Nonsplit Hecke algebras and perverse sheaves

Abstract

Let H be a Hecke algebra arising as an endomorphism algebra of the representation of a Chevalley group G over $F_{q}$ induced by a unipotent cuspidal representation of a Levi quotient L of a parabolic subgroup. We assume that L is not a torus. In this paper we outline a geometric interpretation of the coefficients of the canonical basis of H in terms of perverse sheaves. We illustrate this in detail in the case where the Weyl group of G is ot type $B_{4}$ and that of L is of type $B_{2}$.

1 Introduction

0.1. Let $\mathcal W$ be a Coxeter group such that the set K of simple reflections of $\mathcal W$ is finite. Let $l:\mathcal W{\rightarrow }\mathbf{N}$ be the length function and let $\mathcal L:\mathcal W{\rightarrow }\mathbf{N}$ be a weight function that is, a function such that $\mathcal L(xy)=\mathcal L(x)+\mathcal L(y)$ whenever $x,y\in \mathcal W$ satisfy $l(xy)=l(x)+l(y)$. Let $\mathcal A=\mathbf{Z}[v,v^{-1}]$ where v is an indeterminate. The Hecke algebra of $\mathcal W$ (relative to $\mathcal L$) is the free $\mathcal A$-module $\mathcal H$ with basis $\{\mathcal T_z;z\in W\}$ with the associative algebra structure defined by the rules $(\mathcal T_z+v^{-\mathcal L(z)}(\mathcal T_z-v^{\mathcal L(z)})=0$ if $z\in K$ and $\mathcal T_{xy}=\mathcal T_x\mathcal T_y$ whenever $x,y\in \mathcal W$ satisfy $l(xy)=l(x)+l(y)$. Let $\{c_z;z\in \mathcal W\}$ be the basis of $\mathcal H$ defined in [12, 5.2] in terms of $\mathcal W,\mathcal L$. We have $c_z=\sum _{t\in \mathcal W}p_{t,z}\mathcal T_t$ where $p_{t,z}\in \mathbf{Z}[v^{-1}]$ is zero for all but finitely many t. In the case where $\mathcal H$ is split that is $\mathcal L=l$, we have $p_{t,z}(v)=v^{-l(z)+l(t)}P_{t,z}(v^2)$ where $P_{t,z}$ is the polynomial defined in [6]; in this case, $p_{t,z}$ can be interpreted geometrically in terms of intersection cohomology (at least in the crystallographic case), see [7], and this interpretation has many interesting consequences.

In this paper we are interested in the case where $\mathcal H$ is not split (and not even quasisplit, in the sense of [12, 16.3]). As shown in [8, 9] (resp. in [11]), such $\mathcal H$, with $\mathcal W$ a finite (resp. affine) Weyl group, appear as endomorphism algebras of representations of Chevalley groups over $\mathbf{F}_q$ (resp. $\mathbf{F}_q((\epsilon ))$) induced from unipotent cuspidal representations of a Levi quotient of a parabolic (resp. parahoric) subgroup. Our main goal is to describe the elements $p_{t,z}$ coming from (nonsplit) $\mathcal H$ which appear in this way in representation theory, in geometric terms, involving perverse sheaves. In this paper we outline a strategy to achieve this goal using geometry based on the theory of parabolic character sheaves [13], and we illustrate it in detail in the special case where $\mathcal H$ is the endomorphism algebra of the representation of $SO_9(\mathbf{F}_q)$ induced from a unipotent cuspidal representation of a Levi quotient of type $B_2$ of a parabolic subgroup. (This is the smallest example in which some $p_{t,z}$ can be outside $\mathbf{N}[v^{-1}]$. This $\mathcal H$ is of type $B_2$ with $\mathcal L$ taking the values 1 and 3 at the two simple reflections.) The main effort goes into computing as much as possible of the cohomology sheaves of parabolic character sheaves in this case. (For this we use the complete knowledge of the polynomials $P_{y,w}$ for the Weyl group of type $B_4$, the knowledge of the multiplicity formulas for unipotent character sheaves on $SO_5$ and some additional arguments.) Eventually the various $p_{t,z}$ can be reconstructed from the information contained in the various cohomology sheaves of parabolic character sheaves. I expect that similar results hold for all $\mathcal H$ appearing as above from representation theory. (A conjecture in this direction is formulated in 3.11. It makes more precise a conjecture that was stated in [12, 27.12] before the theory of parabolic character sheaves was available.)

0.2. Notation If X is a finite set, $\sharp X$ denotes the number of elements of X. Let $2^X$ be the set of subsets of X. If $\Gamma$ is a group, $\Gamma '$ is a subset of $\Gamma$ and $\gamma \in \Gamma$, we set ${}^\gamma \Gamma '=\gamma \Gamma \gamma ^{-1}$. Let $\mathbf{k}$ be an algebraically closed field. All algebraic varieties are assumed to be over $\mathbf{k}$. If X is an algebraic variety, $\mathcal D(X)$ denotes the bounded derived category of $\bar{\mathbf{Q}}_l$-sheaves on X (l is a fixed prime number invertible in $\mathbf{k}$). We will largely follow the notation of [2]. If $K\in \mathcal D(X)$ and A is a simple perverse sheaf on X, we write $A\dashv K$ instead of: ”A is a composition factor of $\oplus _{j\in \mathbf{Z}}{}^pH^j(K)$”. For $K\in \mathcal D(X)$, $n\in \mathbf{Z}$ we write $K\langle n\rangle$ instead of K[n](n / 2) (shift, followed by Tate twist). If $f:X{\rightarrow }Y$ is a smooth morphism of algebraic varieties all of whose fibres are irreducible of dimension d and $K\in \mathcal D(Y)$, we set $f^\bigstar (K)=f^*(K)\langle d\rangle$. If X is an irreducible algebraic variety, |X| denotes the dimension of X. For any connected affine algebraic group H, $U_H$ denotes the unipotent radical of H. If $\mathbf{k}$ is an algebraic closure of a finite field $\mathbf{F}_q$ and X is an algebraic variety over $\mathbf{k}$ with a fixed $\mathbf{F}_q$-structure, we shall denote by $\mathcal D_m(X)$ the bounded derived category of mixed $\bar{\mathbf{Q}}_l$-complexes on X.

Assume that $C\in \mathcal D(X)$ and that $\{C_i;i\in \mathcal I\}$ is a family of objects of $\mathcal D(X)$. We shall write $C\Bumpeq \{C_i;i\in \mathcal I\}$ if the following condition is satisfied: there exist distinct elements $i_1,i_2,\dots ,i_s$ in $\mathcal I$, objects $C'_j\in \mathcal D(X)$ ($j=0,1,\dots ,s$) and distinguished triangles $(C'_{j-1},C'_j,C_{i_j})$ for $j=1,2,\dots ,s$ such that $C'_0=0$, $C'_s=C$; moreover, $C_i=0$ unless $i=i_j$ for some $j\in [1,s]$. The same definition can be given with $\mathcal D(X)$ replaced by $\mathcal D_m(X)$.

2 The variety $Z_J$ and its pieces ${}^wZ_J$

1.1. We fix an affine algebraic group $\hat{G}$ whose identity component G is reductive; we also fix a connected component $G^1$ of $\hat{G}$. Let $\mathcal B$ be the variety of Borel subgroups of G. Recall that the Weyl group W of G naturally indexes the set of G-orbits of the simultaneous conjugation action of G on $\mathcal B\times \mathcal B$. We write $\mathcal O_w$ for the G-orbit indexed by $w\in W$. Let $l:W{\rightarrow }\mathbf{N}$ be the length function $w\mapsto l(w)=|\mathcal O_w|-|\mathcal O_1|$; let $I=\{w\in W;l(w)=1\}$. Recall that (W, I) is a finite Coxeter group. Let $\le$ be the standard partial order on W. For any $J\subset I$ let $W_J$ be the subgroup of W generated by J; let $W^J$ (resp. ${}^JW$) be the set of all $w\in W$ such that $l(wa)=l(w)+l(a)$ (resp. $l(aw)=l(a)+l(w)$) for any $a\in W_J$. For $J\subset I$, $K\subset I$, let ${}^KW^J={}^KW\cap W^J$. Any $(W_K,W_J)$-double coset X in W contains a unique element of ${}^KW^J$ denoted by $\min (X)$.

We define an automorphism $\delta :W{\rightarrow }W$ by

$$\begin{aligned} (B,B')\in \mathcal O_w,g\in G^1\implies ({}^gB,{}^gB')\in \mathcal O_{\delta (w)}. \end{aligned}$$

We have $\delta (I)=I$. For $J\subset I$ we write ${}^\delta J$ instead of $\delta (J)$; let $N_{J,\delta }$ be the set of all $w\in W$ such that $wJw^{-1}={}^\delta J$ and w has minimal length in $wW_J=W_{{}^\delta J}w$. We have $N_{J,\delta }\subset {}^{{}^\delta J}W^J$.

1.2. Let $\mathcal P$ be the set of parabolic subgroups of G. For $P\in \mathcal P$ we set $\mathcal B_P=\{B\in \mathcal B;B\subset P\}$. For any $J\subset I$ let $\mathcal P_J$ be the set of all $P\in \mathcal P$ such that

$$\begin{aligned} \{w\in W;(B,B')\in \mathcal O_w\text { for some }B,B'\in \mathcal B_P\}=W_J. \end{aligned}$$

If $J\subset I$ and $P\in \mathcal P_J,g\in G^1$, then ${}^gP\in \mathcal P_{{}^\delta J}$. We have $\mathcal P=\sqcup _{J\subset I}\mathcal P_J$, $\mathcal P_\emptyset =\mathcal B$, $\mathcal P_I=\{G\}$. For $B\in \mathcal B$, $J\subset I$, there is a unique $P\in \mathcal P_J$ such that $B\subset P$; we set $P=B_J$. For $J,K\subset I$, $P\in \mathcal P_K$, $Q\in \mathcal P_J$, the set

$$\begin{aligned} \{w\in W;(B,B')\in \mathcal O_w\text { for some }B\in \mathcal B_P,B'\in \mathcal B_Q\} \end{aligned}$$

is a single $(W_K,W_J)$-double coset in W, hence it contains a unique element in ${}^KW^J$ denoted by $\mathrm{pos}(P,Q)$. We set $P^Q=(P\cap Q)U_P$. We have $P^Q\in \mathcal P_{K\cap \mathrm{Ad}(u)J}$ where $u=\mathrm{pos}(P,Q)$ and $U_{P^Q}=U_P(P\cap U_Q)$ hence

$$\begin{aligned} |U_{P^Q}|=|P\cap U_Q|+|U_P|-|U_P\cap U_Q|. \end{aligned}$$

Note that the condition that P, Q contain a common Levi subgroup is equivalent to the condition that $\hbox {Ad}(u^{-1})K=J$; in this case we have $P^Q=P$, $Q^P=Q$.

1.3. Let $J\subset I$. Following Bédard [1] (see also [13, 2.4, 2.5]), for any $w\in {}^{{}^\delta J}W$ we define a sequence $J=J_0\supset J_1\supset J_2\supset {\dots }$ in $2^I$ and a sequence $w_0,w_1,\dots$ in W nductively by

$J_0=J$,

$J_n=J_{n-1}\cap \delta ^{-1}(\mathrm{Ad}(w_{n-1}J_{n-1}))$, for $n\ge 1$,

$w_n=\min (W_{{}^\delta J}wW_{J_n})$, for $n\ge 0$.

For $n\gg 0$ we have $J_n=J_{n+1}=\cdots$ and $w_n=w_{n+1}=\cdots$; we set $J_\infty ^w=J_n$ for $n\gg 0$ and $w_\infty =w_n$ for $n\gg 0$.

According to loc.cit., this gives a bijection $\kappa :{}^{{}^\delta J}W{\mathop {\rightarrow }\limits ^{\sim }}\mathcal T(J,\delta )$ where $\mathcal T(J,\delta )$ is the set consisting of all sequences $(J_n,w_n)_{n\ge 0}$ in $2^I\times W$ where

$J_n=J_{n-1}\cap \delta ^{-1}(\mathrm{Ad}(w_{n-1})J_{n-1})$ for $n\ge 1$,

$w_n\in {}^{{}^\delta J_n}W^{J_n}$ for $n\ge 0$,

$w_n\in w_{n-1}W_{J_{n-1}}$ for $n\ge 1$.

The inverse bijection $\mathcal T(J,\delta ){\rightarrow }{}^{{}^\delta J}W$ is given by $(J_n,w_n)_{n\ge 0}\mapsto w_\infty$.

Assume that $w\in {}^{{}^\delta J}W$. We have $J^w_\infty =\delta ^{-1}(\mathrm{Ad}(w)J^w_\infty )$. Hence there is a well defined Coxeter group automorphism $\tau _w:W_{{}^\delta (J^w_\infty )}{\rightarrow }W_{{}^\delta (J^w_\infty )}$ given by $x\mapsto \tau _w(x)=w\delta ^{-1}(x)w^{-1}$.

1.4. Let $J\subset I$. We set

$$\begin{aligned} \tilde{Z}_J= & {} \{(P,P',g)\in \mathcal P_J\times \mathcal P_{{}^\delta J}\times G^1;{}^gP=P'\},\\ Z_J= & {} \{(P,P',gU_P);P\in \mathcal P_J,P'\in \mathcal P_{{}^\delta J},gU_P\in G^1/U_P,{}^gP=P'\}. \end{aligned}$$

The variety $Z_J$ is the main object of this paper. (In [13, 3.3] $\tilde{Z}_J,Z_J$ are denoted by $\mathcal Z_{J,\delta }$, $Z_{J,\delta }$.)

Now $G\times G$ acts on $\tilde{Z}_J$ by $(h,h'):(P,P',g)\mapsto ({}^hP,{}^{h'}P',h'gh^{-1})$ and on $Z_J$ by $(h,h'):(P,P',gU_P)\mapsto ({}^hP,{}^{h'}P',h'gh^{-1}U_{{}^hP})$. In this paper we shall restrict these actions to G viewed as the diagonal in $G\times G$. Let $e_J:\tilde{Z}_J{\rightarrow }Z_J$ be the obvious map (an affine space bundle).

Following [13] we will define a partition of $Z_J$ into pieces indexed by ${}^{{}^\delta J}W$.

To any $(P,P',g)\in \tilde{Z}_J$ we associate an element $w_{P,P',g}\in W$ by the requirements (i),(ii) below. (We set $z=\mathrm{pos}(P',P)\in {}^{{}^\delta J}W^J$.)

• (i) $w_{P,P',g}=w_{P_1,P'_1,g}$ where $P_1=P^{{}^{g^{-1}}P}=({}^{g^{-1}}P\cap P)U_P\in \mathcal P_{J\cap \delta ^{-1}(\mathrm{Ad}(z)J)}$, $P'_1=P'{}^P\in \mathcal P_{{}^\delta J\cap \mathrm{Ad}(z)J}$;

• (ii) $w_{P,P',g}=z$ if $z\in N_{J,\delta }$.

These conditions define uniquely $w_{P,P',g}$ by induction on $\sharp J$. If $z\in N_{J,\delta }$ (in particular if $\sharp J=0)$, then $w_{P,P',g}$ is given by (ii) (and (i) is satisfied since $P_1=P$). If $z\notin N_{J,\delta }$, then $\sharp (J\cap \delta ^{-1}(\mathrm{Ad}(z)J))<\sharp J$ and $w_{P,P',g}$ is determined by (i) since $w_{P_1,P'_1,g}$ is known from the induction hypothesis.

From the definitions we see that the map $\tilde{Z}_J{\rightarrow }W$, $(P,P',g)\mapsto w_{P,P',g}$ is the composition $\tilde{Z}_J{\rightarrow }\mathcal T(J,\delta ){\mathop {\rightarrow }\limits ^{\kappa ^{-1}}}{}^{{}^\delta J}W$ (the first map is as in [13, 3.11]); in particular for any $(P,P',g)\in \tilde{Z}_J$ we have $w_{P,P',g}\in {}^{{}^\delta J}W$, $w_{P,P',g}\in W_{{}^\delta J}\mathrm{pos}(P',P)W_J$. For any $w\in {}^{{}^\delta J}W$, we set

$$\begin{aligned} ^w\tilde{Z}_J= & {} \{(P,P',g)\in \tilde{Z}_J;w_{P,P',g}=w\},\\ {}^wZ_J= & {} \{(P,P',gU_P)\in Z_J;w_{P,P',g}=w\}. \end{aligned}$$

The subsets $\{{}^w\tilde{Z}_J;w\in {}^{{}^\delta J}W\}$ are said to be the pieces of $\tilde{Z}_J$; they form a partition of $\tilde{Z}_J$. The subsets $\{{}^wZ_J;w\in {}^{{}^\delta J}W\}$ are said to be the pieces of $Z_J$; they form a partition of $Z_J$. We have ${}^w\tilde{Z}_J=e_J^{-1}({}^wZ_J)$, ${}^wZ_J=e_J({}^w\tilde{Z}_J)$, and ${}^w\tilde{Z}_J,{}^wZ_J$ are stable under the G-actions on $\tilde{Z}_J,Z_J$.

1.5. Let $z\in {}^{{}^\delta J}W^J$ and let $\Omega =W_{{}^\delta J}zW_J$. We set $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$. Let

$$\begin{aligned} \tilde{Z}_{J,\Omega }=\{(P,P',g)\in \tilde{Z}_J;\mathrm{pos}(P',P)=z\}, \end{aligned}$$

a locally closed subvariety of $\tilde{Z}_J$. Let

$$\begin{aligned} \tilde{Z}^\dagger _{J_1,\Omega }=\{(Q,Q',g)\in \tilde{Z}_{J_1};\mathrm{pos}(Q',Q)\in zW_J\}, \end{aligned}$$

a locally closed subvariety of $\tilde{Z}_{J_1}$. By [13, 3.2],

• (a) $\tilde{\alpha }:\tilde{Z}_{J,\Omega }{\rightarrow }\tilde{Z}^\dagger _{J_1,\Omega },\quad (P,P',g)\mapsto (P_1,P'_1,g)\text { where }P'_1=P'{}^P,P_1={}^{g^{-1}}P'_1$

  is a well defined morphism; by [13, 3.6],

• (b) $\tilde{\alpha }$ is an isomorphism. Let

  $$\begin{aligned} Z_{J,\Omega }=\{(P,P',gU_P)\in Z_J;\mathrm{pos}(P',P)=z\}, \end{aligned}$$

  a locally closed subvariety of $Z_J$. Let

  $$\begin{aligned} Z^\dagger _{J_1,\Omega }=\{(Q,Q',gU_Q)\in Z_{J_1};\mathrm{pos}(Q',Q)\in zW_J\}, \end{aligned}$$

  a subvariety of $Z_{J_1}$. Now $\tilde{\alpha }$ induces a morphism $\alpha :Z_{J,\Omega }{\rightarrow }Z^\dagger _{J_1,\Omega }$ given by

• (c) $(P,P',gU_P)\mapsto (P_1,P'_1,gU_{P'}\text { where }P'_1=P'{}^P,P_1={}^{g^{-1}}P'_1.$ From [13, 3.7–3.10] we see that

• (d) $\alpha$ is an affine space bundle with fibres of dimension $|U_P\cap P'|-|U_P\cap U_{P'}|$ for some/any $(P,P')\in \mathcal P_J\times \mathcal P_{{}^\delta J}$ such that $\mathrm{pos}(P',P)=z$.

  Next we note that for $w\in {}^{{}^\delta J}W$ such that $w\in \Omega$, we have

  $$\begin{aligned} {}^w\tilde{Z}_J\subset \tilde{Z}_{J,\Omega }, {}^wZ_J\subset Z_{J,\Omega }, {}^w\tilde{Z}_{J_1}\subset \tilde{Z}^\dagger _{J_1,\Omega }, {}^wZ_{J_1}\subset Z^\dagger _{J_1,\Omega }. \end{aligned}$$

  (We use that $w\in zW_J$.) Using the definitions we have

• (e) ${}^w\tilde{Z}_J=\tilde{\alpha }^{-1}({}^w\tilde{Z}_{J_1}), {}^wZ_J=\alpha ^{-1}({}^wZ_{J_1})$.

  Moreover, using (b), (d) we deduce:

• (f) $\tilde{\alpha }$ restricts to a bijection $\tilde{\vartheta }_{J,w}:{}^w\tilde{Z}_J{\rightarrow }{}^w\tilde{Z}_{J_1}$;

• (g) $\alpha$ restricts to a map ${\vartheta }_{J,w}:{}^wZ_J{\rightarrow }{}^wZ_{J_1}$ all of whose fibres are affine spaces of dimension $|U_P\cap P'|-|U_P\cap U_{P'}|$ for some/any $(P,P')\in \mathcal P_J\times \mathcal P_{{}^\delta J}$ such that $\mathrm{pos}(P',P)=z$.

Proposition 1.6

Let $J\subset I$, $w\in {}^{{}^\delta J}W$. Define $z\in {}^{{}^\delta J}W^J$ by $w\in zW^J$.

• (a) ${}^w\tilde{Z}_J$ (resp. ${}^wZ_J$) is a smooth irreducible locally closed subvariety of $\tilde{Z}_J$ (resp. $Z_J$) of dimension $l(w)+|G|$ (resp. $l(w)+|G|-|\mathcal P_J|$).

• (b) Let $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$. Then $\tilde{\vartheta }_{J,w}:{}^w\tilde{Z}_J{\rightarrow }{}^w\tilde{Z}_{J_1}$ (see 1.5(f)) is an isomorphism and ${\vartheta }_{J,w}:{}^wZ_J{\rightarrow }{}^wZ_{J_1}$ is a smooth morphism all of whose fibres are affine spaces of dimension $|\mathcal P_{J_1}|-|\mathcal P_J|$.

Assume first that $z\in N_{J,\delta }$. Then $z=w$ and

$$\begin{aligned} {}^w\tilde{Z}_J=\{(P,P',g)\in \tilde{Z}_J;\mathrm{pos}(P',P)=z\} \end{aligned}$$

(resp. ${}^wZ_J=\{(P,P',gU_P)\in Z_J;\mathrm{pos}(P',P)=z\}$) is the inverse image of

$$\begin{aligned} {}^zY_J=\{(P,P')\in \mathcal P_J\times \mathcal P_{{}^\delta J};\mathrm{pos}(P',P)=z\} \end{aligned}$$

under the obvious map $\tilde{Z}_J{\rightarrow }\mathcal P_J\times \mathcal P_{{}^\delta J}$ (resp. $Z_J{\rightarrow }\mathcal P_J\times \mathcal P_{{}^\delta J}$), a smooth map with fibres isomorphic to P (resp. $P/U_P$) for $P\in \mathcal P_J$. Since ${}^zY_J$ is smooth, irreducible, locally closed in $\mathcal P_J\times \mathcal P_{{}^\delta J}$, of dimension $l(z)+|\mathcal P_J|$, it follows that in this case ${}^w\tilde{Z}_J$ (resp. ${}^wZ_J$) is a smooth, irreducible, locally closed subvariety of $\tilde{Z}_J$ (resp. $Z_J$) of dimension $l(z)+|\mathcal P_J|+|P|$ (resp. $l(z)+|\mathcal P_J|+|P/U_P|$). Thus (a) follows in this case.

If $z\notin N_{J,\delta }$ then, setting $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$, we have $\sharp J_1<\sharp J$ and we can assume that (a) holds when J is replaced by $J_1$. Using 1.5(e),(b),(d) we deduce that ${}^w\tilde{Z}_J$ (resp. ${}^wZ_J$) is a smooth irreducible locally closed subvariety of $\tilde{Z}_J$ (resp. $Z_J$) of dimension $l(w)+|G|$ (resp. $l(w)+|G|-|\mathcal P_{J_1}|+|U_P\cap P'|-|U_P\cap U_{P'}|$ for some/any $(P,P')\in {}^zY_J$). To complete the proof of (a) it is then enough to note that $|{}^wZ_J|=|{}^w\tilde{Z}_J|-|\mathcal P_J|$. The proof above shows that

$$\begin{aligned} |\mathcal P_{J_1}|-|\mathcal P_J|=|U_P\cap P'|-|U_P\cap U_{P'}| \end{aligned}$$

for some/any $(P,P')\in {}^zY_J$.

Now (b) follows immediately from (a) using 1.5(e), (b), (d).

1.7. Examples. In the case where $J=I$ we can identify $Z_J=G^1$. It has a unique piece, ${}^1Z_J=Z_J$. In the case where $J=\emptyset$, $Z_J$ is a torus bundle over $\mathcal B^2$. The pieces of $Z_J$ are the inverse images of the G-orbits $\mathcal O_w$ ($w\in W$) under $Z_J{\rightarrow }\mathcal B^2$.

Assume now that $\hat{G}=G=GL(V)$ where V is a $\mathbf{k}$-vector space of dimension 3. We can identify the projective space P(V) with $\mathcal P_J$ for a certain 1-element subset J of I. Then $\tilde{Z}_J$ becomes the set of triples $(L,L',g)$ where $L,L'$ are lines in V and $g\in GL(V)$ carries L to $L'$; since $L'$ is determined by L, g we can identify $\tilde{Z}_J$ with the set of pairs (L, g) where L is a line in V and $g\in GL(V)$. For any $r\in [1,3]$ let ${}^r\tilde{Z}_J$ be the set of all $(L,g)\in \tilde{Z}_J$ such that $L,gL,g^2L,\dots$ span an r-dimensional subspace. Then the subsets ${}^r\tilde{Z}_J$ ($r\in [1,3]$) are exactly the pieces of $\tilde{Z}_J$. Now $Z_J$ becomes the set of quadruples $(L,L',\gamma ,\tilde{\gamma })$ where $L,L'$ are lines in V and $\gamma :L{\mathop {\rightarrow }\limits ^{\sim }}L'$, $\tilde{\gamma }:V/L{\mathop {\rightarrow }\limits ^{\sim }}V/L'$ are isomorphisms of vector spaces. Let ${}^1Z_J=\{(L,L',\gamma ,\tilde{\gamma })\in Z_J;L=L'\}$. Let ${}^2Z_J$ be the set of all $(L,L',\gamma ,\tilde{\gamma })\in Z_J$ such that $L\ne L'$ and $\tilde{\gamma }$ carries the image of $L'$ in V / L to the image of L in $V/L'$. Let ${}^3Z_J$ be the set of all $(L,L',\gamma ,\tilde{\gamma })\in Z_J$ such that $L\ne L'$ and such that the following holds: denoting by $L_1$ the image of $L'$ in V / L, by $L'_1$ the image of L in $V/L'$, and setting $L_2=\tilde{\gamma }^{-1}(L'_1)\subset V/L$, $L'_2=\tilde{\gamma }(L_1)\subset V/L'$, we have $L_2\ne L_1$ (hence $L'_2\ne L'_1$) and $\tilde{\gamma }(L_1\oplus L_2)=L'_1\oplus L'_2$. Then ${}^1Z_J,{}^2Z_J,{}^3Z_J$ are exactly the pieces of $Z_J$.

Assume now that $\hat{G}=G=Sp(V)$ where V is a $\mathbf{k}$-vector space of dimension 4 with a given nondegenerate symplectic form $\langle ,\rangle$. We can identify the projective space P(V) with $\mathcal P_J$ for a certain 1-element subset J of I. Then $\tilde{Z}_J$ becomes the set of triples $(L,L',g)$ where $L,L'$ are lines in V and $g\in Sp(V)$ carries L to $L'$; since $L'$ is determined by L, g we can identify $\tilde{Z}_J$ with the set of pairs (L, g) where L is a line in V and $g\in Sp(V)$.

For $r\in [1,2]$ let ${}^r\tilde{Z}_J$ be the set of all $(L,g)\in \tilde{Z}_J$ such that $L,gL,g^2L,\dots$ span an r-dimensional isotropic subspace of V and let ${}^rZ_J'$ be the set of all $(L,g)\in \tilde{Z}_J$ such that $\langle L,gL\rangle =\langle L,g^2L\rangle =\dots =\langle L,g^{r-1}L\rangle =0$, $\langle L,g^rL\rangle \ne 0$. Then the subsets ${}^r\tilde{Z}_J,{}^r\tilde{Z}'_J$ ($r=1,2$) are exactly the pieces of $\tilde{Z}_J$.

Now $Z_J$ becomes the set of quadruples $(L,L',\gamma ,\tilde{\gamma })$ where $L,L'$ are lines in V and $\gamma :L{\mathop {\rightarrow }\limits ^{\sim }}L'$ is an isomorphism of vector spaces and $\tilde{\gamma }:L^\perp /L{\mathop {\rightarrow }\limits ^{\sim }}L'{}^\perp /L'$ is a symplectic isomorphism. Let ${}^1Z_J=\{(L,L',\gamma ,\tilde{\gamma })\in Z_J;L=L'\}$. Let ${}^2Z_J$ be the set of all $(L,L',\gamma ,\tilde{\gamma })\in Z_J$ such that $L\ne L',\langle L,L'\rangle =0$ and $\tilde{\gamma }$ carries the image of $L'$ in $L^\perp /L$ to the image of L in $L'{}^\perp /L'$. Let ${}^2Z'_J$ be the set of all $(L,L',\gamma ,\tilde{\gamma })\in Z_J$ such that $L\ne L'$, $\langle L,L'\rangle =0$ and $\tilde{\gamma }$ does not carry the image of $L'$ in $L^\perp /L$ to the image of L in $L'{}^\perp /L'$. Let ${}^1Z'_J=\{(L,L',\gamma ,\tilde{\gamma })\in Z_J;\langle L,L'\rangle \ne 0\}$. Then the subsets ${}^1Z_J,{}^2Z_J,{}^2Z'_J,{}^1Z'_J$ are exactly the pieces of $Z_J$.

1.8. Let $J\subset I$. Let

$$\begin{aligned} \mathcal B^2_J=\{(B,B',gU_{B_J});(B,B')\in \mathcal B^2,gU_{B_J}\in G^1/U_{B_J},{}^gB=B'\}; \end{aligned}$$

this is well defined since $U_{B_J}\subset B$. Define $\pi _J:\mathcal B^2_J{\rightarrow }Z_J$ by

$$\begin{aligned} (B,B',gU_{B_J})\mapsto (B_J,B'_{{}^\delta J},gU_{B_J}). \end{aligned}$$

For any $y\in W$ let

$$\begin{aligned} \mathcal B^2_{J,y}=\{(B,B',gU_{B_J})\in \mathcal B^2_J;(B,B')\in \mathcal O_y\}, \end{aligned}$$

and let $\pi _{J,y}:\mathcal B^2_{J,y}{\rightarrow }Z_J$ be the restriction of $\pi _J$. The statements (a),(b) below can be deduced from [13, 4.14].

• (a) For any $w\in {}^{{}^\delta J}W$, the image of $\pi _{J,w^{-1}}:\mathcal B^2_{J,w^{-1}}{\rightarrow }Z_J$ is exactly ${}^wZ_J$.

• (b) If $w\in {}^{{}^\delta J}W$ and $x\in W_{{}^\delta (J^w_\infty )}$, then the image of $\pi _{J,w^{-1}x}:\mathcal B^2_{J,w^{-1}x}{\rightarrow }Z_J$ is contained in ${}^wZ_J$. Note that (a) gives an alternative description of ${}^wZ_J$ as the image of $\pi _{J,w^{-1}}$.

1.9. Let $J\subset I$ and let $w\in {}^{{}^\delta J}W$. In [5, 4.6] it is shown that the closure of ${}^wZ_J$ in $Z_J$ is equal to $\cup _{w'\in {}^{d(J)}W;w'\le _Jw}{}^{w'}Z_J$ where $w'\le _Jw$ is defined by the condition: $\delta (u)w'u^{-1}\le w$ for some $u\in W_J$.

3 Unipotent character sheaves on $Z_J$ and on ${}^wZ_J$

2.1. Let $J\subset I$. For $y\in W$ we set $\mathbf{K}^y_J=(\pi _{J,y})_!\bar{\mathbf{Q}}_l\in \mathcal D(Z_J)$. A unipotent character sheaf on $Z_J$ is by definition a simple perverse sheaf A on $Z_J$ such that $A\dashv \mathbf{K}^y_J$ for some $y\in W$. This is a special case of what in [13] is referred to as a parabolic character sheaf. Let $CS(Z_J)$ be the collection of unipotent character sheaves on $Z_J$.

In the case where $J=I$, we can identify $Z_J=Z_I$ with $G^1$. Hence there is a well defined notion of unipotent character sheaf on $G^1$. In this case, for $y\in W$ we have

$$\begin{aligned} \mathcal B^2_{I,y}=\{(B,B',g)\in \mathcal B\times \mathcal B\times G^1;B'={}^gB,(B,B')\in \mathcal O_y\} \end{aligned}$$

and $\pi _{I,y}:\mathcal B^2_{J,y}{\rightarrow }Z_I$ is given by $(B,B',g)\mapsto g$.

In the case where $J=\emptyset$, for any $y\in W$, $\pi _{J,y}$ is the inclusion of

$$\begin{aligned} \{(B,B',gU_B);(B,B')\in \mathcal O_y,g\in G^1,{}^gB=B'\} \end{aligned}$$

into

$$\begin{aligned} Z_\emptyset =\{(B,B',gU_B);(B,B')\in \mathcal B^2,g\in G^1,{}^gB=B'\}. \end{aligned}$$

It follows that $CS(Z_\emptyset )$ consists of the simple perverse sheaves on $Z_\emptyset$ which are (up to shift) the inverse images under $Z_\emptyset {\rightarrow }\mathcal B^2$ of the simple G-equivariant perverse sheaves on $\mathcal B^2$.

2.2. Let $J\subset I$ and let $w\in {}^{{}^\delta J}W$. We define a collection of simple perverse sheaves $CS({}^wZ_J)$ on ${}^wZ_J$ (said to be unipotent character sheaves on ${}^wZ_J$) by induction on $\sharp J$ as follows. We set $z=\min (W_{{}^\delta J}wW_J)$.

Assume first that $z\in N_{J,\delta }$ so that $z=w$. For any $(P,P')\in {}^zY_J$ we denote by $\mathcal S_{P,P'}$ the set of common Levi subgroups of $P,P'$; this is a nonempty set. For any $L\in \mathcal S_{P,P'}$ we denote by ${\hat{L}}$ the normalizer of L in $\hat{G}$. Note that the identity component of ${\hat{L}}$ is L. We set ${\hat{L}}^1=\{g\in G^1;{}^gP=P',{}^gL=L\}\subset {\hat{L}}$. If $g,g'\in {\hat{L}}^1$ then, setting $g'=gh$, we have $h\in P\cap {\hat{L}}$ hence $h\in L$; we see that ${\hat{L}}^1$ is a single connected component of ${\hat{L}}$. We have a diagram

(a) ${\hat{L}}^1{\mathop {\leftarrow }\limits ^{c}}G\times {\hat{L}}^1{\mathop {\rightarrow }\limits ^{c'}}{}^zZ_J$

where

$$\begin{aligned} c(h,g)=g,\quad c'(h,g)=({}^hP,{}^hP',hgh^{-1}U_{{}^hP}). \end{aligned}$$

Now c is a smooth morphism with fibres isomorphic to G and $c'$ is a smooth morphism with fibres isomorphic to $P\cap P'$. By 2.1 (for ${\hat{L}},{\hat{L}}^1$ instead of $\hat{G},G^1$) the notion of unipotent character sheaf on ${\hat{L}}^1$ is well defined. If $A_1\in CS({\hat{L}}^1)$ then $c^\bigstar A_1$ is a simple $(P\cap P')$-equivariant perverse sheaf on $G\times {\hat{L}}^1$ (for the free $P\cap P'$ action $a:(h,g)\mapsto (ha^{-1}, \mathrm{pr}(a)g \mathrm{pr}(a)^{-1})$ where $\mathrm{pr}:P\cap P'{\rightarrow }L$ is the canonical projection), hence it is of the form $c'{}^\bigstar A$ for a well defined (necessarily G-equivariant) simple perverse sheaf A on ${}^zZ_J$. By definition, $CS({}^zZ_J)$ consists of the simple perverse sheaves A obtained as above from some $A_1\in CS({\hat{L}}^1)$. Note that $A_1\mapsto A$ defines a bijection between the set of isomorphism classes of objects in $CS({\hat{L}}^1)$ and the set of isomorphism classes of objects in $CS({}^zZ_J)$. This definition of $CS({}^zZ_J)$ does not depend on the choice of $(P,P')$ in ${}^zY_J$ and that of L in $\mathcal S_{P,P'}$ since the set of triples $(P,P',L)$ as above is a homogeneous G-space.

Next we assume that $z\notin N_{J,\delta }$. Then, setting $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$, we have $\sharp J_1<\sharp J$ so that $CS({}^wZ_{J_1})$ is defined by the induction hypothesis. By definition, $CS({}^wZ_J)$ consists of the simple perverse sheaves of the form ${\vartheta }_{J,w}^\bigstar A$ where $A\in CS({}^wZ_{J_1})$ and ${\vartheta }_{J,w}$ is as in 1.6(b). Note that $A\mapsto {\vartheta }_{J,w}^\bigstar A$ defines a bijection from the set of isomorphism classes of objects in $CS({}^wZ_{J_1})$ and the set of isomorphism classes of objects in $CS({}^wZ_J)$.

This completes the inductive definition of $CS({}^wZ_J)$. Note that if $A\in CS({}^wZ_J)$, then A is G-equivariant.

2.3. Let $J\subset I$ and let $w\in {}^{{}^\delta J}W$. Let $A\in CS({}^wZ_J)$. Let $A^\sharp$ be the unique simple perverse sheaf on $Z_J$ such that $A^\sharp |_{{}^wZ_J}=A$ and $\mathrm{supp}(A^\sharp )$ is the closure in $Z_J$ of $\mathrm{supp}(A)$. (Here $\mathrm{supp}$ denotes support.) Let $CS'(Z_J)$ be the collection of simple perverse sheaves on $Z_J$ that are isomorphic to $A^\sharp$ for some w, A as above. We show that, if $\tilde{A}\in CS'(Z_J)$ and $\tilde{A}\cong A^\sharp$ with w, A as above, then

• (a) w is uniquely determined,

• (b) A is uniquely determined up to isomorphism.

Assume that we have also $\tilde{A}\cong A'{}^\sharp$ where $A'\in CS({}^{w'}Z_J)$. Then $\mathrm{supp}(A)$ and $\mathrm{supp}(A')$ are dense in $\mathrm{supp}(\tilde{A})$. Hence they have nonempty intersection. Since $\mathrm{supp}(A)\subset {}^wZ_J$, $\mathrm{supp}(A')\subset {}^{w'}Z_J$, it follows that ${}^wZ_J\cap {}^{w'}Z_J\ne \emptyset$ so that $w=w'$. We have $\tilde{A}|_{{}^wZ_J}\cong A$, $\tilde{A}|_{{}^wZ_J}\cong A'$ hence $A\cong A'$.

We now state the following result.

Proposition 2.4

Let $J\subset I$.

• (a) We have $CS(Z_J)=CS'(Z_J)$.

• (b) Let $w\in {}^{{}^\delta J}W$ and let $A\in CS(Z_J)$. Let $A'$ be a simple perverse sheaf on ${}^wZ_J$ such that $A'\dashv A|_{{}^wZ_J}$. Then $A'\in CS({}^wZ_J)$.

The proof of (a) appears in [13, 4.13, 4.17]. The proof of (b) appears in [13, 4.12]. We will reprove them here (the proof of 2.4(b) is given in 2.12; the proof of 2.4(a) is given in 2.14). To do so we are using a number of lemmas some of which are more precise than those in [13].

Lemma 2.5

Let $J\subset I$, $z\in N_{J,\delta }$ (so that $z\in {}^{{}^\delta J}W$). Let $y\in W$ and let A be a simple perverse sheaf on ${}^zZ_J$ such that $A\dashv \mathbf{K}^y_J|_{{}^zZ_J}$. Then $A\in CS({}^zZ_J)$.

If $y\notin z^{-1}W_{{}^\delta J}$, then the image of $\pi _{J,y}:\mathcal B^2_{J,y}{\rightarrow }Z_J$ is disjoint from ${}^zZ_J$ hence $\mathbf{K}^y_J|_{{}^zZ_J}=0$. This contradicts the choice of A. Thus we must have $y=z^{-1}y'$ for some $y'\in W_{{}^\delta J}$. We fix $(P,P')\in {}^zY_J$, $L\in \mathcal S_{P,P'}$ (see 2.2). Let $\mathcal B_L$ be the variety of Borel subgroups of L. Let $\mathcal B_{L,y'}=\{(\beta ,\beta ')\in \mathcal B_L^2;(\beta U_{P'},\beta 'U_{P'})\in \mathcal O_{y'}\}$. We have a cartesian diagram

where

$$\begin{aligned}&\displaystyle \Xi =\{(\beta ,\beta ',g)\in \mathcal B_L\times \mathcal B_L\times {\hat{L}}^1;g\beta g^{-1}=\beta ',(\beta ,\beta ')\in \mathcal O_{L,y'}\},\\&\displaystyle \tilde{c}(h,(\beta ,\beta ',g))=(\beta ,\beta ',g),\\&\displaystyle \tilde{c}'(h,(\beta ,\beta ',g))=(hP^{\beta U_{P'}}h^{-1},h\beta 'U_{P'}h^{-1},hgh^{-1}U_{hPh^{-1}}),\\&\displaystyle f(\beta ,\beta ',g)=g,\quad f'(h,(\beta ,\beta ',g)) =(h,g),\\&\displaystyle \quad f''(B,B',gU_{B_J})=(B_J,B'_{{}^\delta J},gU_{B_J}) \end{aligned}$$

and $c,c'$ are as in 2.2(a). It follows that $c^*f_!\bar{\mathbf{Q}}_l=c'{}^*f''_!\bar{\mathbf{Q}}_l$. (Both are equal to $f'_!\bar{\mathbf{Q}}_l$.) We have $f_!\bar{\mathbf{Q}}_l=\mathbf{K}^{y';{\hat{L}}^1}$ (which is defined like $\mathbf{K}^y_J$ by replacing $\hat{G},G^1,J,y$ by ${\hat{L}},{\hat{L}}^1,{}^\delta J,y'$) and $f''_!\bar{\mathbf{Q}}_l=\mathbf{K}^{z^{-1}y'}_J|_{{}^zZ_J}$. Thus we have

• (a) $c'{}^*(\mathbf{K}^{z^{-1}y'}_J|_{{}^zZ_J})=c^*(\mathbf{K}^{y';{\hat{L}}^1}).$

  Since c is smooth with fibres isomorphic to G and $c'$ is smooth with fibres isomorphic to $P\cap P'$, it follows that

• (b) $c'{}^\bigstar \oplus _j{}^pH^j(\mathbf{K}^{z^{-1}y'}_J|_{{}^zZ_J})=c^\bigstar \oplus _j{}^pH^j(\mathbf{K}^{y';{\hat{L}}^1})$

  so that $c'{}^\bigstar A\dashv c^\bigstar \mathbf{K}^{y';{\hat{L}}^1}$ hence there exists a simple perverse sheaf C on ${\hat{L}}^1$ such that $c'{}^\bigstar A=c^\bigstar C$ and $C\dashv \mathbf{K}^{y';{\hat{L}}^1}$. Thus $C\in CS({\hat{L}}^1)$ and from the definitions we see that $A\in CS({}^zZ_J)$. The lemma is proved.

2.6. For any $y_1,y_2,y_3$ in W and any $i\in \mathbf{Z}$ we set

$$\begin{aligned} R^i_{y_1,y_2,y_3}=H^i_c(\{B\in \mathcal B;(B_1,B)\in \mathcal O_{y_1},(B,B'_1)\in \mathcal O_{y_2}\},\bar{\mathbf{Q}}_l) \end{aligned}$$

where $(B_1,B'_1)\in \mathcal O_{y_3}$ is fixed. This is a $\bar{\mathbf{Q}}_l$-vector space independent of the choice of $(B_1,B'_1)$ (since G acts on $\mathcal O_{y_3}$ transitively with connected isotropy groups).

For $J\subset I, u\in W$ we define $\tilde{p}_{J,u}:{}^{u^{-1}}\tilde{Z}_\emptyset {\rightarrow }\tilde{Z}_J$ by $(B,B',g)\mapsto (B_J,B'_{{}^\delta J},g)$ and we set $\tilde{\mathbf{K}}^u_J=(\tilde{p}_{J,u})_!\bar{\mathbf{Q}}_l\in \mathcal D(\tilde{Z}_J)$; we have $\tilde{\mathbf{K}}^u_J=e_J^*\mathbf{K}^u_J$.

Lemma 2.7

Let $J\subset I$. Let $y^*\in W^{{}^\delta J}$, $y_*\in W_{{}^\delta J}$, $y=y^*y_*$. Let $z\in {}^{{}^\delta J}W^J$ and let $\Omega =W_{{}^\delta J}zW_J$. We set $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$. Let $\tilde{\alpha }:\tilde{Z}_{J,\Omega }{\mathop {\rightarrow }\limits ^{\sim }}\tilde{Z}^\dagger _{J_1,\Omega }$ be as in 1.5(b).

• (a) In $\mathcal D(\tilde{Z}_{J,\Omega })$ we have

  $$\begin{aligned} \tilde{\mathbf{K}}^y_J|_{\tilde{Z}_{J,\Omega }}\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \tilde{\alpha }^*(\tilde{\mathbf{K}}^{y'y^*}_{J_1}|_{\tilde{Z}^\dagger _{J_1,\Omega }})[-i];y'\in W_J,i\in \mathbf{Z}\}. \end{aligned}$$

• (b) Let $w\in \Omega \cap {}^{{}^\delta J}W$. Let ${\vartheta }_{J,w}:{}^wZ_J{\mathop {\rightarrow }\limits ^{\sim }}{}^wZ_{J_1}$ be as in 1.6(b). In $\mathcal D({}^wZ_J)$ we have

  $$\begin{aligned} \mathbf{K}^y_J|_{{}^wZ_J}\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*} \otimes {\vartheta }_{J,w}^*(\mathbf{K}^{y'y^*}_{J_1}|_{{}^wZ_{J_1}})[-i];y'\in W_J,i\in \mathbf{Z}\}. \end{aligned}$$

• (c) In the setup of (b) let $\tau _w:W_{{}^\delta (J^w_\infty )}{\rightarrow }W_{{}^\delta (J^w_\infty )}$ be as in 1.3. Let $x\in W_{{}^\delta (J^w_\infty )}$. We have

  $$\begin{aligned} \mathbf{K}^{w^{-1}x}_J|_{{}^wZ_J}={\vartheta }_{J,w}^*(\mathbf{K}^{w^{-1}\tau _w(x)}_{J_1}|_{{}^wZ_{J_1}}). \end{aligned}$$

We prove (a). Assume first that $y^{-1}\notin \Omega$. In this case, the image of $\tilde{p}_{J,y}:{}^{y^{-1}}\tilde{Z}_\emptyset {\rightarrow }\tilde{Z}_J$ is disjoint from $\tilde{Z}_{J,\Omega }$, hence $\tilde{\mathbf{K}}^y_J|_{\tilde{Z}_{J,\Omega }}=0$. Moreover, for any $y'\in W_J$, the image of $\tilde{p}_{J_1,y'y^*}:{}^{(y'y^*)^{-1}}\tilde{Z}_\emptyset {\rightarrow }\tilde{Z}_{J_1}$ is disjoint from $\tilde{Z}^\dagger _{J_1,\Omega }$. (We have $y^{*-1}\notin \Omega$. If $y^{*-1}y'{}^{-1}\in W_{{}^\delta J_1}zW_J$, then $y^{*-1}\in W_{{}^\delta J_1}zW_J\subset \Omega$, a contradiction.) Thus $\tilde{\mathbf{K}}^{y'y^*}_{J_1}|_{\tilde{Z}^\dagger _{J_1,\Omega }}=0$ and (a) holds.

We now assume that $y^{-1}\in \Omega$. We set

$$\begin{aligned} E^y_J= & {} \{(B_1,B,B'_1,g)\in \mathcal B\times \mathcal B\times \mathcal B\times G^1; (B_1,B)\in \mathcal O_{\delta ^{-1}(y_*)},\\&(B,B'_1)\in \mathcal O_{y^*},B'_1={}^gB_1\},\\ {}^{y^{*-1}W_J}\tilde{Z}_\emptyset= & {} \cup _{y'\in W_J}{}^{(y'y^*)^{-1}}\tilde{Z}_\emptyset \subset \tilde{Z}_\emptyset . \end{aligned}$$

Note that $\theta :E^y_J{\rightarrow }{}^{y^{-1}}\tilde{Z}_\emptyset$, $(B_1,B,B'_1,g)\mapsto (B,{}^gB,g)$ is a well defined isomorphism. Define $k:E^y_J{\rightarrow }{}^{y^{*-1}W_J}\tilde{Z}_\emptyset$ by $(B_1,B,B'_1,g)\mapsto (B_1,B'_1,g)$.

Now $\tilde{p}_{J,y}:{}^{y^{-1}}Z_\emptyset {\rightarrow }\tilde{Z}_J$ factors as ${}^{y^{-1}}Z_\emptyset {\mathop {\rightarrow }\limits ^{\phi }}\tilde{Z}_{J,\Omega }{\mathop {\rightarrow }\limits ^{j}}\tilde{Z}_J$ where j is the inclusion and ${}^{y^{*-1}W_J}\tilde{Z}_\emptyset {\rightarrow }\tilde{Z}_{J_1}$ (restriction of $\tilde{p}_{J_1}$) factors as ${}^{y^{*-1}W_J}\tilde{Z}_\emptyset {\mathop {\rightarrow }\limits ^{\psi }}\tilde{Z}^\dagger _{J_1,\Omega }{\mathop {\rightarrow }\limits ^{j_1}}\tilde{Z}_{J_1}$ where $j_1$ is the inclusion. We have a commutative diagram

From the definitions we have

$$\begin{aligned} \kappa _!\bar{\mathbf{Q}}_l\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \tilde{j}_{y'!}\bar{\mathbf{Q}}_l[-i]; y'\in W_J,i\in \mathbf{Z}\}, \end{aligned}$$

where $\tilde{j}_{y'}:{}^{(y'y^*)^{-1}}\tilde{Z}_\emptyset {\rightarrow }{}^{y^{*-1}W_J}\tilde{Z}_\emptyset$ is the obvious inclusion. It follows that

$$\begin{aligned} \psi _!\kappa _!\bar{\mathbf{Q}}_l\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \psi _!\tilde{j}_{y'!}\bar{\mathbf{Q}}_l[-i]; y'\in W_J,i\in \mathbf{Z}\}, \end{aligned}$$

that is,

$$\begin{aligned} \tilde{\alpha }_!\phi _!\theta _!\bar{\mathbf{Q}}_l\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \psi _!\tilde{j}_{y'!}\bar{\mathbf{Q}}_l[-i]; y'\in W_J,i\in \mathbf{Z}\}. \end{aligned}$$

Since $\theta$ is an isomorphism we have $\theta _!\bar{\mathbf{Q}}_l=\bar{\mathbf{Q}}_l$; since $\tilde{\alpha }$ is an isomorphism we have $\tilde{\alpha }^*\tilde{\alpha }_!=1$, hence

$$\begin{aligned} \phi _!\bar{\mathbf{Q}}_l\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \tilde{\alpha }^*\psi _!\tilde{j}_{y'!}\bar{\mathbf{Q}}_l[-i]; y'\in W_J,i\in \mathbf{Z}\}. \end{aligned}$$

Since $j^*j_!=1, j_1^*j_{1!}=1$ and $j_!\phi _!\bar{\mathbf{Q}}_l=(\tilde{p}_{J,y})_!\bar{\mathbf{Q}}_l=\tilde{\mathbf{K}}^y_J$,

$$\begin{aligned} j_{1!}\psi _!\tilde{j}_{y'!}\bar{\mathbf{Q}}_l=(\tilde{p}_{J_1,y'y^*})_!\bar{\mathbf{Q}}_l=\tilde{\mathbf{K}}^{y'y^*}_{J_1}, \end{aligned}$$

we have

$$\begin{aligned} j^*\tilde{\mathbf{K}}^y_J\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \tilde{\alpha }^*j_1^*(\tilde{\mathbf{K}}^{y'y^*}_{J_1})[-i];y'\in W_J,i\in \mathbf{Z}\} \end{aligned}$$

and (a) is proved.

We prove (b). We have a commutative diagram

where $h,h_1$ are the inclusions. Applying $h^*$ to the relation $\Bumpeq$ in (a) we obtain

$$\begin{aligned} \tilde{\mathbf{K}}^y_J|_{{}^w\tilde{Z}_J}\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes h^*\tilde{\alpha }^*j_1^*(\tilde{\mathbf{K}}^{y'y^*}_{J_1})[-i]; y'\in W_J,i\in \mathbf{Z}\}. \end{aligned}$$

Let $\tilde{\vartheta }_{J,w}$ be as in 1.6(b). Using that $h^*\tilde{\alpha }^*j_1^*=\tilde{\vartheta }_{J,w}^*h_1^*j_1^*$ and that $j_1h_1:{}^w\tilde{Z}_{J_1}{\rightarrow }\tilde{Z}_{J_1}$ is the inclusion we obtain

$$\begin{aligned} \tilde{\mathbf{K}}^y_J|_{{}^w\tilde{Z}_J}\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \tilde{\vartheta }_{J,w}^*(\tilde{\mathbf{K}}^{y'y^*}_{J_1}|_{{}^w\tilde{Z}_{J_1}})[-i];y'\in W_J,i\in \mathbf{Z}\} \end{aligned}$$

or equivalently

$$\begin{aligned} (e_J^*\mathbf{K}^y_J)|_{{}^w\tilde{Z}_J}\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes \tilde{\vartheta }_{J,w}^*((e_{J_1}^*\mathbf{K}^{y'y^*}_{J_1})|_{{}^w\tilde{Z}_{J_1}})[-i];y'\in W_J, i\in \mathbf{Z}\}. \end{aligned}$$

Let $e_{J,w}:{}^w\tilde{Z}_J{\rightarrow }{}^wZ_J$ (resp. $e_{J_1,w}:{}^w\tilde{Z}_{J_1}{\rightarrow }{}^wZ_{J_1}$) be the restriction of $e_J$ (resp. $e_{J_1}$). We have $(e_J^*\mathbf{K}^y_J)|_{{}^w\tilde{Z}_J}=e_{J,w}^*(\mathbf{K}^y_J|_{{}^wZ_J})$ and

$$\begin{aligned} \tilde{\vartheta }_{J,w}^*((e_{J_1}^*\mathbf{K}^{y'y^*}_{J_1})|_{{}^w\tilde{Z}_{J_1}})= \tilde{\vartheta }_{J,w}^*((e_{J_1,w}^*(\mathbf{K}^{y'y^*}_{J_1}|_{{}^wZ_{J_1}}))= e_{J,w}^*{\vartheta }_{J,w}^*(\mathbf{K}^{y'y^*}_{J_1}|_{{}^wZ_{J_1}}), \end{aligned}$$

hence

$$\begin{aligned} e_{J,w}^*(\mathbf{K}^y_J|_{{}^wZ_J})\Bumpeq \{R^i_{\delta ^{-1}(y_*),y^*;y'y^*}\otimes e_{J,w}^*{\vartheta }_{J,w}^*(\mathbf{K}^{y'y^*}_{J_1}|_{{}^wZ_{J_1}})[-i];y'\in W_J,i\in \mathbf{Z}\}. \end{aligned}$$

Applying $(e_{J,w})_!$ and using that $(e_{J,w})_!e_{J,w}^*=\langle -2|\mathcal P_J|\rangle$ we see that (b) holds.

We prove (c). We shall use notation in the proof of (a) with $y^*=w^{-1}$, $y_*=x$, $y=w^{-1}x$. We have $\delta ^{-1}(x)w^{-1}=w^{-1}\tau _w(x)$. Since $w^{-1}\in W^{{}^\delta J}$ and $\tau _w(x)\in W_{{}^\delta J}$ we have

$$\begin{aligned} l(\delta ^{-1}(x)w^{-1})= & {} l(w^{-1}\tau _w(x))=l(w^{-1}) +l(\tau _w(x))=l(w^{-1})+l(x)\nonumber \\&=l(\delta ^{-1}(x))+l(w^{-1}). \end{aligned}$$

Hence if $(B_1,B,B'_1,g)\in E^y_J$, then B is uniquely determined by $B_1,B'_1$ and k is an isomorphism of $E^y_J$ onto the subspace

$$\begin{aligned} {}^{w\delta ^{-1}(x)^{-1}}\tilde{Z}_\emptyset ={}^{\tau _w(x)^{-1}w}\tilde{Z}_\emptyset \end{aligned}$$

of ${}^{y^{*-1}W_J}\tilde{Z}_\emptyset$. Hence we have $j_{1!}\tilde{\alpha }_!\phi _!\theta _!\bar{\mathbf{Q}}_l=j_{1!}\psi _!k_!\bar{\mathbf{Q}}_l=\tilde{\mathbf{K}}^{w^{-1}\tau _w(x)}_{J_1}$. Since $\theta$ is an isomorphism we have $\theta _!\bar{\mathbf{Q}}_l=\bar{\mathbf{Q}}_l$. Hence $j_{1!}\tilde{\alpha }_!\phi _!\bar{\mathbf{Q}}_l=\tilde{\mathbf{K}}^{w^{-1}\tau _w(x)}_{J_1}$. Since $j_1^*j_{1!}=1$, $\tilde{\alpha }^*\tilde{\alpha }_!=1$ we deduce $\phi _!\bar{\mathbf{Q}}_l=\tilde{\alpha }^*j_1^*\tilde{\mathbf{K}}^{w^{-1}\tau _w(x)}_{J_1}$. We have $j_!\phi _!\bar{\mathbf{Q}}_l=\tilde{\mathbf{K}}^{w^{-1}x}_J$. Since $j^*j_!=1$, we deduce $\phi _!\bar{\mathbf{Q}}_l=j^*\tilde{\mathbf{K}}^{w^{-1}x}_J$. Thus

$$\begin{aligned} j^*\tilde{\mathbf{K}}^{w^{-1}x}_J=\tilde{\alpha }^*j_1^*\tilde{\mathbf{K}}^{w^{-1}\tau _w(x)}_{J_1}. \end{aligned}$$

Applying $h^*$ (notation in the proof of (b)) we obtain

$$\begin{aligned} \tilde{\mathbf{K}}^{w^{-1}x}_J|_{{}^w\tilde{Z}_J}=\tilde{\vartheta }_{J,w}^*(\tilde{\mathbf{K}}^{w^{-1}\tau _w(x)}_{J_1}|_{{}^w\tilde{Z}_{J_1}}). \end{aligned}$$

From this we deduce as in the proof of (b) that (c) holds. The lemma is proved.

Lemma 2.8

Let $J\subset I, w\in {}^{{}^\delta J}W$. Let $y\in W$. Let A be a simple perverse sheaf on ${}^wZ_J$ such that $A\dashv (\mathbf{K}_y|_{{}^wZ_J})$. Then $A\in CS({}^wZ_J)$.

We argue by induction on $\sharp J$. Let $z=\min (W_{{}^\delta J}wW_J)$.

Assume first that $z\in N_{J,\delta }$ so that $z=w$. If $y\notin z^{-1}W_{{}^\delta J}$, then the image of $\pi _{J,y}:\mathcal B^2_{J,y}{\rightarrow }Z_J$ is disjoint from ${}^wZ_J$ hence $\mathbf{K}^y|_{{}^wZ_J}=0$. This contradicts the choice of A. Thus we must have $y=z^{-1}y'$ for some $y'\in W_{{}^\delta J}$. Then with the notation in 2.5(b) we have

$$\begin{aligned} c'{}^\bigstar \oplus _j{}^pH^j(\mathbf{K}^{z^{-1}y'}|_{{}^wZ_J})=c^\bigstar \oplus _j{}^pH^j(\mathbf{K}^{y';{\hat{L}}^1}). \end{aligned}$$

Hence $c'{}^\bigstar A\dashv c^\bigstar \oplus _j{}^pH^j(\mathbf{K}^{y';{\hat{L}}^1})$ and there exists a simple perverse sheaf $A_1$ on ${\hat{L}}^1$ such that $c'{}^\bigstar A=c^\bigstar A_1$ and $A_1\dashv \mathbf{K}^{y';{\hat{L}}^1}$. Thus $A_1\in CS({\hat{L}}^1)$ and from the definitions we see that $A\in CS({}^wZ_J)$.

Next we assume that $z\notin N_{J,\delta }$. Then, setting $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$, we have $\sharp J_1<\sharp J$. We can write uniquely $y=y^*y_*$ as in 2.7. From 2.7(b) we see that there exists $y'\in W_J$ such that $A\dashv {\vartheta }_{J,w}^*\left( \mathbf{K}^{y'y^*}_{J_1}|_{{}^wZ_{J_1}}\right)$. Hence there exists a simple perverse sheaf $A_1$ on ${}^wZ_{J_1}$ such that $A={\vartheta }_{J,w}^\bigstar A_1$ and $A_1\dashv \mathbf{K}^{y'y^*}_{J_1}|_{{}^wZ_{J_1}}$. By the induction hypothesis, we have $A_1\in CS({}^wZ_{J_1})$. Hence $A\in CS({}^wZ_J)$. The lemma is proved.

Lemma 2.9

Let $J\subset I$ and let $w\in {}^{{}^\delta J}W$. Let $A\in CS({}^wZ_J)$.

• (a) There exists $x\in W_{{}^\delta (J^w_\infty )}$ such that $A\dashv \mathbf{K}^{w^{-1}x}_J|_{{}^wZ_J}$.

• (b) There exists $x\in W_{{}^\delta (J^w_\infty )}$ such that $A^\sharp \dashv \mathbf{K}^{w^{-1}x}_J$. In particular, $A^\sharp \in CS(Z_J)$. Thus we have $CS'(Z_J)\subset CS(Z_J)$.

We prove (a) by induction on $\sharp J$. Let $z=\min (W_{{}^\delta J}wW_J)$. Let $\tau _w:W_{{}^\delta (J^w_\infty )}{\rightarrow }W_{{}^\delta (J^w_\infty )}$ be as in 1.3.

Assume first that $z\in N_{J,\delta }$ so that $z=w$ and $J=J^w_\infty$. With notation in 2.2(a) we have $c'{}^\bigstar A=c^\bigstar A_1$ where $A_1\in CS({\hat{L}}^1)$ so that $A_1\dashv \mathbf{K}^{x;{\hat{L}}^1}$ for some $x\in W_{{}^\delta J}$. Since

$$\begin{aligned} c'{}^\bigstar \oplus _j{}^pH^j(\mathbf{K}^{w^{-1}x}|_{{}^wZ_J})=c^\bigstar \left( \oplus _j{}^pH^j(\mathbf{K}^{x;{\hat{L}}^1})\right) \end{aligned}$$

(see 2.5(b)) we see that $c'{}^\bigstar A\dashv c'{}^\bigstar \mathbf{K}^{w^{-1}x}|_{{}^wZ_J}$ hence $A\dashv \mathbf{K}^{w^{-1}x}|_{{}^wZ_J}$. Thus (a) holds in this case.

Next we assume that $z\notin N_{J,\delta }$. Then, setting $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$ we have $\sharp J_1<\sharp J$. We have $A={\vartheta }_{J,w}^\bigstar A_1$ where $A_1\in CS({}^wZ_{J_1})$. By the induction hypothesis, there exists $x\in W_{{}^\delta (J^w_\infty )}$ such that $A_1\dashv \mathbf{K}^{w^{-1}x}_{J_1}|_{{}^wZ_{J_1}}$. Hence

$$\begin{aligned} A={\vartheta }_{J,w}^*A_1\dashv {\vartheta }_{J,w}^*\left( \mathbf{K}^{w^{-1}x}_{J_1}|_{{}^wZ_{J_1}}\right) . \end{aligned}$$

Using 2.7(c) with x replaced by $\tau _w^{-1}(x)$, we deduce that $A\dashv \mathbf{K}^{w^{-1}\tau _w^{-1}(x)}_J|_{{}^wZ_J}$. This proves (a).

We prove (b). Let x be as in (a). Applying [14, 36.3(c)] with $Y,Y',C$ replaced by ${}^wZ_J,Z_J,\mathbf{K}^{w^{-1}x}_J|_{{}^wZ_J}$, we deduce that (b) holds. (We use the fact that, if $i:{}^wZ_J{\rightarrow }Z_J$ is the inclusion, then $i_!(\mathbf{K}^{w^{-1}x}_J|_{{}^wZ_J})=\mathbf{K}^{w^{-1}x}_J$, which follows from 1.8(b).)

2.10. Let $J\subset I$. For $y\in W$ let $\bar{\mathcal O}_y$ be the closure of $\mathcal O_y$ in $\mathcal B^2$. The closure of $\mathcal B^2_{J,y}$ in $\mathcal B^2_J$ is

$$\begin{aligned} \bar{\mathcal B}^2_{J,y}=\{(B,B',gU_{B_J})\in \mathcal B^2_J;(B,B')\in \bar{\mathcal O}_y\}. \end{aligned}$$

Let $\mathcal K_{J,y}$ be the intersection cohomology complex of $\bar{\mathcal B}^2_{J,y}$ with coefficients in $\bar{\mathbf{Q}}_l$, extended by zero on $\mathcal B^2_J-\bar{\mathcal B}^2_{J,y}$. Let $\bar{\mathbf{K}}^y_J=\pi _{J!}\mathcal K_{y,J}$ with $\pi _J:\mathcal B^2_J{\rightarrow }Z_J$ as in 1.8. For $y'\in W$ let $\iota _{y'}:\mathcal B^2_{J,y'}{\rightarrow }\mathcal B^2_J$ be the inclusion. By [7] we have

• (a) $\mathcal K_{y,J}\Bumpeq \{\mathcal V_{y',y}^i\otimes \iota _{y'!}\bar{\mathbf{Q}}_l\langle -i\rangle ;y'\in W,y'\le y,i\in 2\mathbf{Z}\}.$

  where $\mathcal V_{y',y}^i$ are $\bar{\mathbf{Q}}_l$-vector spaces such that

• (b) $\sum _{i\in 2\mathbf{Z}}\dim \mathcal V_{y',y}^iv^i=P_{y',y}(v^2)$

  and $P_{y',y}$ is the polynomial in [6]. Applying $\pi _{J!}$ we obtain

• (c) $\bar{\mathbf{K}}^y_J\Bumpeq \{\mathcal V_{y',y}^i\otimes \mathbf{K}^{y'}_J\langle -i\rangle ;y'\in W,y'\le y,i\in 2\mathbf{Z}\}.$

  Note that

• (d) $|\bar{\mathcal B}^2_{J,y}|=l(y)+|G|-|\mathcal P_J|.$

2.11. Let $J\subset I$ and let A be a simple perverse sheaf on $Z_J$. We show that conditions (i),(ii) below are equivalent:

• (i) $A\in CS(Z_J)$;

• (ii) We have $A\dashv \bar{\mathbf{K}}^y_J$ for some $y\in W$.

Assume that (ii) holds. Then from 2.10(c) we see that $A\dashv \mathbf{K}^{y'}_J$ for some $y'\in W$, $y'\le y$ hence (i) holds. Conversely, assume that (i) holds. We can find $y\in W$ such that $A\dashv \mathbf{K}^y_J$ for some $y\in W$ and such that for any $y'\in W$ such that $y'<y$. Using this and 2.10(c) we see that (ii) holds. (We use that $\dim \mathcal V_{y,y}^0=1$ and $\mathcal V_{y,y}^i=0$ for $i\ne 0$.)

2.12. We prove 2.4(b). By 2.11, A is a composition factor of ${}^pH^j(\bar{\mathbf{K}}^y_J)$ for some $y\in W$ and some $j\in \mathbf{Z}$. By the decomposition theorem, ${}^pH^j(\bar{\mathbf{K}}^y_J)$ is a semisimple perverse sheaf and ${}^pH^j(\bar{\mathbf{K}}^y_J)[-j]$ is a direct summand of $\bar{\mathbf{K}}^y_J$. It follows that A[j] is a direct summand of $\bar{\mathbf{K}}^y_J$. Hence $A'\dashv \bar{\mathbf{K}}^y_J|_{{}^wZ_J}$. Using 2.10(c) we deduce that $A'\dashv \mathbf{K}^{y'}_J|_{{}^wZ_J}$ for some $y'\le y$. Using 2.8 we deduce that $A'\in CS({}^wZ_J)$. This proves 2.4(b).

Lemma 2.13

Let $J\subset I$. Let $A\in CS(Z_J)$. There exist $w\in {}^{{}^\delta J}W$ and $A'\in CS({}^wZ_J)$ such that $A=A'{}^\sharp$. In particular, $CS(Z_J)\subset CS'(Z_J)$.

The subsets $\{\mathrm{supp}(A)\cap {}^wZ_J;w\in {}^{{}^\delta J}W\}$ form a partition of $\mathrm{supp}(A)$ into locally closed subvarieties. Hence we can find $w\in {}^{{}^\delta J}W$ such that $\mathrm{supp}(A)\cap {}^wZ_J$ is open dense in $\mathrm{supp}(A)$. We set $A'=A|_{{}^wZ_J}$. Then $A'$ is a simple perverse sheaf on ${}^wZ_J$ and $A'\in CS({}^wZ_J)$ (see 2.4(b)). From the definitions we have $A=A'{}^\sharp$. The lemma is proved.

2.14. Let $J\subset I$. Since $CS'(Z_J)\subset CS(Z_J)$ (see 2.9(b)) and $CS(Z_J)\subset CS'(Z_J)$ (see 2.13) we see that 2.4(a) holds.

2.15. For any $J\subset I$ and any $w\in {}^{d(J)}W$ we choose $(P,P')\in \mathcal P_{J^w_\infty }\times \mathcal P_{{}^\delta (J^w_\infty )}$ and a common Levi subgroup $L^w$ of $P,P'$. Define ${\hat{L}}^w,{\hat{L}}^{w1}$ as ${\hat{L}},{\hat{L}}^1$ in 2.2 with J, L replaced by $J^w_\infty ,L^w$. We shall denote by $\underline{CS}({\hat{L}}^{w1})$ a set of representatives for the objects in $CS({\hat{L}}^{w1})$. For $C\in \underline{CS}({\hat{L}}^{w1})$ we denote by $\tilde{C}$ the object of $CS({}^wZ_{J^w_\infty })$ such that $c'{}^\bigstar \tilde{C}=c^\bigstar C$ (where $c,c'$ are as in 2.5 with J replaced by $J^w_\infty$); we set

$$\begin{aligned} C_w={\vartheta }_{J,w}^\bigstar {\vartheta }_{J_1,w}^\bigstar {\vartheta }_{J_2,w}^\bigstar \dots (\tilde{C})\in CS({}^wZ_J) \end{aligned}$$

where $(J_n,w_n)_{n\ge 0}=\kappa (w)$. Then $C_w^\sharp :=(C_w)^\sharp$ is defined as in 2.3; note that $C_w^\sharp \in CS(Z_J)$ by 2.4(a). From 2.4 and the definitions we see that

• (a) $\{C_w;C\in \underline{CS}({\hat{L}}^{w1})\}$ is a set of representatives for the isomorphism classes of objects in $CS({}^wZ_J)$;

• (b) $\{C_w^\sharp ; w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1})\}$ is a set of representatives for the isomorphism classes of objects in $CS(Z_J)$.

Let $\mathfrak {K}_J$ be the free $\mathcal A$-module ($\mathcal A$ as in 0.1) with basis $\{C_w^\sharp ;w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1})\}$. For $w\in {}^{{}^\delta J}W$ let ${}^w\mathfrak {K}_J$ be the free $\mathcal A$-module with basis $\{C_w;C\in \underline{CS}({\hat{L}}^{w1})\}$. Let $\mathfrak {K}({\hat{L}}^{w1})$ be the free $\mathcal A$-module with basis $\{C;C\in \underline{CS}({\hat{L}}^{w1})\}$.

Let $\xi \mapsto \xi _w$ be the $\mathcal A$-module isomorphism $\mathfrak {K}({\hat{L}}^{w1}){\mathop {\rightarrow }\limits ^{\sim }}{}^w\mathfrak {K}_J$ such that $C\mapsto C_w$ for any $C\in \underline{CS}({\hat{L}}^{w1})$.

4 Mixed structures

3.1. In this section we assume that $\mathbf{k}$ is an algebraic closure of a finite field $\mathbf{F}_q$ with q elements and that we are given an $\mathbf{F}_q$-rational structure on $\hat{G}$ with Frobenius map $F:\hat{G}{\rightarrow }\hat{G}$ such that $F(G^1)=G^1$ and the restriction of F to G is the Frobenius map of an $\mathbf{F}_q$-split rational structure on G. Then for any $J\subset I$, we have $P\in \mathcal P_J\implies F(P)\in \mathcal P_J$ and $P\mapsto F(P)$ is the Frobenius map of an $\mathbf{F}_q$-rational structure on $\mathcal P_J$; moreover, $(P,P',gU_P)\mapsto (F(P),F(P'),F(g)U_{F(P)})$ is the Frobenius map of an $\mathbf{F}_q$-rational structure on $Z_J$. For any $w\in W$ we have $(B,B')\in \mathcal O_w\implies (F(B),F(B'))\in \mathcal O_w$ and $(B,B')\mapsto (F(B),F(B'))$ is the Frobenius map of an $\mathbf{F}_q$-rational structure on $\mathcal O_w$. For any $J\subset I$ and any $w\in {}^{{}^\delta J}W$, ${}^wZ_J$ is a subvariety of $Z_J$ defined over $\mathbf{F}_q$; we choose $P,P',L^w$ as in 2.15 in such a way that $F(P)=P$, $F(P')=P'$, $F(L^w)=L^w$, $F({\hat{L}}^{w1})={\hat{L}}^{w1}$ (notation of 2.2 with J replaced by $J^w_\infty$). We shall assume (as we may, by replacing if necessary q by a power of q) that for any $J\subset I$, any $w\in {}^{{}^\delta J}W$, and any $C\in \underline{CS}({\hat{L}}^{w1})$, we can find an isomorphism $\phi _C:F^*C{\mathop {\rightarrow }\limits ^{\sim }}C$ which makes C into a pure complex of weight 0; we shall assume that such a $\phi _C$ has been chosen.

Let $J\subset I$, $w\in {}^{{}^\delta J}W$. For $C\in \underline{CS}({\hat{L}}^{w1})$ let $\tilde{C}\in CS({}^wZ_{J^w_\infty })$, $C_w\in CS({}^wZ_J)$, $C_w^\sharp \in CS(Z_J)$ be as in 2.15. Then $\tilde{C},C_w,C_w^\sharp$ inherit from C mixed structures which are pure of weight 0.

3.2. Let $J\subset I$. Let $\omega$ (resp. $\omega _J$) be the element of maximal length in W (resp. $W_J$). Let $\tilde{B}\in \mathcal B$. Let $\tilde{P}=\tilde{B}_{\omega J\omega }$, $\tilde{U}=U_{\tilde{P}}$. Now $\tilde{U}$ acts by conjugation on

$$\begin{aligned} \mathcal U=\{B\in \mathcal B;\mathrm{pos}(\tilde{B},B)\in \omega W_J\}, \end{aligned}$$

an open subset of $\mathcal B$. For any $P\in \mathcal P_J$ such that $\mathrm{pos}(\tilde{B},P)=\omega \omega _J$ we define $\mathcal B_P\times \tilde{U}{\rightarrow }\mathcal U$ by $(B_1,u)\mapsto uB_1u^{-1}$. We show:

• (a) This map is a bijection (in fact an isomorphism).

Assume that $B'\in \mathcal B_P,B''\in \mathcal B_P$ and $u'\in \tilde{U},u''\in \tilde{U}$ satisfy $u'B'u'{}^{-1}=u''B''u''{}^{-1}$. Setting $u=u''{}^{-1}u'\in \tilde{U}$ we have $uB'u^{-1}=B''$ hence $uPu^{-1}=P$ and $u\in P$. Now $\tilde{P},P$ are opposed parabolic subgroups so that $P\cap \tilde{U}=\{1\}$. Thus $u=1$ so that $u'=u''$ and $B'=B''$. We see that our map is injective. Let $B\in \mathcal U$. We have $B\in \mathcal B_{P'}$ where $P'\in \mathcal P_J,\mathrm{pos}(\tilde{B},P')=\omega \omega _J$. Now $\tilde{U}$ acts transitively (by conjugation) on $\{P_1\in \mathcal P_J,\mathrm{pos}(\tilde{B},P_1)=\omega \omega _J\}$ hence there exists $u\in \tilde{U}$ such that $uPu^{-1}=P'$. Setting $B_1=u^{-1}Bu$ we have $B_1\in \mathcal B_P$ and our map takes $(B_1,u)$ to B; thus it is surjective hence bijective. We omit the proof of the fact that it is an isomorphism.

Let $L=\tilde{P}\cap P$; this is a common Levi subgroup of $\tilde{P}$ and P. Let S be the identity component of the centre of L. We can find a one-parameter subgroup $\lambda :\mathbf{k}^*{\rightarrow }S$ such that $\lim _{t\mapsto 0}\lambda (t)u\lambda (t)^{-1}=1$ for any $u\in \tilde{U}$. We define an action $t:B\mapsto \lambda (t)B\lambda (t)^{-1}$ of $\mathbf{k}^*$ on $\mathcal U$. (This is well defined since $\lambda (t)\in \tilde{B}$.) Under the isomorphism (a) this action becomes the action of $\mathbf{k}^*$ on $\mathcal B_P\times \tilde{U}$ given by $t:(B_1,u)\mapsto (B_1,\lambda (t)u\lambda (t)^{-1})$. (To see this we must check that $\lambda (t)u\lambda (t)^{-1}B_1\lambda (t)u^{-1}\lambda (t)^{-1}=\lambda (t)uB_1u^{-1}\lambda (t)^{-1}$ for $B_1\in \mathcal B_P$; this holds since $\lambda (t)\in B_1$.) Now $\lim _{t\mapsto 0}(B_1,\lambda (t)u\lambda (t)^{-1})=(B_1,1)$ for any $(B_1,u)\in \mathcal B_P\times \tilde{U}$. Hence $t:B\mapsto \lambda (t)B\lambda (t)^{-1}$ is a flow on $\mathcal U$ which contracts $\mathcal U$ to its fixed point set $\mathcal B_P$. We have $\mathcal B\times \mathcal P_J=\sqcup _{y\in W^J}\mathcal O'_y$ where $\mathcal O'_y$ is the image of $\mathcal B^2_{yW_J}:=\cup _{a\in W_J}\mathcal O_{ya}$ under the obvious map $\mathcal B\times \mathcal B{\rightarrow }\mathcal B\times \mathcal P_J$. This is exactly the decomposition of $\mathcal B\times \mathcal P_J$ into G-orbits where G acts by simultaneous conjugation. Hence $\mathcal B^2_{yW_J}$ is a locally closed subvariety of $\mathcal B^2$ for any $y\in W^J$.

3.3. Let $J\subset I$. We now fix $y\in W^J$. Let $\tilde{B}\subset \tilde{P}$ be as in 3.2. Let $B_1\in \mathcal B$ be such that $(\tilde{B},B_1)\in \mathcal O_\omega$ and let $P=(B_1)_J$; then $\mathrm{pos}(\tilde{B},P)=\omega \omega _J$. We define $S,\lambda$ in terms of $\tilde{P},P$ as in 3.2. We choose $B^*\in \mathcal B$ as follows: we note that $T_1:=\tilde{B}\cap B_1$ is a maximal torus of G containing S (since $S\subset \tilde{B}$, $S\subset B_1$) and we choose $B^*$ so that $T_1\subset B^*$ and $(\tilde{B},B^*)\in \mathcal O_{\omega y^{-1}}$. We have $S\subset B^*$. Since $(B^*,\tilde{B})\in \mathcal O_{y\omega }$, $(\tilde{B},B_1)\in \mathcal O_\omega$ and $B^*,\tilde{B},B_1$ contain a common maximal torus, we have $(B^*,B_1)\in \mathcal O_{y\omega \omega }=\mathcal O_y$. Hence for any $B\in \mathcal B_P$, we have $(B^*,B)\in \mathcal O_{ya}$ with $a\in W_J$. In other words, we have $\mathcal B_P\subset \cup _{a\in W_J}\mathcal B_{ya}$, where for any $z\in W$ we set $\mathcal B_z=\{B\in \mathcal B;(B^*,B)\in \mathcal O_z\}$. As in 3.2, we set $\mathcal U=\{B\in \mathcal B;\mathrm{pos}(\tilde{B},B)\in \omega W_J\}$. Note that $\cup _{a\in W_J}\mathcal B_{ya}\subset \mathcal U$. (If $B\in \mathcal B$ satisfies $\mathrm{pos}(B^*,B)\in yW_J$ then for some $B'\in \mathcal B$ we have $\mathrm{pos}(B^*,B')=y,\mathrm{pos}(B',B)\in W_J$, hence $\mathrm{pos}(\tilde{B},B')=\omega$ and $\mathrm{pos}(\tilde{B},B)\in \omega W_J$.)

For $z\in W$ we set $\bar{\mathcal B}_z=\{B\in \mathcal B;(B^*,B)\in \bar{\mathcal O}_z\}$. Let $w\in W$. We show:

• (a) $\bar{\mathbf{K}}^w|_{\mathcal B^2_{yW_J}}$ is pure of weight 0.

Let $\bar{\mathcal K}^w$ be the intersection cohomology complex of $\bar{\mathcal B}_w$ with coefficients in $\bar{\mathbf{Q}}_l$; it is naturally a pure complex of weight 0. Using the transitivity of the simultaneous conjugation action of G on $\mathcal O'_y$ and the fact the fibre of $\mathcal B^2_{yW_J}{\rightarrow }\mathcal O'_y$ at $(B^*,P)\in \mathcal O'_y$ may be identified with $\mathcal B_P$, we see that it is enough to show that:

• (b) $\bar{\mathcal K}^w|_{\bar{\mathcal B}_w\cap \mathcal B_P}$ is pure of weight 0.

We can assume that $\bar{\mathcal B}_w\cap \mathcal B_P\ne \emptyset$. Since $\bar{\mathcal B}_w\cap \mathcal U$ is open in $\bar{\mathcal B}_w$, we have $\bar{\mathcal K}^w|_{\bar{\mathcal B}_w\cap \mathcal U}=K$ where K is the intersection cohomology complex of $\bar{\mathcal B}_w\cap \mathcal U$ and it is enough to show that $K|_{\bar{\mathcal B}_w\cap \mathcal B_P}$ is pure of weight 0 (recall that $\mathcal B_P\subset \mathcal U$). For any $z\in W$, $\mathcal B_z\cap \mathcal U$ is stable under the $\mathbf{k}^*$-action $t:B\mapsto \lambda (t)B\lambda (t)^{-1}$ on $\mathcal U$ (we use that $\lambda (t)\in B^*$ for any t). Hence $\bar{\mathcal B}_w\cap \mathcal U$ is stable under this $\mathbf{k}^*$-action on $\mathcal U$. Since the $\mathbf{k}^*$-action on $\mathcal U$ is a contraction to its fixed point set $\mathcal B_P$ and $\bar{\mathcal B}_w\cap \mathcal U$ is closed in $\mathcal U$ and $\mathbf{k}^*$-stable, we deduce that the $\mathbf{k}^*$-action on $\bar{\mathcal B}_w\cap \mathcal U$ is a contraction to $\bar{\mathcal B}_w\cap \mathcal B_P$ so that (b) follows from the “hyperbolic localization theorem” [3]. This proves (a).

Proposition 3.4

Let $J\subset I$ and let $z\in N_{J,\delta }$.

• (a) For $y\in W$, $\bar{\mathbf{K}}^y_J|_{{}^zZ_J}$ (with its natural mixed structure) is pure of weight 0.

• (b) If $w\in {}^{{}^\delta J}W$, $C\in CS({\hat{L}}^{w1})$, then $C_w^\sharp |_{{}^zZ_J}$ (with its natural mixed structure) is pure of weight 0.

We prove (a). We have ${}^zZ_J=\{(P,P',gU_P)\in Z_J;\mathrm{pos}(P',P)=z\}$. We have a diagram

$$\begin{aligned} \mathcal B^2_{z^{-1}W_{{}^\delta J}}{\mathop {\leftarrow }\limits ^{c}}\Xi {\mathop {\rightarrow }\limits ^{d}}{}^zZ_J \end{aligned}$$

where $\Xi =\{(B,B',gU_{B_J})\in \mathcal B^2_J; (B,B')\in \cup _{a\in W_{{}^\delta J}}\mathcal O_{z^{-1}a}\}$ and $c(B,B',gU_{B_J})=(B,B')$, $d(B,B',gU_{B_J})=(B_J,B'_{{}^\delta J},gU_{B_J})$. Now $\Xi$ is the inverse image of ${}^zZ_J$ under $\pi _J:\mathcal B^2_J{\rightarrow }Z_J$ and d is the restriction of $\pi _J$. Moreover $\pi _J$ is proper hence d is proper. Note also that c is smooth. From the definitions we see that $\bar{\mathbf{K}}^y_J|_{{}^zZ_J}=d_!c^*(\bar{\mathbf{K}}^y|_{\mathcal B^2_{z^{-1}W_{{}^\delta J}}})$. It remains to note that $\bar{\mathbf{K}}^y|_{\mathcal B^2_{z^{-1}W_{{}^\delta J}}}$ is pure of weight 0 (see 3.3(a) with J replaced by ${}^\delta J$), that $c^*$ maps pure complexes of weight zero to pure complexes of weight zero (since c is smooth) and $d_!$ maps pure complexes of weight zero to pure complexes of weight zero (by Deligne’s theorem applied to the proper map d).

We prove (b). We can find $y\in W$ and $j\in \mathbf{Z}$ such that $C_w^\sharp$ appears in ${}^pH^j(\bar{\mathbf{K}}^y_J)$ (with mixed structures being not necessarily compatible). Since $\bar{\mathbf{K}}^y_J$ and $\bar{\mathbf{K}}^y_J|_{{}^zZ_J}$ are pure of weight 0 it follows that ${}^pH^j(\bar{\mathbf{K}}^y_J)$ and ${}^pH^j(\bar{\mathbf{K}}^y_J)|_{{}^zZ_J}$ are pure of weight j. We can find a nonzero mixed vector space V of pure weight j such that $V\otimes C_w^\sharp$ is a direct summand of ${}^pH^j(\bar{\mathbf{K}}^y_J)$ (respecting the mixed structures). Then $V\otimes C_w^\sharp |_{{}^zZ_J}$ is a direct summand of ${}^pH^j(\bar{\mathbf{K}}^y_J)|_{{}^zZ_J}$ (respecting the mixed structures). Hence $C_w^\sharp |_{{}^zZ_J}$ is pure of weight 0.

Remark. More generally for $J\subset I$, $y\in W$, $w\in {}^{{}^\delta J}W$, we expect that $\bar{\mathbf{K}}^y_J|_{{}^wZ_J}$ (with its natural mixed structure) is pure of weight 0.

3.5. Let $J\subset I$. Let $y\in W$. Since $\bar{\mathbf{K}}^y_J\langle |\bar{\mathcal B}^2_{J,y}|\rangle$ is pure of weight 0, we have for any $j\in \mathbf{Z}$:

• (a) ${}^pH^{-j}(\bar{\mathbf{K}}^y_J\langle |\bar{\mathcal B}^2_{J,y}|\rangle )= \oplus _{w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1})}V_{y,w,C,j}(j/2)\otimes C_w^\sharp$

  in $\mathcal D_m(Z_J)$, where $V_{y,w,C,j}$ are $\bar{\mathbf{Q}}_l$-vector spaces of pure weight 0. Moreover, by the relative hard Lefschetz theorem [2, 5.4.10], for any y, w, C, j we have:

• (b) $V_{y,w,C,j}=V_{y,w,C,-j}.$

  Hence if $t\in N_{J,\delta }$, we have

• (c) $\bar{\mathbf{K}}^y_J|_{{}^tZ_J}\langle |\bar{\mathcal B}^2_{J,y}|\rangle \Bumpeq \{V_{y,w,C,j}\otimes C_w^\sharp |_{{}^tZ_J}\langle j\rangle ;w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1}),j\in \mathbf{Z}\}.$

  Since for $w\in {}^{{}^\delta J}W$, $C_w^\sharp |_{{}^tZ_J}$ is pure of weight 0 (see 3.4) we have

• (d) $C_w^\sharp |_{{}^tZ_J}\Bumpeq \{{}'V{}_{t,w,j'}^{C',C}\otimes C'_t\langle j'\rangle ; C'\in \underline{CS}({\hat{L}}^{t1}),j'\in \mathbf{N}\}$

  in $\mathcal D_m({}^tZ_J)$, where ${}'V_{t,w,j'}^{C',C}$ are $\bar{\mathbf{Q}}_l$-vector spaces of pure weight 0. Note that if $w=t\in N_{J,\delta }$, we have $C_w^\sharp |_{{}^tZ_J}=C_t$ hence ${}'V_{t,t,j'}^{C',C}$ is $\bar{\mathbf{Q}}_l$ if $C'=C,j'=0$ and ${}'V_{t,t,j'}^{C',C}$ is 0 otherwise. From (c),(d) we deduce

• (e) $\begin{array}{l}\bar{\mathbf{K}}^y_J|_{{}^tZ_J}\langle |\bar{\mathcal B}^2_{J,y}|\rangle \Bumpeq \{V_{y,w,C,j}\otimes {}'V_{t,w,j'}^{C',C}\otimes C'_t\langle j+j'\rangle ;\\ w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1}),C'\in \underline{CS}({\hat{L}}^{t1}),j\in \mathbf{Z},j'\in \mathbf{N}\}\end{array}$

in $\mathcal D_m({}^tZ_J)$.

3.6. Let $J\subset I$. For any $w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1})$ and any $K\in \mathcal D_m(Z_J)$ we set

$$\begin{aligned} (C_w^\sharp :K)=\sum _{j,h\in \mathbf{Z}}(-1)^j(\text { multiplicity of } C_w^\sharp \text { in }{}^pH^j(K)_h)(-v)^h\in \mathcal A. \end{aligned}$$

Here, for any mixed perverse sheaf R, $R_h$ denotes the subquotient of R of pure weight h. We set

• (a) $\chi (K)=\sum _{w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1})}(C_w^\sharp :K)C_w^\sharp \in \mathfrak {K}_J.$

  For any $w\in {}^{{}^\delta J}W,C\in \underline{CS}({\hat{L}}^{w1})$ and any $K'\in \mathcal D_m({}^wZ_J)$ we set

• (b) $(C_w:K')=\sum _{j,h\in \mathbf{Z}}(-1)^j(\text {multiplicity of } C_w \text { in } {}^pH^j(K')_h)(-v)^h\in \mathcal A.$

  We set

  $$\begin{aligned} \chi _w(K')=\sum _{C\in \underline{CS}({\hat{L}}^{w1})}(C_w:K')C_w\in {}^w\mathfrak {K}_J. \end{aligned}$$

  From 2.10(b),(c) we deduce for $y\in W$:

• (c) $\chi _w(\bar{\mathbf{K}}^y_J\langle |\bar{\mathcal B}^2_{J,y}|\rangle )v^{|\bar{\mathcal B}^2_{J,y}|}= \chi _w(\bar{\mathbf{K}}^y_J)=\sum _{y'\in W;y'\le y}P_{y',y}(v^2)\chi _w(\mathbf{K}^{y'}_J).$

  (We use that $\mathcal V_{y',y}$ in 2.10(b) are pure of weight 0, see [7].)

  For $t\in N_{J,\delta }$, $w\in N_{J,\delta }$, $C\in \underline{CS}({\hat{L}}^{w1})$, we have

• (d) $\chi _t(C^\sharp _w|_{{}^tZ})=\sum _{C'\in \underline{CS}({\hat{L}}^{t1})}\Phi _{t,w}^{C',C}C'_t$

  where

  $$\begin{aligned} \Phi _{t,w}^{C',C}=\sum _{j'\in \mathbf{N}}\dim {}'V_{t,w,j'}^{C',C}v^{-j'}\in \mathbf{N}[v^{-1}] \end{aligned}$$

  specializes to the polynomial $P_{y,w}$ of [6] (in the case where $J=\emptyset$, $\delta =1$). Note that $\Phi _{t,w}^{C',C}$ is 1 if $t=w,C=C'$ and is in $v^{-1}\mathbf{N}[v^{-1}]$ otherwise.

  For $t\in N_{J,\delta }$, $y\in W$ we set

• (e) $[y]_{(t)}=v^{-|{}^tZ_J|}\chi _t(\bar{\mathbf{K}}^{z^{-1}u}_J).$

3.7. Let $J\subset I$, $z\in N_{J,\delta }$, $y\in W$. For $u\in W_{{}^\delta J}$ we define $\bar{\mathbf{K}}^{u;{\hat{L}}^{z1}}$ like $\bar{\mathbf{K}}^y_J$ by replacing $\hat{G},G^1,J,y$ by ${\hat{L}}^z,{\hat{L}}^{z1},{}^\delta J,u$. We define

$$\begin{aligned} \widetilde{[u]}=v^{-|L^z|}\chi (\mathbf{K}^{u;{\hat{L}}^{z1}})\in \mathfrak {K}({\hat{L}}^{z1}),\quad [u]=v^{-|L^z|}\chi (\bar{\mathbf{K}}^{u;{\hat{L}}^{z1}})\in \mathfrak {K}({\hat{L}}^{z1}), \end{aligned}$$

like $\chi (\mathbf{K}^y_J)$ and $\chi (\bar{\mathbf{K}}^y_J)$ by replacing $\hat{G},G^1,J,y$ by ${\hat{L}}^z,{\hat{L}}^{z1},{}^\delta J,u$.

From the proof of 2.5 we obtain:

If $y\notin z^{-1}W_{{}^\delta J}$, then $\chi _z(\mathbf{K}^y_J)=0$. If $y=z^{-1}u$ with $u\in W_{{}^\delta J}$, then

$$\begin{aligned} \chi _z(\mathbf{K}^y_J)=v^{|{}^zZ_J|}\sum _{C\in \underline{CS}({\hat{L}}^{z1})}f_{u,C}C_z \end{aligned}$$

where $f_{u,C}\in \mathcal A$ are given by $\chi (\mathbf{K}^{u;{\hat{L}}^{z1}})=v^{|L^z|}\sum _{C\in \underline{CS}({\hat{L}}^{z1})}f_{u,C}C$. Thus we have

• (a) $v^{-|{}^zZ_J|}\chi _z(\mathbf{K}^y_J)=v^{-|L^z|}(\chi (\mathbf{K}^{u;{\hat{L}}^{z1}}))_z.$

  Let $z,t\in N_J$. Using (a), 3.6(c), we see that for $u\in W_{{}^\delta J}$ we have

• (b) $\chi _t(\bar{\mathbf{K}}^{z^{-1}u}_J)=\sum _{u'\in W_{{}^\delta J};t^{-1}u'\le z^{-1}u}P_{t^{-1}u',z^{-1}u}(v^2)v^{|{}^tZ_J|}\widetilde{u'}_t.$

  Using 3.6(c) for ${\hat{L}}^{z1}$ instead of $G^1$ we see that for $u'\in W_{{}^\delta J}$ we have

  $$\begin{aligned}{}[u']=\sum _{u''\in W_{{}^\delta J};u''\le u'}P_{u'',u'}(v^2)\widetilde{[u'']}, \end{aligned}$$

  hence

  $$\begin{aligned} \widetilde{[u']}=\sum _{u''\in W_{{}^\delta J};u''\le u'}P'_{u'',u'}(v^2)[u''] \end{aligned}$$

  where $(P'_{u'',u'}(v^2))$ is the matrix inverse to $(P_{u'',u'}(v^2))$ (with $u',u''$ in $W_{{}^\delta J}$). Introducing this in (b) we obtain

• (c) $[z^{-1}u]_{(t)}=\sum _{u''\in W_{{}^\delta J}} \left( \sum _{u'\in W_J;u''\le u',t^{-1}u'\le z^{-1}u}P'_{u'',u'}(v^2)P_{t^{-1}u',z^{-1}u}(v^2)\right) [u'']_t.$

3.8. Let H be the Hecke algebra of W. As an $\mathcal A$-module, H has a basis $\{T_w;w\in W\}$. The multiplication satisfies $(T_s+1)(T_s-v^2)=0$ if $s\in I$, $T_wT_{w'}=T_{ww'}$ if $l(ww')=l(w)+l(w')$. For any $J\subset I$ we define an $\mathcal A$-linear map $\mu _J:H{\rightarrow }H$ by

(a) $\mu _J(T_y)=T_{\delta ^{-1}(y_*)}T_{y^*}$

for any $y\in W$ where $y^*\in W^{{}^\delta J},y_*\in W_{{}^\delta J}$ are uniquely determined by $y=y^*y_*$.

Now let $w\in {}^{{}^\delta J}W$ and let $(J_n,w_n)_{n\ge 0}=\kappa (w)$. Let $n_0\ge 0$ be the smallest integer such that $J_{n_0}=J_{n_0+1}=\dots$. Let $H_w$ be the Hecke algebra defined in terms of $W_{{}^\delta (J^w_\infty )}$ in the same way as H was defined in terms of W. The automorphism $\tau _w:W_{{}^\delta (J^w_\infty )}{\rightarrow }W_{{}^\delta (J^w_\infty )}$ induces an algebra automorphism $H_w{\rightarrow }H_w$ ($T_y\mapsto T_{\tau _w(y)}$ for $y\in W_{{}^\delta (J^w_\infty )}$) denoted again by $\tau _w$.

We define a $\mathcal A$-linear function $e_w:H{\rightarrow }H_w$ by $e_w(T_y)=T_{y_1}$ if $y=w^{-1}y_1$, $y_1\in W_{{}^\delta (J^w_\infty )}$ and $e_w(T_y)=0$ for all other $y\in W$. For $n\ge n_0$ we define an $\mathcal A$-linear function $E_{w,J,n}:H{\rightarrow }H_w$ by

(b) $E_{w,J,n}(T_y)=e_w(\mu _{J_{n-1}}\dots \mu _{J_2}\mu _{J_1}\mu _{J_0}(T_y)),\quad (y\in W).$

(If $n=n_0=0$ we have $E_{w,J,n}(T_y)=e_w(T_y)$.) We show that for $y\in W$ and $n\ge n_0$ we have

(c) $E_{w,J,n+1}(T_y)=\tau _w(E_{w,J,n}(T_y)).$

It is enough to show that for $y\in W$ we have

(d) $e_w(\mu _{J^w_\infty }(T_y))=\tau _w(e_w(T_y)).$

We have $y=y'y''$ where $y'\in W^{{}^\delta (J^w_\infty )},y''\in W_{{}^\delta (J^w_\infty )}$ and

$$\begin{aligned} e_w(\mu _{J^w_\infty }(T_y))=e_w(T_{\delta ^{-1}(y'')}T_{y'}),\\ e_w(T_y)=\delta _{y',w^{-1}}T_{y''}. \end{aligned}$$

Now $T_{\delta ^{-1}(y'')}T_{y'}$ is an $\mathcal A$-linear combination of terms $T_{\tilde{y}y'}$ with $\tilde{y}\in W_{J^w_\infty }$. If $e_w(T_{\tilde{y}y'})\ne 0$, then $\tilde{y}y'\in w^{-1}W_{{}^\delta (J^w_\infty )}=W_{J^w_\infty }w^{-1}$ hence $y'=w^{-1}$. Thus, if $y'\ne w^{-1}$, both sides of (d) are zero. Now assume that $y'=w^{-1}$. We have

$$\begin{aligned} e_w(T_{\delta ^{-1}(y'')}T_{y'})=e_w(T_{\delta ^{-1}(y'')}T_{w^{-1}})=e_w(T_{w^{-1}}T_{\tau _w(y'')})=T_{\tau _w(y'')}=\tau _w(e_w(T_y)). \end{aligned}$$

This proves (d) hence also (c).

Proposition 3.9

Let $J\subset I$ and let $w\in {}^{{}^\delta J}W$; let $n\ge n_0$ ($n_0$ as in 3.8). Recall the isomorphism $\mathfrak {K}({\hat{L}}^{w1}){\mathop {\rightarrow }\limits ^{\sim }}{}^w\mathfrak {K}_J$, $\xi \mapsto \xi _w$ in 2.15. For $y\in W$ we have

(a) $\chi _w(\mathbf{K}^y_J)=v^{\phi (w,J)}\sum _{y'\in W_{{}^\delta (J^w_\infty )}}f_{y',n}\chi (\mathbf{K}^{y',{\hat{L}}^{w1}})_w$

where $\phi (w,J)\in \mathbf{Z}$ is explicitly computable and $f_{y',n}\in \mathcal A$ are given by

$$\begin{aligned} E_{w,J,n}(T_y)=\sum _{y'\in W_{{}^\delta (J^w_\infty )}}f_{y',n}T_{y'}\in H_w, \end{aligned}$$

see 3.8.

We argue by induction on $\sharp J$. Let $z=\min (W_{{}^\delta J}wW_J)$. Assume first that $z\in N_{J,\delta }$ so that $z=w$ and $n_0=0$. If $y\notin z^{-1}W_{{}^\delta J}$ then, as in the proof of 2.8 we have $\mathbf{K}^y|_{{}^wZ_J}=0$ hence $\chi _w(\mathbf{K}^y_J)=0$. From the definition, in this case we have $E_{z,J,0}(T_y)=0$. Using 3.8(c) we deduce that $E_{z,J,n}(T_y)=0$ for any $n\ge 0$ hence (a) holds in this case. Thus we can assume that $y=z^{-1}y'$ with $y'\in W_{{}^\delta J}$. From the definition, in this case we have $E_{z,J,0}(T_y)=T_{y'}$. Using 3.8(c) we deduce that for $n\ge 0$ we have $E_{z,J,n}(T_y)=\tau _w^n(T_{y'})=T_{\tau _w^n(y')}$. Using 3.7(a) we have $\chi _z(\mathbf{K}^{z^{-1}y'}_J)=v^{\phi (z,J)}\chi (\mathbf{K}^{y',{\hat{L}}^{z1}})_z$ where $\phi (z,J)=|{}^zZ_J|-|L^z|$. It remains to use that $\chi _z(\mathbf{K}^{z^{-1}y'}_J)=\chi _z(\mathbf{K}^{z^{-1}\tau _z^n(y')}_J)$ which follows from 2.7(c).

Next we assume that $z\notin N_{J,\delta }$. Then, setting $J_1=J\cap \delta ^{-1}(\mathrm{Ad}(z)J)$, we have $\sharp J_1<\sharp J$. We can write uniquely $y=y^*y_*$ as in 2.7. From 2.7(b) we see that

$$\begin{aligned} \chi _w(\mathbf{K}^y_J)=v^{\phi '(w,J)}\sum _{y'\in W_J}g_{y'}\chi _w(\mathbf{K}^{y'y^*}_{J_1}) \end{aligned}$$

where $T_{\delta ^{-1}(y_*)}T_{y^*}=\sum _{y'\in W_J}g_{y'}T_{y'y^*}$ in H (with $g_{y'}\in \mathcal A$) and $\phi '(w,J)\in \mathbf{Z}$. By the induction hypothesis we have

$$\begin{aligned} \chi _w(\mathbf{K}^{y'y^*}_{J_1})=v^{\phi (w,J_1)} \sum _{\tilde{y}'\in W_{{}^\delta (J^w_\infty )}}\tilde{f}_{\tilde{y}',n-1}\chi \left( \mathbf{K}^{\tilde{y}',{\hat{L}}^{w1}}\right) _w, \end{aligned}$$

where

$$\begin{aligned} E_{w,J_1,n-1}(T_{y'y^*})=\sum _{\tilde{y}'\in W_{{}^\delta (J^w_\infty )}}\tilde{f}_{\tilde{y}',n-1}T_{\tilde{y}'}\in H_w \end{aligned}$$

and $\phi (w,J_1)\in \mathbf{Z}$. It follows that

$$\begin{aligned} \chi _w(\mathbf{K}^y_J)=v^{\phi '(w,J)+\phi (w,J_1)}\sum _{y'\in W_J,\tilde{y}'\in W_{{}^\delta (J^w_\infty )}}g_{y'}\tilde{f}_{\tilde{y}',n-1} \chi \left( \mathbf{K}^{\tilde{y}',{\hat{L}}^{w1}}\right) _w. \end{aligned}$$

We have

$$\begin{aligned}&\sum _{y'\in W_J,\tilde{y}'\in W_{{}^\delta (J^w_\infty )}}g_{y'}\tilde{f}_{\tilde{y}',n-1}T_{\tilde{y}'}= \sum _{y'\in W_J}g_{y'}E_{w,J_1,n-1}(T_{y'y^*})\\= & {} \sum _{y'\in W_J}g_{y'}e_w(\mu _{J_{n-1}}\dots \mu _{J_2}\mu _{J_1}(T_{y'y^*}))\\= & {} e_w(\mu _{J_{n-1}}\dots \mu _{J_2}\mu _{J_1}(T_{\delta ^{-1}(y_*)}T_{y^*}))\\= & {} e_w(\mu _{J_{n-1}}\dots \mu _{J_2}\mu _{J_1}\mu _J(T_y))=E_{w,J,n}(T_y). \end{aligned}$$

Thus (a) holds with $\phi (w,J)=\phi '(w,J)+\phi (w,J_1)$.

Proposition 3.10

Let $J\subset I$ and let $\omega _J\in W_J$ be as in 3.2. Let $w\in {}^{{}^\delta J}W$.

• (a) If $z\in N_{J,\delta }$, then $\bar{\mathbf{K}}^{w^{-1}\delta (\omega _J)}_J|_{{}^zZ_J}$ is a direct sum of complexes of the form $\bar{\mathbf{Q}}_l\langle j\rangle$ with $j\in \mathbf{Z}$.

• (b) Let $C=\bar{\mathbf{Q}}_l\langle |L^w|\rangle \in \underline{CS}({\hat{L}}^{w1})$. Then for some $j\in \mathbf{Z}$, $C^\sharp _w\langle j\rangle$ is a direct summand of $\bar{\mathbf{K}}^{w^{-1}\delta (\omega _J)}_J$.

• (c) If C is as in (b) and z is as in (a), then $C^\sharp _w|_{{}^zZ_J}$ is a direct sum of complexes of the form $\bar{\mathbf{Q}}_l\langle j\rangle$ with $j\in \mathbf{Z}$.

We prove (a). Using 3.4(a) and 3.6(c) we see that it is enough to show that

$$\begin{aligned} \sum _{y'\in W;y'\le w^{-1}\delta (\omega _J)}P_{y',w^{-1}\omega _J}(v^2)\chi _z(\mathbf{K}^{y'}_J)\in \mathcal A\bar{\mathbf{Q}}_l\langle |{}^zZ_J|\rangle . \end{aligned}$$

Since $w^{-1}\in W^{{}^\delta J}$, the last sum is equal to

$$\begin{aligned} \sum _{y'\in W^{{}^\delta J};y'\le w^{-1}\delta (\omega _J)}P_{y',w^{-1}\omega _J}(v^2) \sum _{u\in W_{{}^\delta (J)}}\chi _z\left( \mathbf{K}^{y'u}_J\right) . \end{aligned}$$

(We use a standard property of the polynomials $P_{y',y}$.) Hence it is enough to show that

$$\begin{aligned} \sum _{u\in W_{{}^\delta (J)}}\chi _z(\mathbf{K}^{y'u}_J)\in \mathcal A\bar{\mathbf{Q}}_l\langle |{}^zZ_J|\rangle \end{aligned}$$

for any $y'\in W^{{}^\delta J}$. By arguments in 3.7, the left-hand side is zero unless $y'=z^{-1}$ in which case it equals

$$\begin{aligned} v^{|{}^zZ_J|-|L^z|}\sum _{u\in W_{{}^\delta (J)}}\left( \chi (\mathbf{K}^{u;{\hat{L}}^{z1}})\right) _z= v^{|{}^zZ_J|-|L^z|}\left( \chi \left( \bar{\mathbf{K}}^{\delta (\omega _J);{\hat{L}}^{z1}}\right) \right) _z. \end{aligned}$$

It remains to use that

$$\begin{aligned} \chi (\bar{\mathbf{K}}^{\delta (\omega _J);{\hat{L}}^{z1}})\in \mathcal A\bar{\mathbf{Q}}_l\langle |{\hat{L}}^{z1}|\rangle . \end{aligned}$$

We omit the proof of (b). Now (c) follows immediately from (a) and (b).

3.11. In this subsection we assume that $\hat{G}=G$ is simple adjoint, hence $G^1=G$ and $\delta =1$. We fix $J\subset I$ and we write $N_J$ instead of $N_{J,\delta }=\{w\in W;wJw^{-1}=J\}$, a subgroup of W with unit element e. We assume that $J\ne I$ and that we are given a cuspidal object $A_1$ of $\underline{CS}(L^e)$ (with $L^e$ as in 3.1). It follows that $L^e$ modulo its centre is simple or $\{1\}$. We will attach to each $w\in N_J$ a (not identically zero) map $\iota _w:\underline{CS}({\hat{L}}^{w1}){\rightarrow }\mathbf{Z}$ (said to be a cuspidal map) well defined up to multiplication by $\pm 1$. Let $F_w:L^e{\rightarrow }L^e$ be the Frobenius map for an $\mathbf{F}_q$-rational structure on $L^e$ whose action on the Weyl group $W_J$ of $L^e$ (or of $L^w$) is induced by the conjugation action of the connected component ${\hat{L}}^{w1}$ of ${\hat{L}}^w$ on $L^w$ (see 2.15). As in [15] we identify $\underline{CS}({\hat{L}}^{w1})$ with $\underline{cs}(L^e,F_w)$, a set of representatives for the isomorphism classes of unipotent representations of the finite reductive group $(L^e)^{F_w}$. Then $\iota _w$ becomes a map $\underline{cs}(L^e,F_w){\rightarrow }\mathbf{Z}$. Now $A_1$ gives rise to a cuspidal object $A_w$ of $\underline{cs}(L^w)$ and as in [15] this corresponds to a unipotent cuspidal representation $\pi _w$ of $L^w(\mathbf{F}_q)$ (with respect to a split Frobenius map). According to [10, 4.23], $\rho _w$ has an associated two-sided cell $\mathbf{c}$ of $W_J$ and it corresponds to a pair (x, r) where x is an element of a certain finite group $\Gamma$ attached to $\mathbf{c}$ and r is an irreducible representation of $Z_\Gamma (x)$. Moreover, $\mathbf{c}$ gives rise to a subset $\underline{CS}({\hat{L}}^{w1})_\mathbf{c}$ of $\underline{CS}({\hat{L}}^{w1})$ or equivalently to a subset $\underline{cs}(L^e,F_w)_\mathbf{c}$ of $\underline{cs}(L^e,F_w)$ in natural bijection [10, 4.23] with the set $\bar{M}=\bar{M}(\Gamma \subset \tilde{\Gamma })$ defined as follows: $\Gamma \subset \tilde{\Gamma }$ is a certain embedding of $\Gamma$ as a normal subgroup into a finite group $\tilde{\Gamma }$ such that $\tilde{\Gamma }/\Gamma$ is cyclic of order $\le 3$ with a given $\Gamma$-coset $\Gamma ^1$ whose image in $\tilde{\Gamma }/\Gamma$ generates $\tilde{\Gamma }/\Gamma$; $\bar{M}(\Gamma \subset \tilde{\Gamma })$ consists of all pairs (y, s) where $y\in \Gamma ^1$ is defined up to conjugation in $\tilde{\Gamma }$ and s is an irreducible representation of $Z_\Gamma (y)$, the centralizer of y in $\Gamma$, up to isomorphism. Our function $\iota _w$ is required to be zero on the complement of $\underline{cs}(L^e,F_w)_\mathbf{c}$, hence it can be viewed as a function $\iota _w:\bar{M}(\Gamma \subset \tilde{\Gamma }){\rightarrow }\mathbf{Z}$. Let $M(\tilde{\Gamma })$ be the set consisting of all pairs $(y',s')$ where $y'\in \tilde{\Gamma }$ is defined up to conjugation in $\tilde{\Gamma }$ and $s'$ is an irreducible representation of $Z_{\tilde{\Gamma }}(y')$, the centralizer of $y'$ in $\tilde{\Gamma }$, up to isomorphism. We can find an irreducible representation $r'$ of $Z_{\tilde{\Gamma }}(x)$ whose restriction to $Z_\Gamma (x)$ is r.

Let $\{,\}:M(\tilde{\Gamma })\times M(\tilde{\Gamma }){\rightarrow }\bar{\mathbf{Q}}_l$ be the pairing [10, 4.14.3]. We define $j_{r'}:\bar{M}(\Gamma \subset \tilde{\Gamma }){\rightarrow }\mathbf{Z}$ by $j_{r'}(y,s)=\{(x,r'),(y,s')\}$ where $s'$ is an irreducible representation of $Z_{\tilde{\Gamma }}(y)$ whose restriction to $Z_\Gamma (y)$ is s. Since $x\in \Gamma$, $j_{r'}(y,s)$ is independent of the choice of $s'$. We can choose $r'$ so that $j_{r'}(y,s)$ takes values in $\mathbf{Q}$. We define $\iota _w:\bar{M}(\Gamma \subset \tilde{\Gamma }){\rightarrow }\mathbf{Z}$ as $cj_{r'}$ where c is a rational number $>0$ such that $\iota _w$ takes values in $\mathbf{Z}$ and there is no integer $d>1$ dividing all values of $\iota _w$. In the case where $\tilde{\Gamma }/\Gamma$ has order 1 or 3, $\iota _w$ is unique. In the case where $\tilde{\Gamma }/\Gamma$ has order 2, $\iota _w$ is unique up to multiplication by $\pm 1$. We state some conjectures.

Conjecture 1. For any $t\le z$ in $N_J$ there is a (necessarily unique) $X_{t,z}\in \mathbf{Z}[v^{-1}]$ such that

$$\begin{aligned} \sum _{C\in \underline{CS}({\hat{L}}^{z1})_\mathbf{c}}\iota _z(C)\Phi _{t,z}^{C',C}=X_{t,z}\iota _t(C') \end{aligned}$$

for any $C'\in \underline{CS}({\hat{L}}^{t1})_\mathbf{c}$ where $\Phi _{t,z}^{C',C}\in \mathbf{N}[v^{-1}]$ are as in 3.6.

An equivalent statement is that in ${}^t\mathfrak {K}_J$ we have

$$\begin{aligned}\sum _{C\in \underline{CS}({\hat{L}}^{z1})_\mathbf{c}}\iota _z(C)\chi _t(C\sharp _z|_{{}^tZ_J})=X_{t,z}\sum _{C'\in \underline{CS}({\hat{L}}^{t1})_\mathbf{c}} \iota _t(C')C'_t\end{aligned}$$

modulo $\sum _{C'\in \underline{CS}({\hat{L}}^{t1})-\underline{CS}({\hat{L}}^{t1})_\mathbf{c}}\mathcal AC'_t$.

Conjecture 2. For any $t\le z$ in $N_J$, the matrix $(\Phi _{t,z}^{C',C})_{(C',C)\in \underline{CS}({\hat{L}}^{t1})_\mathbf{c}\times \underline{CS}({\hat{L}}^{z1})_\mathbf{c}}$ is square and invertible.

For any $i\in I-J$ we have $i\in N_J$. It follows that $\omega _{J\cup i}\omega _J=\omega _J\omega _{J\cup i}\in N_J$ hence $\tau _i=\omega _{J\cup i}\omega _J=\omega _J\omega _{J\cup i}$ has square 1. It is known that $N_J$ together with $\{\tau _i;i\in I-J\}$ is a Coxeter group. Let $a:W{\rightarrow }\mathbf{N}$ be the a-function of W, l, see [12, 13.6]. For $i\in I-J$ we set $c_i=a(x\tau _i)-a(x')$ where $x,x'\in \mathbf{c}$; this is independent of the choice of $x,x'$ by [12, 9.13, P11]. There is a unique weight function $\mathcal L:N_J{\rightarrow }\mathbf{N}$ such that $\mathcal L(\tau _i)=c_i$ for all $i\in I-J$. Hence the Hecke algebra $\mathcal H$ associated to $N_J,\mathcal L$ and the elements $p_{t,z}$ (for t, z in $N_J$) are well defined as in 0.1.

Conjecture 3. For any $t\le z$ in $N_J$ we have $X_{t,z}=e_{t,z}p_{t,z}$ where $e_{t,w}=\pm 1$.

5 An example

4.1. In this section we assume that we are in the setup of 3.1 and that $\hat{G}=G^1=G$ is simply connected; we also assume that W is of type $B_4$. We have $\delta =1$. We shall denote the elements of I as $s_i$ ($i=1,2,3,4$) where the notation is chosen so that $(s_1s_2)^4=(s_2s_3)^3=(s_3s_4)^3=1$ and $s_is_j=s_js_i$ if $i-j\in \{\pm 2,\pm 3\}$. An element $w\in W$ with reduced expression $s_{i_1}s_{i_2}\dots s_{i_m}$ will be denoted as $i_1i_2\dots i_m$. In particular we write i instead of $s_i$; the unit element of W is denoted by $\emptyset$. We set $J=\{1,2\}\subset I$. The elements of ${}^JW^J$ are

$$\begin{aligned}&\emptyset ,3,4,34,43,343,3243,32123,321234,321243,432123,\\&3212343,3432123,4321234,34321234,32123432123, 321234321234. \end{aligned}$$

Now $N_J:=N_{J,1}$ is the subgroup of W consisting of the elements

$$\begin{aligned}&\emptyset ,e=4,f=32123,fe=321234,ef=432123,\\&efe=4321234,fef=32123432123,efef=fefe=321234321234. \end{aligned}$$

It is a Coxeter group (of order 8) with generators e, f which satisfy $(ef)^4=1$. We define a weight function $\mathcal L:N_J{\rightarrow }\mathbf{N}$ by

$$\begin{aligned} \emptyset \mapsto 0,e\mapsto 1,f\mapsto 3,fe\mapsto 4,ef\mapsto 4,efe\mapsto 5,fef\mapsto 7,efef\mapsto 8 \end{aligned}$$

and a homomorphism $\epsilon :N_J\mapsto \{\pm 1\}$ by $\epsilon (e)=1$, $\epsilon (f)=-1$. Note that $\mathcal L$ coincides with the weight function defined in 3.11 in terms of $W, W_J$ and the two-sided cell $\mathbf{c}=\{1,2,12,21,121,212\}$ of $W_J$.

If $z\in N_J$, then the Weyl group of $L^z$ is $W_J=\{\emptyset ,1,2,12,21,121,212,1212\}$. In our case we have $L^z={\hat{L}}^{z1}$. Also $L^z$ is independent of z (in $N_J$) up to an inner automorphism; hence we can use the notation L instead of $L^z$.

4.2. The objects of $\underline{CS}(L)$ can be denoted by $1,\rho ,\sigma ,\sigma ',\theta ,S$. Here $1,\rho ,\sigma ,\sigma ',S$ are perverse sheaves on L with support equal to L which are generically local systems (up to shift) of rank 1, 2, 1, 1, 1; $\theta$ is a cuspidal character sheaf on L. They can be characterized by the equalities

$[\emptyset ]=1+2\rho +\sigma +\sigma '+S$;

$[1]=(1+v^2)(1+\rho +\sigma )$;

$[2]=(1+v^2)(1+\rho +\sigma ')$;

$[12]=v^2(\rho +\sigma +\sigma '+\theta )+(1+v^2)^21$;

$[21]=v^2(\rho +\sigma +\sigma '+\theta )+(1+v^2)^21$;

$[121]=(v^2+v^4)(\sigma '+\theta )+(1+v^2)(1+v^4)1$;

$[212]=(v^2+v^4)(\sigma +\theta )+(1+v^2)(1+v^4)1$;

$[1212]=(1+v^2)^2(1+v^4)1$.

(Recall from 3.7 that for $u\in W_J$ we have $[u]=v^{-|L|}\chi (\bar{\mathbf{K}}^{u;L})$.)

4.3. Let $z,t\in N_J$, $u\in W_J$. In our case the explicit values of $P_{t^{-1}u',z^{-1}u}(v^2)$ in 3.7(c) can be found in the tables of [4]; moreover in 3.7(c) we have $P'_{u',u''}(v^2)=(-1)^{l(u')+l(u'')}$. Hence the coefficients of $[z^{-1}u]_{(t)}$ in 3.7(c) are explicitly known. In subsections 4.4-4.10 we give for any z, t in $N_J$ the explicit values of $[z^{-1}u]_{(t)}$, with $u\in W_J-\{\emptyset ,1212\}$, as an $\mathcal A$-linear combination of elements $[u'']_t$ with $u''\in W_J$. For $\xi ,\xi '$ in ${}^t\mathfrak {K}_J$ we write $\xi \sim \xi '$ if $\xi -\xi '\in \mathcal A1_t$.

4.4. Assume that (t, z) satisfies either $t=z$ or that it is one of

$$\begin{aligned}&(\emptyset ,4), (32123,432123), (32123,321234),(32123,4321234),\\&(432123,4321234),(321234,4321234), (32123432123,321234321234) \end{aligned}$$

that is,

$$\begin{aligned} (\emptyset ,e),(f,ef),(f,fe),(f,efe),(ef,efe),(fe,efe),(fef,efef). \end{aligned}$$

Note that $l(z)-l(t)=\mathcal L(z)-\mathcal L(t)$, $\epsilon (z)\epsilon (t)=1$. From 3.7(c) we have

$[z^{-1}(121)]_{(t)}=[121]_t$,

$[z^{-1}(212)]_{(t)}=[212]_t$,

$[z^{-1}(12)]_{(t)}=[12]_t$,

$[z^{-1}(21)]_{(t)}=[21]_t$,

$[z^{-1}(2)]_{(t)}=[2]_t$,

$[z^{-1}(1)]_{(t)}=[1]_t$.

Hence, using 4.2, we have

$[z^{-1}(121)]_{(t)}\sim (v^2+v^4)(\theta _t+\sigma '_t)$,

$[z^{-1}(212)]_{(t)}\sim (v^2+v^4)(\theta _t+\sigma _t)$,

$[z^{-1}(12)]_{(t)}\sim v^2(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(21)]_{(t)}\sim v^2(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(2)]_{(t)}\sim (1+v^2)(\rho _t+\sigma '_t)$,

$[z^{-1}(1)]_{(t)}\sim (1+v^2)(\rho _t+\sigma _t)$.

4.5. Assume that (t, z) is one of

$$\begin{aligned}&(\emptyset ,32123),(\emptyset ,432123), (4,432123),(\emptyset ,321234),(4,321234),\\&(32123,321234321234),(432123,321234321234),(321234,321234321234),\\&(4321234,321234321234), (432123,32123432123), (321234,32123432123) \end{aligned}$$

that is, one of

$$\begin{aligned}&(\emptyset ,f),(\emptyset ,ef),(e,ef), (\emptyset ,fe),(e,fe),\\&(f,efef), (ef,efef), (fe,efef), (efe,efef), (ef, fef), (fe,fef). \end{aligned}$$

Note that $l(z)-l(t)=\mathcal L(z)-\mathcal L(t)+2$, $\epsilon (z)\epsilon (t)=-1$. From 3.7(c) we have

$[z^{-1}(121)]_{(t)}=v^2[121]_t+[1212]_t$,

$[z^{-1}(212)]_{(t)}=v^4[2]_t+[1212]_t$,

$[z^{-1}(12)]_{(t)}=v^2[12]_t+[1212]_t$,

$[z^{-1}(21)]_{(t)}=v^2[21]_t+[1212]_t$,

$[z^{-1}(2)]_{(t)}=[212]_t$,

$[z^{-1}(1)]_{(t)}=v^2[1]_t+[1212]_t$.

Hence, using 4.2, we have

$[z^{-1}(121)]_{(t)}\sim (v^4+v^6)(\theta _t+\sigma '_t)$,

$[z^{-1}(212)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma '_t)$,

$[z^{-1}(12)]_{(t)}\sim v^4(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(21)]_{(t)}\sim v^4(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(2)]_{(t)}\sim (v^2+v^4)(\theta _t+\sigma _t)$,

$[z^{-1}(1)]_{(t)}\sim (v^2+v^4)(\rho _t+\sigma _t)$.

4.6. Assume that (t, z) is one of $(\emptyset ,4321234), (4,4321234)$ that is, one of $(\emptyset ,efe),(e,efe)$.

Note that $l(z)-l(t)=\mathcal L(z)-\mathcal L(t)+2=-1$, $\epsilon (z)\epsilon (t)=-1$. From 3.7(c) we have

$[z^{-1}(121)]_{(t)}=(v^2+v^4)[121]_t+(1+v^2)[1212]_t$,

$[z^{-1}(212)]_{(t)}=(v^4+v^6)[2]_t+(1+v^2)[1212]_t$,

$[z^{-1}(12)]_{(t)}=(v^2+v^4)[12]_t+(1+v^2)[1212]_t$,

$[z^{-1}(21)]_{(t)}=(v^2+v^4)[21]_t+(1+v^2)[1212]_t$,

$[z^{-1}(2)]_{(t)}=(1+v^2)[212]_t$,

$[z^{-1}(1)]_{(t)}=(v^2+v^4)[1]_t+(1+v^2)[1212]_t$.

Hence, using 4.2, we have

$[z^{-1}(121)]_{(t)}\sim (v^2+v^4)^2(\theta _t+\sigma '_t)$,

$[z^{-1}(212)]_{(t)}\sim (v^4+v^6)(1+v^2)(\rho _t+\sigma '_t)$,

$[z^{-1}(12)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(21)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(2)]_{(t)}\sim (1+v^2)(v^2+v^4)(\theta _t+\sigma _t)$,

$[z^{-1}(1)]_{(t)}\sim (1+v^2)(v^2+v^4)(\rho _t+\sigma _t)$.

4.7. Assume that (t, z) is one of

$$\begin{aligned} (\emptyset ,321234321234), (4,321234321234),(4,32123432123) \end{aligned}$$

that is, one of $(\emptyset ,efef),(e,efef),(e,fef)$.

Note that $l(z)-l(t)=\mathcal L(z)-\mathcal L(t)+4$, $\epsilon (z)\epsilon (t)=1$. From 3.7(c) we have

$[z^{-1}(121)]_{(t)}=v^4[121]_t$,

$[z^{-1}(212)]_{(t)}=v^4[212]_t$,

$[z^{-1}(12)]_{(t)}=v^4[12]_t$,

$[z^{-1}(21)]_{(t)}=v^4[21]_t$,

$[z^{-1}(2)]_{(t)}=v^4[2]_t$,

$[z^{-1}(1)]_{(t)}=v^4[1]_t$.

Hence, using 4.2, we have

$[z^{-1}(121)]_{(t)}\sim (v^6+v^8)(\theta _t+\sigma '_t)$,

$[z^{-1}(212)]_{(t)}\sim (v^6+v^8)(\theta _t+\sigma _t)$,

$[z^{-1}(12)]_{(t)}\sim v^6(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(21)]_{(t)}\sim v^6(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(2)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma '_t)$,

$[z^{-1}(1)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma _t)$.

4.8. Assume that $(t,z)=(\emptyset ,32123432123)$ that is, $(\emptyset ,fef)$.

Note that $l(z)-l(t)=\mathcal L(z)-\mathcal L(t)+4$, $\epsilon (z)\epsilon (t)=1$. From 3.7(c) we have

$[z^{-1}(121)]_{(t)}=v^8[1]_t+v^4[121]_t$,

$[z^{-1}(212)]_{(t)}=(v^4+v^6)[212]_t$,

$[z^{-1}(12)]_{(t)}=(v^4+v^6)[12]_t$,

$[z^{-1}(21)]_{(t)}=(v^4+v^6)[21]_t$,

$[z^{-1}(2)]_{(t)}=(v^4+v^6)[2]_t$,

$[z^{-1}(1)]_{(t)}=v^4[121]_t+v^4[1]_t$.

Hence, using 4.2, we have

$[z^{-1}(121)]_{(t)}\sim (v^8+v^{10})(\rho _t+\sigma _t)+(v^6+v^8)(\theta _t+\sigma '_t)$,

$[z^{-1}(212)]_{(t)}\sim (v^4+v^6)(v^2+v^4)(\theta _t+\sigma _t)$,

$[z^{-1}(12)]_{(t)}\sim (v^6+v^8)(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(21)]_{(t)}\sim (v^6+v^8)(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(2)]_{(t)}\sim (v^4+v^6)(1+v^2)(\rho _t+\sigma '_t)$,

$[z^{-1}(1)]_{(t)}\sim (v^6+v^8)(\theta _t+\sigma '_t)+(v^4+v^6)(\rho _t+\sigma _t)$.

4.9. Assume that $(t,z)=(32123,32123432123)$ that is, (f, fef).

Note that $l(z)-l(t)=\mathcal L(z)-\mathcal L(t)+2$, $\epsilon (z)\epsilon (t)=-1$. From 3.7(c) we have

$[z^{-1}(121)]_{(t)}=v^6[1]_t+v^2[121]_t$,

$[z^{-1}(212)]_{(t)}=(v^4+v^6)[2]_t$,

$[z^{-1}(12)]_{(t)}=(v^2+v^4)[12]_t$,

$[z^{-1}(21)]_{(t)}=(v^2+v^4)[21]_t$,

$[z^{-1}(2)]_{(t)}=(1+v^2)[212]_t$,

$[z^{-1}(1)]_{(t)}=v^2[121]_t+v^2[1]_t$.

Hence, using 4.2, we have

$[z^{-1}(121)]_{(t)}\sim (v^6+v^8)(\rho _t+\sigma _t)+(v^4+v^6)(\theta _t+\sigma '_t)$,

$[z^{-1}(212)]_{(t)}\sim (v^4+v^6)(1+v^2)(\rho _t+\sigma '_t)$,

$[z^{-1}(12)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(21)]_{(t)}\sim (v^4+v^6)(\rho _t+\sigma _t+\sigma '_t+\theta _t)$,

$[z^{-1}(2)]_{(t)}\sim (1+v^2)(v^2+v^4)(\theta _t+\sigma _t)$,

$[z^{-1}(1)]_{(t)}\sim (v^4+v^6)(\theta _t+\sigma '_t)+(v^2+v^4)(\rho _t+\sigma _t)$.

4.10. If (t, z) in $N_J$ is not as in 4.4-4.9, then we have $[z^{-1}u]_{(t)}=0$ for any $u\in W_J$.

4.11. Let $z,t\in N_J$, $u\in W_J$. We set

$$\begin{aligned}{}[z^{-1}u]'_{(t)}=\chi _t(\bar{\mathbf{K}}^{z^{-1}u}_J\langle |\bar{\mathcal B}^2_{J,z^{-1}u}|\rangle ). \end{aligned}$$

Using 2.10(d) we have

(a) $[z^{-1}u]'_{(t)}=v^{-|\bar{\mathcal B}^2_{J,z^{-1}u}|+|{}^tZ_J|}[z^{-1}u]_{(t)}=v^{-l(z)+l(t)-l(u)}[z^{-1}u]_{(t)}.$

Let $C\in \underline{CS}(L)-\{1\}$, let $t,z\in N_J$ and let $u\in W_J-\{\emptyset ,1212\}$. From 4.4-4.10 we see that the following result holds:

(b) The coefficient of $C_t$ in $[z^{-1}u]'_{(t)}$ is in $\mathbf{N}[v^{-1}]$. More precisely, this coefficient is:

• (i) 0 or $1+v^{-2}$ if (t, z) is as in 4.4 with $t\ne z$, $u\in \{1,2,121,212\}$;

• (ii) in $v^{-1}\mathbf{N}[v^{-1}]$ if t, z, u are not as in (i) but $t\ne z$.

4.12. Let $z,t\in N_J$, $u\in W_J$. Using 4.11(a) and 3.5(e) with $y=z^{-1}u$ we deduce

$$\begin{aligned} v^{-l(z)+l(t)-l(u)}[z^{-1}u]_{(t)}=\sum _{C'\in \underline{CS}(L)}N^{z,t,u}_{C'}C'_t \end{aligned}$$

where

$$\begin{aligned} N^{z,t,u}_{C'}=\sum _{w\in {}^JW,C\in \underline{CS}(L),j\in \mathbf{Z},j'\in \mathbf{N}} \dim V_{z^{-1}u,w,C,j}\dim {}'V_{t,w,j'}^{C',C})v^{-j-j'}. \end{aligned}$$

If $z=t$, in the previous sum we must have $w=z$. Note that $\dim {}'V_{z,z,j'}^{C',C}=0$ unless $C=C'$, $j'=0$ and we have

(a) $N^{z,z,u}_{C'}=\sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,z,C',j}v^{-j}.$ We show:

Proposition 4.13

Let $z,t\in N_J$. Assume that $z\ne t$, $j\in \mathbf{Z}$, $u\in W_J-\{\emptyset ,1212\}$ and $C'\in \underline{CS}(L)-\{1\}$. Then $V_{z^{-1}u,t,C',j}=0$ (notation of 3.5).

We first note that if t, z, u are as in 4.11(i), then from the definitions we have

• (a) ${}'V_{t,z,1}^{C',C'}={}'V_{z,z,0}^{C',C'}$.

For $\xi ,\xi '$ in $\mathcal A$ we write $\xi \ge \xi '$ if $\xi -\xi '\in \mathbf{N}[v,v^{-1}]$. Note that $N^{z,t,u}_{C'}$ is $\ge$ than the corresponding sum in which $(w,C,j')$ is restricted

to $(t,C',0)$ (if t, z, u are as in 4.11(ii));

to $(t,C',0)$ or to $(z,C',1)$ (if t, z, u are as in 4.11(i)).

Thus

$$\begin{aligned} N^{z,t,u}_{C'}\ge \sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}\dim {}'V_{t,t,0}^{C',C'}v^{-j}= \sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}v^{-j} \end{aligned}$$

(if t, z, u are as in 4.11(ii)) and

$$\begin{aligned}&N^{z,t,u}_{C'}\ge \sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}\dim {}'V_{t,t,0}^{C',C'}v^{-j}\\&\qquad + \sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,z,C',j}\dim {}'V_{t,z,1}^{C',C'}v^{-j-1} =\sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}\dim {}'V_{t,t,0}^{C',C'}v^{-j}\\&\qquad + \sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,z,C',j}\dim {}'V_{z,z,0}^{C',C'}v^{-j-1} =\sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}v^{-j}+v^{-1}N^{z,z,u}_{C'} \end{aligned}$$

(if t, z, u are as in 4.11(i)). (We have used (a) and 4.12(a).) If t, z, u are as in 4.11(i)), we have $N^{z,t,u}_{C'}=v^{-1}N^{z,z,u}_{C'}$ (see 4.4) and we deduce that

$$\begin{aligned} 0\ge \sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}v^{-j}; \end{aligned}$$

hence $V_{z^{-1}u,t,C',j}=0$ for all j. If t, z, u are as in 4.11(ii)), we see using 3.5(b) that the sum $\sum _{j\in \mathbf{Z}}\dim V_{z^{-1}u,t,C',j}v^{-j}$ is either zero, or

for some $j, v^j$ and $v^{-j}$ both appear in it with $>0$ coefficient.

In the last case it follows that $v^j$ and $v^{-j}$ both appear in $N^{z,t,u}_{C'}$ with $>0$ coefficient. This is not compatible with the inclusion $N^{z,t,u}_{C'}\in v^{-1}\mathbf{N}[v^{-1}]$. Thus we must have $\sum _{j\in \mathbf{Z}}\dim (V_{z^{-1}u,t,C',j})v^{-j}=0$ hence $V_{z^{-1}u,t,C',j}=0$ for all j. The proposition is proved.

Proposition 4.14

Let $z\in N_J$, $u\in W_J-\{\emptyset ,1212\}$, $j\in \mathbf{Z}$. Let $t\in {}^JW$ be such that $\sharp J^t_\infty <\sharp J$. Assume that $C\in \underline{CS}({\hat{L}}^{t1})$ is not isomorphic to $\bar{\mathbf{Q}}_l\langle |L^t|\rangle$. Then $V_{z^{-1}u,t,C,j}=0$ (notation of 3.5(a)).

The existence of C guarantees that $L^t$ is not a torus. Thus $J^t_\infty$ consists of a single element i (equal to 1 or 2) and $L^t$ has semisimple rank 1. Hence C is uniquely determined and it appears with coefficient 1 in $v^{-|L^t|}\chi (\mathbf{K}^{\emptyset ,L^t})$ and with coefficient $-1$ in $v^{-|L^t|}\chi (\mathbf{K}^{i,L^t})$. Hence the coefficient of $C_t$ in $v^{-|\bar{\mathcal B}^2_{J,y}|}\chi _t(\mathbf{K}^y_J)$ is explicitly computable from 3.9(a) for any $y\in W$. Using now 3.6(c) (in which the polynomials $P_{y',y}(v^2)$ are explicitly known from [4]) we see that the coefficient of $C_t$ in $v^{-|\bar{\mathcal B}^2_{J,y}|}\chi _t(\bar{\mathbf{K}}^y_J)$ is explicitly computable for any $y\in W$. In particular, the coefficient of $C_t$ in $v^{-|\bar{\mathcal B}^2_{J,z^{-1}u}|}\chi _t(\bar{\mathbf{K}}^{z^{-1}u}_J)$ is explicitly computable. We find that this coefficient is in $v^{-1}\mathbf{Z}[v^{-1}]$. On the other hand this coefficient is equal to $\sum _{j\in \mathbf{Z}}\dim (V_{z^{-1}u,t,C,j})v^{-j}$ which is invariant under the involution $v\mapsto v^{-1}$ of $\mathcal A$. This forces $\dim (V_{z^{-1}u,t,C,j})$ to be zero for any j. The proposition is proved.

Proposition 4.15

Let $z\in N_J$, $u\in W_J-\{\emptyset ,1212\}$.

• (a) We have

  $$\begin{aligned} \bar{\mathbf{K}}^{z^{-1}u}_J\langle |\bar{\mathcal B}^2_{J,z^{-1}u}|\rangle \cong \oplus _{C\in \underline{CS}(L)-\{1\},j\in \mathbf{Z}}V_{z^{-1}u,z,C,j}\otimes C_z^\sharp \langle j\rangle \oplus K' \end{aligned}$$

  where

  $$\begin{aligned} K'\cong \oplus _{w\in {}^JW,j\in \mathbf{Z}}\tilde{V}_{w,j}\otimes (\bar{\mathbf{Q}}_l\langle |L^w|\rangle )_w^\sharp \langle j \rangle \end{aligned}$$

  and $\tilde{V}_{w,j}$ are certain $\bar{\mathbf{Q}}_l$-vector spaces.

• (b) For any $t\in N_J$ we have (with $\sim$ as in 4.3):

  $$\begin{aligned}{}[z^{-1}u]'_{(t)}\sim \sum _{C\in \underline{CS}(L)-\{1\},j\in \mathbf{Z}}\dim (V_{z^{-1}u,z,C,j})v^{-j}\chi _t(C_z^\sharp ).\end{aligned}$$

(a) follows from 4.13, 4.14; (b) follows from (a) using 3.10(a).

4.16. In the setup of 4.15(b), the integers $\dim (V_{z^{-1}u,z,C,j})$ (with $C\ne 1$) can be obtained from 4.4 (with $t=z$). Thus we can rewrite 4.15(b) as follows (recall that $z,t\in N_J$; we set $\zeta =v^{-l(z)+l(t)}$ and we take $u\in \{1,2,121,212\}$):

$$\begin{aligned} \zeta v^{-3}[z^{-1}121]_t\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}[z^{-1}212]_t\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}[z^{-1}2]_t\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}[z^{-1}1]_t\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

Using now the formulas in 4.4–4.10 we deduce that the following hold.

If (t, z) are as in 4.4, then

$$\begin{aligned} \zeta v^{-3}(v^2+v^4)(\theta _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}(v^2+v^4)(\theta _t+\sigma _t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}(1+v^2)(\rho _t+\sigma '_t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}(1+v^2)(\rho _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

If (t, z) are as in 4.5, then

$$\begin{aligned} \zeta v^{-3}(v^4+v^6)(\theta _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}(v^4+v^6)(\rho _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}(v^2+v^4)(\theta _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}(v^2+v^4)(\rho _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

If (t, z) are as in 4.6, then

$$\begin{aligned} \zeta v^{-3}(v^2+v^4)^2(\theta _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}(v^4+v^6)(1+v^2)(\rho _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}(1+v^2)(v^2+v^4)(\theta _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}(1+v^2)(v^2+v^4)(\rho _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

If (t, z) are as in 4.7, then

$$\begin{aligned} \zeta v^{-3}(v^6+v^8)(\theta _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}(v^6+v^8)(\theta _t+\sigma _t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}(v^4+v^6)(\rho _t+\sigma '_t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}(v^4+v^6)(\rho _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

If (t, z) are as in 4.8, then

$$\begin{aligned} \zeta v^{-3}((v^8+v^{10})(\rho _t+\sigma _t)+(v^6+v^8)(\theta _t+\sigma '_t))\sim v^{-3}(v^2+v^4) (\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}(v^4+v^6)(v^2+v^4)(\theta _t+\sigma _t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}(v^4+v^6)(1+v^2)(\rho _t+\sigma '_t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}((v^6+v^8)(\theta _t+\sigma '_t)+(v^4+v^6)(\rho _t+\sigma _t))\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

If (t, z) are as in 4.9, then

$$\begin{aligned} \zeta v^{-3}((v^6+v^8)(\rho _t+\sigma _t)+(v^4+v^6)(\theta _t+\sigma '_t))\sim v^{-3}(v^2+v^4) (\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-3}(v^4+v^6)(1+v^2)(\rho _t+\sigma '_t)\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ \zeta v^{-1}(1+v^2)(v^2+v^4)(\theta _t+\sigma _t)\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ \zeta v^{-1}((v^4+v^6)(\theta _t+\sigma '_t)+(v^2+v^4)(\rho _t+\sigma _t))\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

If (t, z) are as in 4.10, then

$$\begin{aligned} 0\sim v^{-3}(v^2+v^4) (\chi _t(\theta _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ 0\sim v^{-3}(v^2+v^4)(\chi _t(\theta _z^\sharp )+\chi _t(\sigma _z^\sharp )),\\ 0\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma '{}_z^\sharp )),\\ 0\sim v^{-1}(1+v^2)(\chi _t(\rho _z^\sharp )+\chi _t(\sigma _z^\sharp )). \end{aligned}$$

4.17. Let $z,t\in N_J$. Let ${}^t\mathfrak {K}_J^+=\sum _{C'\in \underline{CS}(L)}\mathbf{N}[v,v^{-1}]C'_t.$ From 3.5(d) we have (for $C\in \underline{CS}(L)$):

$$\begin{aligned} \chi _t(C_z^\sharp )=\sum _{C'\in \underline{CS}(L),j'\in \mathbf{Z}}\dim {}'V_{t,z,j'}^{C',C}v^{j'}C'_t, \end{aligned}$$

hence

(a) $\chi _t(C_z^\sharp )\in {}^t\mathfrak {K}_J^+.$

Using (a) we can extract from the formulas in 4.16 the following facts about $\chi _t(C_z^\sharp )$. (As in 4.16 we set $\zeta =v^{-l(z)+l(t)}$).

If (t, z) are as in 4.4, then

$$\begin{aligned} \chi _t(\rho _z^\sharp )\sim \zeta \rho _t,\quad \chi _t(\sigma _z^\sharp )\sim \zeta \sigma _t,\quad \chi _t(\sigma '{}_z^\sharp )\sim \zeta \sigma '_t,\quad \chi _t(\theta _z^\sharp )\sim \zeta \theta _t. \end{aligned}$$

If (t, z) are as in 4.5, then

$$\begin{aligned} \chi _t(\rho _z^\sharp )\sim \zeta v^2\sigma _t,\quad \chi _t(\sigma _z^\sharp )\sim \zeta v^2\rho _t,\quad \chi _t(s'{}_z^\sharp )\sim \zeta v^2\theta _t,\quad \chi _t(\theta _z^\sharp )\sim \zeta v^2\sigma '_t. \end{aligned}$$

If (t, z) are as in 4.6, then

$$\begin{aligned}&\chi _t(\rho _z^\sharp )\sim \zeta (v^2+v^4)\sigma _t,\quad \chi _t(\sigma _z^\sharp )\sim \zeta (v^2+v^4)\rho _t,\quad \\&\chi _t(\sigma '{}_z^\sharp )\sim \zeta (v^2+v^4)\theta _t,\quad \chi _t(\theta _z^\sharp )\sim \zeta (v^2+v^4)\sigma '_t. \end{aligned}$$

If (t, z) are as in 4.7, then

$$\begin{aligned} \chi _t(\rho _z^\sharp )\sim \zeta v^4\rho _t,\quad \chi _t(\sigma _z^\sharp )\sim \zeta v^4\sigma _t,\quad \chi _t(\sigma '{}_z^\sharp )\sim \zeta v^4\sigma '_t, \chi _t(\theta _z^\sharp )\sim \zeta v^4\theta _t. \end{aligned}$$

If (t, z) are as in 4.8, then

$$\begin{aligned}&\chi _t(\rho _z^\sharp )\sim \zeta (v^4\rho _t+v^6\sigma '_t),\quad \chi _t(\sigma _z^\sharp )\sim \zeta (v^4\sigma _t+v^6\theta _t),\quad \\&\chi _t(\sigma '{}_z^\sharp )\sim \zeta (v^4\sigma '_t+v^6\rho _t),\quad \chi _t(\theta _z^\sharp )\sim \zeta (v^4\theta _t+v^6\sigma _t). \end{aligned}$$

If (t, z) are as in 4.9, then

$$\begin{aligned}&\chi _t(\rho _z^\sharp )\sim \zeta (v^2\sigma _t+v^4\theta _t),\quad \chi _t(\sigma _z^\sharp )\sim \zeta (v^2\rho _t+v^4\sigma '_t),\quad \\&\chi _t(\sigma '{}_z^\sharp )\sim \zeta (v^2\theta _t+v^4\sigma _t),\quad \chi _t(\theta _z^\sharp )\sim \zeta (v^2\sigma '_t+v^4\rho _t). \end{aligned}$$

If (t, z) are as in 4.10, then

$$\begin{aligned} \chi _t(\rho _z^\sharp )\sim 0,\quad \chi _t(\sigma _z^\sharp )\sim 0,\quad \chi _t(\sigma '{}_z^\sharp )\sim 0,\quad \chi _t(\theta _z^\sharp )\sim 0. \end{aligned}$$

4.18. Let $z,t\in N_J$. Using the results in 4.17 we see that

(a) $\chi _t(\rho _z^\sharp )-\chi _t(\sigma _z^\sharp )-\chi _t(\sigma '{}_z^\sharp )+\chi _t(\theta _z^\sharp )\sim X_{t,z}(\rho _t-\sigma _t-\sigma '_t+\theta _t),$

where $X_{t,z}\in \mathcal A$ is as follows:

$X_{t,z}=v^{-l(z)+l(t)}$ if (t, z) are as in 4.4;

$X_{t,z}=-v^{-l(z)+l(t)} v^2$ if (t, z) are as in 4.5;

$X_{t,z}=-v^{-l(z)+l(t)} (v^2+v^4)$ if (t, z) are as in 4.6;

$X_{t,z}=v^{-l(z)+l(t)} v^4$ if (t, z) are as in 4.7;

$X_{t,z}=v^{-l(z)+l(t)} (v^4-v^6)$ if (t, z) are as in 4.8;

$X_{t,z}=-v^{-l(z)+l(t)}(v^2-v^4)$ if (t, z) are as in 4.9;

$X_{t,z}=0$ if (t, z) are as in 4.10.

It follows that

$X_{t,z}=\epsilon (z)\epsilon (t)v^{-\mathcal L(z)+\mathcal L(t)}$ if (t, z) are as in 4.4, 4.5 or 4.7;

$X_{t,z}=\epsilon (z)\epsilon (t)v^{-\mathcal L(z)+\mathcal L(t)} (1+v^2)$ if (t, z) are as in 4.6;

$X_{t,z}=\epsilon (z)\epsilon (t)v^{-\mathcal L(z)+\mathcal L(t)} (1-v^2)$ if (t, z) are as in 4.8 or 4.9;

$X_{t,z}=0$ if (t, z) are as in 4.10.

4.19. Define $\mathcal H,p_{t,z}$ as in 0.1 in terms of $\mathcal W=N_J$, $\mathcal L$. According to [12, 7.6] we have:

• (i) $p_{t,z}=v^{-\mathcal L(z)+\mathcal L(t)} (1+v^2)$ if $z=efe$ and $t\in \{\emptyset ,e\}$;

• (ii) $p_{t,z}=v^{-\mathcal L(z)+\mathcal L(t)} (1-v^2)$ if $z=fef$ and $t\in \{\emptyset ,f\}$;

• (iii) $p_{t,z}=v^{-\mathcal L(z)+\mathcal L(t)}$ if $t\le z$ in the usual partial order of $N_J$ with (t, z) not as in (i),(ii);

• (iv) $p_{t,z}=0$ if .

We can now restate the result in 4.18 as follows.

$$\begin{aligned} \chi _t(\rho _z^\sharp )-\chi _t(\sigma _z^\sharp )-\chi _t(\sigma '{}_z^\sharp )+\chi _t(\theta _z^\sharp )\sim p_{t,z}\epsilon (z)\epsilon (t)(\rho _t-\sigma _t-\sigma '_t+\theta _t). \end{aligned}$$

We see that Conjectures 1,2,3 in 3.11 hold in our case.