_m \right) \le \ell\left(\til s w_1 \ldots w_m \right) + j -1 = \ell\left( w_1 \ldots w_m \right)- (r-j+1)
				\end{eq}
				whence $\ell\left(u_1\ldots u_r x_1 \ldots x_{r'}v_1 \ldots v_r\right) = \ell\left( w_1 \ldots w_m \right)- (r-j+1)$. Applying the exchange condition for $(W, S)$, there is either an $1\le i\le r$ such that
				\begin{eq}
								u_r x_1 \ldots  x_{j-1} = x_1 \ldots x_j
				\end{eq}
				or there is an $1\le i\le r'$ such that
				\begin{eq}
								u_r x_1 \ldots x_{r'} v_1 \ldots v_{i-1} = x_1  \ldots x_{r'} v_1 \ldots v_i.
				\end{eq}
				The first case cannot really happen since $x_1 \ldots x_{r'}$ is a reduced word. because otherwise $\ell\left( u_rx_1\ldots x_{r'} \right) = \ell\left( x_1\ldots \ha x_{i} \ldots x_{r'} \right) \le r'$, absurd. Thus
				\begin{eq}
								u_r x_1 \ldots x_{r'} v_1 \ldots v_{r} = x_1  \ldots x_{r'} v_1 \ldots \ha v_{i_1} \ldots v_r.
				\end{eq}
				with $i_1 = i$. Now  Applying again the exchange condition and reasoning on the reducedness of $u_r x_1 \cdots x_r$, we have
				\begin{eq}
								u_{r-1}u_r x_1 \ldots x_{r'} v_1 \ldots v_{r} = u_{r-1}x_1  \ldots x_{r'} v_1 \ldots \ha v_{i_1} \ldots v_r = x_1  \ldots x_{r'} v_1 \ldots \ha v_{i_1} \ldots v_{i_2} \ldots v_r
				\end{eq}
				for some $1\le i_2 \le m$ and $i_2\neq i_1$. Continuing on, we arrive at
				\begin{eq}
								u_1\ldots u_r x_1 \ldots x_{r'} v_1 \ldots v_{r} = x_1  \ldots x_{r'}
				\end{eq}
				so that
				\begin{eq}
								\til s w_1 \ldots w_m = u_1\ldots u_r x_1 \ldots x_{r'} v_1 \ldots v_{r} = x_1  \ldots x_{r'} = w_1 \ldots w_{m-1}
				\end{eq}
				or equivalently $\til s w_1 \ldots w_{m-1} = w_1 \ldots w_m$, since $\til s^2 = e$. This proves the proposition.
				\par
\end{proof}
For $w\in \til W$, we denote $\til\ell(w)\in \bfN\cup \left\{ \infty \right\}$ the length of $w$ in the Coxeter group $\left( \til W, \til S \right)$ (in contrary to the length $\ell(w)$ in the Coxeter group $(W, S)$). This is defined to be
\begin{eq}
				\til\ell(w) = \inf\left\{ q\in \bfN ; \exists w_1 \ldots w_q\in \til S,\; w = w_1 \ldots w_q \right\}\in \bfN\cup \left\{ \infty \right\}.
\end{eq}
\begin{theo}\label{theo:strrelcox}
				In the above notations,
				\begin{enumerate}[label=(\roman*)]
								\item
												the pair $\left( \til W, \til S \right)$ forms a Coxeter system.
								\item
												Then for any $w, y\in \til W$, we have
												\begin{eq}
																\til\ell(y) + \til \ell(w) = \til \ell(yw) \quad \text{if and only if}\quad \ell(y) + \ell(w) = \ell(yw).
												\end{eq}
				\end{enumerate}
\end{theo}
\begin{proof}
	Let us first verify that $\til S$ generates $\til W$. Let $w\in \til W$. We prove by induction on $\ell(w)$ that $w$ is in the subgroup generated by $\til S$. When $\ell(w) = 0$, it is trivially true. Suppose that $w\neq e$. Take any $s\in D(w)$. Then $s\in S \setminus \Sigma$, and by~\autoref{prop:dichotomy} we have
				\begin{eq}
								T_{\Sigma\cup \left\{ s \right\}} \setminus T_{\Sigma} \subseteq T(w).
				\end{eq}
				From this, we see that $T(w) = T(w\til s)\sqcup \left( T_{\Sigma\cup \left\{ s \right\}} \setminus T_{\Sigma} \right)$, so
				\begin{eq}
								\ell(w\til s) = \ell\left( w \right) - \ell\left( \til s \right) < \ell\left( w \right)
				\end{eq}
				By induction hypothesis, $w\til s$ belongs to the subgroup generated by $\til S$, and so is $w$. In particular, $\til \ell(w) < \infty$ for all $w\in \til W$.\par

				We prove (ii). It suffices to treat the case where $y = \til s\in \til S$ since the general case follows by induction on $\til \ell(y)$. Let $w = w_1 \ldots w_q\in \til W$ be a decomposition in $(\til W, \til S)$ such that $w_i\in \til S$, and $q = \til \ell(w)$ is minimal. \par

				Suppose first that $\ell(\til sw) < \ell(\til s) + \ell(w)$. Then~\autoref{prop:exchange} implies that there is an index $1\le j\le q$ such that $\til sw_1 \ldots w_{j-1} = w_j \ldots w_q$, and hence
				\begin{eq}
								\til sw = w_1 \ldots {w_{j-1}}{w_{j+1}} \ldots w_q
				\end{eq}
				which implies $\til\ell(\til sw) = \til\ell(w) - 1$. \par

				Similarly, if we replace $w$ with $\til sw$ in the above argument, we will have that $\ell(w) < \ell(\til s) + \ell(\til sw)$ implies $\til\ell(w) = \til \ell(\til sw) - 1$.\par

				According to~\autoref{prop:dichotomy}, either $\ell(\til sw) = \ell(w) + \ell(\til s)$ or $\ell(\til sw) = \ell(w) - \ell(\til s)$ should hold. Therefore we have the equivalences of conditions:

				\begin{equation}\label{equa:length}
								\begin{split}
												\til \ell(\til sw) = \til \ell(w) + 1 \Longleftrightarrow \ell(\til sw) = \ell(w) + \ell(\til s) \\
												\til \ell(\til sw) = \til \ell(w) - 1 \Longleftrightarrow \ell(\til sw) = \ell(w) - \ell(\til s).
								\end{split}
				\end{equation}
				This proves (ii). \par

				In order to show that $\left( \til W, \til S \right)$ is a Coxeter system, it suffices to verify the exchange condition~\cite[4.1.6, Th\'eor\`eme 1]{Bourbaki}. This follows immediately from the above equivalences of conditions~\autoref{equa:length} and~\autoref{prop:exchange}.

\end{proof}
Since $\til W \cong N_W\left(W_{\Sigma}\right) / W_{\Sigma}$, the resulting Coxeter system $\left( \til W, \til S \right)$ or the quotient
\begin{eq}
				\left( N_W\left(W_{\Sigma}\right) / W_{\Sigma}, \til S\cdot W_{\Sigma} / W_{\Sigma} \right)
\end{eq}
is called the {\bf relative Coxeter group}.

\subsection{Coxeter complex}
Let $\frakF$ be the Coxeter complex of $(W,S)$, which is defined by
\begin{eq}
				\frakF = \left\{ yW_{\Sigma}\subseteq W\;;\; \Sigma\subsetneq S,\; y\in W^{\Sigma}\right\}
\end{eq}
equipped with a partial order $\left(\frakF, \le \right)$ opposite to inclusions. Let $\frakC\subseteq \frakF$ be the set of maximal elements in this partial order, consisting of the singletons. The spherical part of the Coxeter complex to the subset
\begin{eq}
				\frakF^{\sph} = \left\{ yW_{\Sigma}\in \frakF\;;\; \# W_{\Sigma} < \infty \right\}
\end{eq}
equipped with the induced partial order. Then clearly $\frakC\subseteq \frakF^{\sph}$. Elements of $\frakC$ are called {\bf chambers}.\par

The group $W$ acts on $\frakF$ and $\frakF^{\sph}$ by left translation. The restriction of this action on $\frakC$ is simply transitive. Let $yW_{\Sigma}\in \frakF$ be a facet, then the stabiliser of it is described by
\begin{eq}
				\Stab_W\left(yW_{\Sigma}  \right) = yW_{\Sigma}y^{-1}.
\end{eq}
In particular, if $yW_{\Sigma}\in \frakF^{\sph}$, then it has finite stabiliser. \par

We remark that in the case where $(W, S)$ is of finite or affine type, $\frakF$ and $\frakF^{\sph}$ coincide. In the case of finite type, $\frakF$ has a geometric realisation as a sphere of dimension $\# S - 1$, while in the case of affine type, $\frakF$ can be realised as an euclidean space of dimension $\# S - 1$.

\subsection{Fixed sub-complex}
Now we fix $\Sigma\subsetneq S$ such that $W_{\Sigma}$ is finite. Consider the sub-complex $\frakF^{\sph}(\Sigma) = \left(\frakF^{\sph}\right)^{W_{\Sigma}}$ consisting of the facets fixed by the subgroup $W_{\Sigma}$. The fixed facets $y'W_{\Sigma'}\in \frakF^{\sph}\left( \Sigma \right)$ are characterised by the property that $W_{\Sigma}\subseteq \Stab_W(y'W_{\Sigma'}) = y'W_{\Sigma'}y'^{-1}$. Let $\frakC(\Sigma)\subseteq \frakF^{\sph}(\Sigma)$ be the set of maximal elements, called chambers in $\frakF^{\sph}\left( \Sigma \right)$. Then the elements $y'W_{\Sigma'}\in \frakF^{\sph}(\Sigma)$ are those such that $y'W_{\Sigma'}y'^{-1} = W_{\Sigma}$. \par

\begin{theo}\label{theo:fixed}
				Suppose that $\Sigma\subsetneq S$ satisfies the condtions of~\autoref{subsec:relCox}. Let $y'W_{\Sigma'}\in \frakC(\Sigma)$ be any chamber. Then $\Sigma' = \Sigma$. Moreover, the relative Coxeter group $\til W = \rmN(\Sigma, \Sigma) / W_{\Sigma}$ acts simply transitively on $\frakC\left( \Sigma \right)$.
\end{theo}
\begin{proof}
	Applying~\autoref{prop:lien}, we obtain sequences $\{\Sigma_i\}_{i=1}^q$ and $\{y_i\}_{i=1}^q$ which have the properties listed in the proposition. In particular, $y_i = w_0^{\Sigma_i\cup\left\{ s \right\}}w_0^{\Sigma_i}$ holds. Since $w_0^{\Sigma_1\cup\left\{ s \right\}}$ normalises $W_{\Sigma}$, so does $y_1$. Hence $\Sigma_2 = \Sigma_1 = \Sigma$. By induction, we can show that $\Sigma_i = \Sigma$ and $y_i\in \rmN(\Sigma, \Sigma)$ for all $i$. Thus $\Sigma' = \Sigma$ and that $y'W_{\Sigma}$ and $W_{\Sigma}$ are connected by $\rmN\left( \Sigma, \Sigma \right)$.
\end{proof}

\section{Affine root systems and double affine Hecke algebras}
We recall the notion of affine root systems. The reference is~\cite{macdonald03}.
\subsection{Affine reflections on euclidean spaces}
Let $\bbE$ be an euclidean space of dimension $n > 0$ and let $V$ be its vector space of translations. In particular, $V$ is equipped with a positive definite scalar product $\langle\relbar,\relbar\rangle: V\times V\to \bfR$. The dual space $V^*$ is identified with $V$ via the scalar product $\langle\relbar, \relbar\rangle$. Let $\bfC[\bbE]^{\le 1}$ be the space of affine functions on $\bbE$. We have a map of gradient $\partial: \bfC[\bbE]^{\le 1}\to V$ whose kernel is the set of constant functions. The space $\bfC[\bbE]^{\le 1}$ which is equipped with a scalar product $\langle f, g\rangle = \langle \partial f, \partial g\rangle$. For any non-constant function $f\in \bfC[\bbE]^{\le 1}$, let $f^{\vee} = 2f / |f|^2$ and define the reflection with respect to the zero hyperplane of $f$:
\begin{eq}
	s_f:\bbE\to \bbE,\quad s_f(x) = x - f(x)\partial f
\end{eq}
and
\begin{eq}
	s_f:\bfC[\bbE]^{\le 1}\to \bfC[\bbE]^{\le 1},\quad s_f(g) = g - \langle f^{\vee}, g\rangle f.
\end{eq}

\subsection{Affine root systems}

An affine root system on $\bbE$ is a subset $S\subset \bfC[\bbE]^{\le 1}$ satisfying the following conditions:
\begin{enumerate}
	\item
		$S$ spans $\bfC[\bbE]^{\le 1}$ and the elements of $S$ are non-constant functions on $\bbE$.
	\item
		$s_a(b)\in S$ for all $a, b\in S$
	\item
		$\langle a^{\vee}, b\rangle\in\bfZ$ for all $a, b\in S$.
	\item
		the group $W_S$ of auto-isometries on $\bbE$ generated by $\{s_a\;;\; a\in S\}$ acts properly on $\bbE$.
\end{enumerate}
The group $W_S$ is called the affine Weyl group (or simply the Weyl group of $S$). An affine root system $(\bbE, S)$ is irreducible if there is no partition $S = S_1\sqcup S_2$ with $\langle \relbar, \relbar\rangle\mid_{S_1\times S_2} = 0$ and $S_1\neq \emptyset$ and $S_2\neq\emptyset$; it is reduced if $a\in S$ implies $2a\notin S$. \par

Let $(\bbE, S)$ be an affine root system. The set $R = \partial(S)\subset V^*$ is a finite root system on $V$. Let $P_R = \rmP(R)\subset V$ denote the weight lattice, $Q_R = \bfZ R$ the root lattice, $P^\vee_R = P(R^{\vee})$ the coweight lattice and $Q^\vee_R = \bfZ R^{\vee}$ the coroot lattice. \par

Conversely, let $(V, R)$ be an irreducible finite root system, reduced or not. Let $P_R$ be the weight lattice, $Q_R$ the root lattice. \par
We define the affinisation of $(V, R)$ to be the affine root system $(\bbE, S)$ with $\bbE = V$ and
\begin{eq}
	S = \langle \alpha + n\;;\; n\in \bfZ, \alpha\in R, \alpha\notin 2Q_R\rangle \sqcup\langle \alpha + 2m + 1\;;\; m\in \bfZ, \alpha\in R, \alpha\in 2Q_R\rangle.
\end{eq}
Given a base $\Delta_0\subset R$, we form $\Delta = \Delta_0 \cup \left\{ a_0 \right\}$, where $a_0 = 1 - \theta$ with $\theta\in R$ being the highest root with respect to the base $\Delta_0$. It follows that $\delta = 1$. From now on, we will suppose that $(\bbE, S)$ comes from a finite root system $(V,R)$ in this sense. \par

\subsection{Affine Weyl group}\label{subsec:sysaff-fini}

Let $(\bbE, S)$ be an irreducible affine root system. A base of $S$ is an $\bfR$-linearly independent subset $\Delta\subset S$ such that
\begin{enumerate}
	\item
		$S\subset \bfN \Delta \cup -\bfN \Delta$.
	\item
		The set $\bigcap_{a\in \Delta}\left\{ x\in \bbE\;;\; a(x) > 0 \right\}$ is non-empty.
\end{enumerate}
The $W_S$-action on $S$ induces a simple transitive $W_S$-action on the set of bases of $S$. \par

We choose $a_0\in \Delta$ such that the image of the set $\Delta_0 = \Delta\setminus \left\{ a_0 \right\}$ under the total differential $\partial:S\to R$ forms a base for the finite root system $R$ in the usual sense. This choice yields a base point $O\in \bbE$ by $a(O) = 0$ for $a\in \Delta_0$. We will identify $\Delta_0$ with its image $\partial \Delta_0\in R$. It follows that the parabolic Coxeter subgroup $\langle s_a \;;\; a\in \Delta_0\rangle$ of $W_S$ is isomorphic to the finite Weyl group $W_R = W\left( R, \Delta_0 \right) = \langle s_{\alpha}\;;\; \alpha\in \Delta_0\rangle$ and there is an isomorphism
\begin{eq}
	Q^{\vee}_R  \rtimes W_R &\cong W_S  \\
	(\mu, w) &\mapsto X^\mu w,
\end{eq}
where the element $X^{\mu}$ acts on $S$ by $a\mapsto a - \langle \partial a, \mu\rangle$.  The extended affine Weyl group is defined to be $\tilde W_S = P^{\vee}_R\rtimes W_R$.
It acts on $S$ by extending the $W_S$ action by the same formula $X^{\mu} a =  a - \langle \partial a, \mu\rangle$ for $\mu\in P^{\vee}_R$. \par

The length function $\ell:\til W_S\to \bfN$ is by $\ell(w) = \#\left( S^+ \cap w^{-1} S^- \right)$. This extends the usual length function on the Coxeter group $W_S$ with respect to the set of generators $\left\{s_a  \right\}_{a\in \Delta}$.

 \par

 \subsection{Alcoves}\label{subsec:alcoves}
For every affine root $a\in S$, let $H_a = \left\{ \lambda\in \bbE\;;\; a(\lambda) = 0 \right\}$ be the vanishing locus of $a$.
The affine hyperplanes $\{H_a\}_{a\in S}$ yield a simplicial cellular decomposition of $\bbE$. The open cells are called \textbf{alcoves}. Thus the set of connected componenents
\begin{eq}
	\pi_0\left( \bbE \setminus \bigcup_{a\in S} H_a \right)
\end{eq}
is the set of alcoves. The affine Weyl group $W_S$ acts simply transitively on it. The fundamental alcove is defined to be $\nu_0 = \bigcap_{a\in \Delta}\left\{ x\in \bbE\;;\; a(x) > 0 \right\}$.

\subsection{Degenerate double affine Hecke algebra $\bbH$}\label{subsec:dDAHA}
Let $h = \left\{ h_a \right\}_{a\in S}$ be a $\til W_S$-invariant family of complex numbers. We suppose that $h_a\neq 0$ for all $a\in S$, although this is not strictly indispensable. The degenerate double affine Hecke algebra with parameters $h$ attached to the affine root system $S$ is the associative unital $\bfC$-algebra on the vector space $\bbH = \bfC W_S \otimes \bfC[\bbE]$ whose multiplication satisfies following properties:
\begin{itemize}
	\item
		Each of the subspaces $\bfC W_S$ and $\bfC[\bbE]$ is given the usual ring structure, so that they are subalgebras of $\bbH$.
	\item
		$w\in \bfC W_S$ and $f\in \bfC[\bbE]$ multiply by juxtaposition: $(w\otimes 1)(1\otimes f) = w\otimes f$.
	\item
		$a\in \Delta$ and $f\in \bfC[\bbE]$ satisfy the following:
		\begin{eq}
			\left(s_{a} \otimes 1 \right)(1 \otimes f) - \left(1 \otimes s_a(f) \right)(s_{a} \otimes 1) = 1\otimes h_{a} \frac{f - s_a(f)}{a}.
		\end{eq}
\end{itemize}

\subsection{Category $\calO$}\label{subsec:OH}
For each $\lambda\in \bbE_{\bfC}$, let $\frakm_{\lambda}\subset \bfC[\bbE]$ be the maximal ideal generated by $f(\lambda) - \lambda$ for all $f\in \bfC[\bbE]$.
Given any module $M\in \bbH\Mod$, for each $\lambda\in \bbE_{\bfC}$ consider the generalised $\lambda$-weight space in $M$:
\begin{eq}
	M_{\lambda} = \bigcup_{N\ge 0}\left\{ a\in M\;;\;\frakm_{\lambda}^N a = 0\right\}.
\end{eq}
For any $\lambda_0\in \bbE_{\bfC}$, we define $\calO_{\lambda_0}\left( \bbH \right)$ to be the full subcategory of $\bbH\mof$ consisting of those $M\in\bbH\mof$ such that
\begin{eq}
	M = \bigoplus_{\lambda\in W_S\, \lambda_0} M_{\lambda}.
\end{eq}
In other words, the polynomial subalgebra $\bfC[\bbE]$ acts locally finitely on $M$ with eigenvalues in the $W_S$-orbit of $\lambda_0\in \bbE_{\bfC}$. \par

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