2.1 Postnikov diagrams in triangulated categories

Let ${\mathcal {D}}$ be a triangulated category. We will need the following lemma:

Lemma 2.1. Consider a diagram in ${\mathcal {D}}$

where $(u,v,w)$ and $(u',v',w')$ are distinguished triangles. Suppose that

$$ \begin{align*}\mathrm{Hom}(B,A')=0,\hspace{.5cm}\text{ and }\hspace{.5cm}\mathrm{Hom}(A,B'[-1])=0. \end{align*} $$

Then there exists a unique morphism $C\rightarrow C'$ which extends the above diagram into a morphism of distinguished triangles.

Proof. Only the unicity needs to be proved (the existence follows from the axioms of a triangulated category). We are thus reduced to prove that if we have a morphism of triangles,

then $\beta =0$ .

As $\beta \circ u=0$ , there exists $\gamma :A\rightarrow C'$ such that $\gamma \circ v=\beta $ . Since

$$ \begin{align*}v'\circ\gamma\circ v=0 \end{align*} $$

there exists $\varepsilon :B\rightarrow A'$ such that $v'\circ \gamma =\varepsilon \circ w$ . By assumption, $\varepsilon =0$ and so $v'\circ \gamma =0$ .

There exists thus $\overline {\gamma }:A\rightarrow B'[-1]$ such that $u'\circ \overline {\gamma }=\gamma $ . By assumption, we must have $\overline {\gamma }=0$ , and so $\gamma =0$ from which we get that $\beta =0$ .

A Postnikov diagram $\Lambda $ [Reference Orlov16] consists of a complex

in ${\mathcal {D}}$ , called the base and denoted by $\Lambda _b$ , together with a finite sequence of distinguished triangles

with $i=1,\dots ,m$ , such that $\partial _i=\alpha _{i-1}\circ d_i$ for all i. We visualize $\Lambda $ as

(2.1)

The length of $\Lambda $ is the integer m.

The object $C_m$ will be called the outcome of the Postnikov diagram $\Lambda $ .

Notice that if ${\mathcal {D}}$ is equipped with a nondegenerate t-structure, then we can define the Postnikov diagram $\Lambda (K)$ with outcome K for any complex $K\in {\mathcal {D}}^{[n,n+m]}$ as

(2.2)

using the distinguished triangles

The construction of $\Lambda (K)$ is functorial in K.

Remark 2.2. Given $K\in {\mathcal {D}}^{[n,n+m]}$ , then $\Lambda (K)$ is the unique (up to a unique isomorphism) Postnikov diagram of the form (2.1) with outcome K such that

(2.3) $$ \begin{align} C_i\in{\mathcal{D}}^{\leq n-m+2i}\hspace{1cm}\text{and}\hspace{1cm} A_i\in {\mathcal{D}}^{\geq n-m+2i} \end{align} $$

for all $i=1,\dots ,m$ . This follows from [Reference Beilinson, Bernstein and Deligne2, Proposition 1.3.3(ii)].

Lemma 2.3. Given $K\in {\mathcal {D}}^{[n,n+m]}$ , the Postnikov diagram $\Lambda (K)$ is the unique one (up a unique isomorphism) that completes the subdiagram

Lemma 2.4. Assume that we have a Postnikov diagram with the notation as in diagram (2.1) and that for each $i=0,\dots ,m-2$ , any $j\geq i+2$ and any $k\geq 1$ we have

$$ \begin{align*}\mathrm{Hom}(A_j,A_i[-k])=0. \end{align*} $$

Then if $(d^{\prime }_m,d^{\prime }_{m-1},\dots ,d^{\prime }_1)$ is a sequence of arrows $d^{\prime }_i:A_i\rightarrow C_{i-1}$ such that $\partial _i=\alpha _{i-1}\circ d^{\prime }_i$ for all i, then

$$ \begin{align*}(d^{\prime}_m,\dots,d^{\prime}_1)=(d_m,\dots,d_1). \end{align*} $$

Proof. We need to prove that

$$ \begin{align*}\mathrm{Hom}(A_{i+2},C_i[-1])=0 \end{align*} $$

for all $i=0,\dots ,m-2$ .

Since we have a distinguished triangle

we have an exact sequence

By assumption, the right-hand side is $0$ , and so we are reduced to prove that

$$ \begin{align*}\mathrm{Hom}(A_{i+2},C_{i-1}[-2])=0. \end{align*} $$

Repeating the argument as many times as needed, we end up proving that $\mathrm {Hom}(A_{i+2},C_0[-1-i])=0$ , which follows also from the assumption as $C_0=A_0$ .

Proposition 2.5. Assume given a complex $\Lambda _b$ in ${\mathcal {D}}$

(2.4)

and suppose that it satisfies

(2.5) $$ \begin{align} \mathrm{Hom}(A_j,A_i)=0,\hspace{.5cm}\text{ for all } j<i, \end{align} $$

and

(2.6) $$ \begin{align} \mathrm{Hom}(A_j,A_i[-k])=0,\hspace{.5cm}\text{ for all }j>i\text{ and } k\geq 1. \end{align} $$

Then the complex (2.4) can be completed in a unique way (up to a unique isomorphism) into a Postnikov diagram $\Lambda $ in ${\mathcal {D}}$ .

Proof. The proof goes by induction on $0\leq r <m$ . If $r=0$ , then this is obvious. Suppose that we have proved the proposition up to the rank r. Therefore, we have the complex $C_r$ . By Lemma 2.4, there exists a unique map $d_{r+1}$ such that the following diagram commutes:

We complete $d_{r+1}$ into a distinguished triangle

Now we notice that

$$ \begin{align*}\mathrm{Hom}(A_{r+1},C_r[-1])=0,\hspace{.5cm}\text{ and }\hspace{.5cm}\mathrm{Hom}(C_r,A_{r+1})=0. \end{align*} $$

Indeed, the complexes $C_r$ being successive extensions of the complexes $A_i$ , with $i=0,\dots ,r$ , this follows from the assumptions (2.5) and (2.6) .

By Lemma 2.1, such a $C_{r+1}$ is thus unique (up to a unique isomorphism).

2.2 Stacks and sheaves: notation and convention

Let k be an algebraic closure of a finite field. In these notes, unless specified, our stacks are k-algebraic stacks of finite type, and quotient stacks are denoted by $[X/G]$ if G is an algebraic group over k acting a k-scheme X. If $X=\mathrm {Spec}(k)$ , we put $\mathrm {B}(G)=[X/G]$ .

For a morphism $f:\mathcal {X}\rightarrow {\mathcal {Y}}$ of stacks, we have the usual functors

$$ \begin{align*}f_*=Rf_*:D_{\mathrm{c}}^+(\mathcal{X})\rightarrow D_{\mathrm{c}}^+({\mathcal{Y}}),\hspace{1cm} f_!=Rf_!:D_{\mathrm{c}}^-(\mathcal{X})\rightarrow D_{\mathrm{c}}^-({\mathcal{Y}}),\end{align*} $$

$$ \begin{align*}f^*, f^!:D_{\mathrm{c}}^{\bullet}({\mathcal{Y}})\rightarrow D_{\mathrm{c}}^{\bullet}(\mathcal{X}),\hspace{.5cm}(\text{with } \bullet\in\{\emptyset,-,+,b\}) \end{align*} $$

between derived categories of constructible ${\overline {\mathbb {Q}}_{\ell }}$ -sheaves (see [Reference Laszlo and Olsson9]).

We will use freely the properties of these functors (adjunction, projection formula, base change,…). When there is no ambiguity, we will sometimes write $K|_{\mathcal {X}}$ instead of $f^*(K)$ .

Except in some rare occasions, we will only need the category $D_{\mathrm {c}}^{\mathrm {b}}$ .

We will denote by $\overline {\mathbb {Q}}_{\ell ,\, \mathcal {X}}$ the constant sheaf on $\mathcal {X}$ . If there are no ambiguities, we will sometimes denote it simply by ${\overline {\mathbb {Q}}_{\ell }}$ to alleviate the notation.

For two ${\mathcal {Y}}$ -stacks $\mathcal {X}$ and $\mathcal {X}'$ , we define the external tensor product of $K\in D_{\mathrm {c}}^-(\mathcal {X})$ and $K'\in D_{\mathrm {c}}^-(\mathcal {X}')$ above ${\mathcal {Y}}$ as

$$ \begin{align*}K\boxtimes_{\mathcal{Y}} K':=\mathrm{pr}_1^*K\otimes\mathrm{pr}_2^*K'\in D_{\mathrm{c}}^-(\mathcal{X}\times_{\mathcal{Y}}\mathcal{X}'). \end{align*} $$

where $\mathrm {pr}_1$ and $\mathrm {pr}_2$ are the two projections.

We consider the auto-dual perversity p and we denote by $\mathcal {M}(\mathcal {X})$ the full subcategory of $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ of perverse sheaves on $\mathcal {X}$ . Then, if $\mathcal {X}$ is an equidimensional stack with smooth dense open substack ${\mathcal {U}}$ , the intersection cohomology complex on $\mathcal {X}$ with coefficient in a local system $\mathcal {E}$ on U is denoted by ${\mathrm {IC}}_{\mathcal {X}}(\mathcal {E})$ ;its restriction to ${\mathcal {U}}$ is $\mathcal {E}$ . If $\mathcal {E}={\overline {\mathbb {Q}}_{\ell }}$ , we will simply write ${\mathrm {IC}}_{\mathcal {X}}$ instead of ${\mathrm {IC}}_{\mathcal {X}}({\overline {\mathbb {Q}}_{\ell }})$ . Recall that ${\mathrm {IC}}_{\mathcal {X}}(\mathcal {E})[\mathrm {dim}\,\mathcal {X}]\in \mathcal {M}(\mathcal {X})$ is the image of

$$ \begin{align*}{^p}{\mathcal{H}}^0(j_!\mathcal{E}[\mathrm{dim}\,\mathcal{X}])\rightarrow {^p}{\mathcal{H}}^0(j_*\mathcal{E}[\mathrm{dim}\,\mathcal{X}]), \end{align*} $$

where $j:{\mathcal {U}}\rightarrow \mathcal {X}$ is the inclusion. Recall also that if $\mathcal {X}'\rightarrow \mathcal {X}$ is a small resolution of singularities (representable, proper, birational, $\mathcal {X}'$ smooth), then $f_*{\overline {\mathbb {Q}}_{\ell }}={\mathrm {IC}}_{\mathcal {X}}$ .

If $D_{\mathcal {X}}$ denotes the Verdier dual, then $D_{\mathcal {X}}({\mathrm {IC}}_{\mathcal {X}}(\mathcal {E}))={\mathrm {IC}}_{\mathcal {X}}(\mathcal {E}^{\vee })[2\mathrm {dim}(\mathcal {X})](\mathrm {dim}(\mathcal {X}))$ , where $\mathcal {E}^{\vee }$ denotes the dual local system.

Proposition 2.8. Let $\mathcal {X}$ be an equidimensional algebraic stack, and let ${\mathcal {U}}$ be a dense open smooth substack of $\mathcal {X}$ . Suppose that we have a short exact sequence

in the category of perverse sheaves on $\mathcal {X}$ . If both $A'$ and $A"$ are the intermediate extension of their restriction to ${\mathcal {U}}$ , then A is also the intermediate extension of its restriction to ${\mathcal {U}}$ .

Proof. Let j be the inclusion of ${\mathcal {U}}$ in $\mathcal {X}$ . From $j_!j^*\rightarrow 1\rightarrow j_*j^*$ and the fact that A is a perverse sheaf, we have a commutative diagram:

from which we can identify $j_{!*}j^*A$ as a subquotient of A. In particular,

$$ \begin{align*}\mathrm{length}(j_{!*}j^*A)\leq \mathrm{length}(A). \end{align*} $$

It is thus enough to prove that

(2.7) $$ \begin{align} \mathrm{length}(j_{!*}j^*A)\geq \mathrm{length}(A). \end{align} $$

We have the commutative diagram of perverse sheaves

where the top and bottom horizontal sequences are exact. We thus deduce that the middle sequence is exact at $A'$ and $A"$ and the composition $A'\rightarrow A"$ is zero, from which we deduce the inequality (2.7) as

$$ \begin{align*} \mathrm{length}(A)=\mathrm{length}(A')+\mathrm{length}(A").\\[-36pt] \end{align*} $$

As we assume that k is the algebraic closure of a finite field and that all our stacks are of finite type, any stack we consider in this paper will be defined over some finite subfield of k.

If $\mathcal {X}$ is defined over finite subfield $k_o\subset k$ , we denote by $\mathcal {X}_o$ the corresponding $k_o$ -structure on $\mathcal {X}$ (if no confusion arises) and we denote by $\mathrm {Frob}_o:\mathcal {X}\rightarrow \mathcal {X}$ the induced geometric Frobenius on $\mathcal {X}$ .

Let $K,K'\in D_{\mathrm {c}}^b(\mathcal {X})$ . We recall that $H_c^i(\mathcal {X},K)$ (resp. $\mathrm {Ext}^i(K,K')=\mathrm {Hom}(K,K'[i])$ ) is said to be pure of weight r if there exists a finite subfield $k_o$ of k such that $\mathcal {X}$ and K (resp. K and $K'$ ) are defined over $k_o$ , and the eigenvalues of the induced automorphism $\mathrm {Frob}_o^*$ on $H_c^i(\mathcal {X},K)$ (resp. $\mathrm {Ext}^i(K,K')$ ) are algebraic numbers whose complex conjugates are all of absolute value $|k_o|^{r/2}$ . It is well known that this notion of weight does not depend on the choice of $k_o$ . In particular, the Tate twist ${\overline {\mathbb {Q}}_{\ell }}(i)$ makes sense over k.

Let $\mathcal {X}_o$ be a $k_o$ -structure on $\mathcal {X}$ and $K_o, K^{\prime }_o\in D_{\mathrm {c}}^b(\mathcal {X}_o)$ , and let us denote by K and $K'$ the induced complexes in $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ . Then the following sequence [Reference Beilinson, Bernstein and Deligne2, (5.1.2.5)]

(2.8)

is exact for all i.

Notice that if $\mathrm {Ext}^j(K,K')=0$ for odd values of j, then it follows from the above exact sequence that for all i,

$$ \begin{align*}\mathrm{Ext}^{2i}(K_o,K^{\prime}_o)\rightarrow\mathrm{Ext}^{2i}(K,K')^{\mathrm{Frob}_o} \end{align*} $$

is an isomorphism.

2.3 Cohomology of $\mathrm {B}(T)$

Let T be a rank d torus over k. Recall that for any k-scheme S, the category $\mathrm {B}(T)(S)$ is the category of T-torsors over S, and the algebraic stack $\mathrm {B}(T)$ is of dimension $-d$ .

The cohomology $H^{\bullet }(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})$ is concentrated in nonnegative even degrees, and the morphism

$$ \begin{align*}c_{\mathrm{can}}:X^*(T)\rightarrow H^2(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})(1) \end{align*} $$

given by Chern classes induces an isomorphism

$$ \begin{align*}X^*(T)\otimes {\overline{\mathbb{Q}}_{\ell}}(-1)\simeq H^2(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

As ${\overline {\mathbb {Q}}_{\ell }}$ -algebras, $H^{2\bullet }(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})$ is isomorphic to $\mathrm {Sym}^{\bullet }\big (X^*(T)\otimes {\overline {\mathbb {Q}}_{\ell }}(-1)\big )$ .

By duality, $H_c^{\bullet }(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})$ is thus concentrated in even degrees $\leq -2\,\mathrm {dim}(T)$ , and $H_c^{2i}(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})$ is pure of weight $2i$ .

2.4 W-equivariant complexes

Let W be a finite group and a stack $\mathcal {X}$ . An action of W on a complex $K\in D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ is a group homomorphism

$$ \begin{align*}\theta: W\longrightarrow \mathrm{Aut}(K).\\[-15pt] \end{align*} $$

In this case we define the W-invariant part $K^W\rightarrow K$ in $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ of $(K,\theta )$ as follows.

Notice first that in any additive category, if for some morphism $e:A\rightarrow A$ , there exist two morphisms $u:A\rightarrow A'$ and $v:A'\rightarrow A$ such that $v\circ u=e$ and $u\circ v=1_{K'}$ (in which case e is an idempotent which is said to be split), then $1-e$ admits a kernel which is v and a cokernel which is u. In particular, $(A',u,v)$ is unique up to a unique isomorphism and we call it the splitting of e.

Since $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ is a triangulated category with bounded t-structure, by the main theorem of [Reference Le and Chen10], every idempotent element e of $\mathrm {End}(K)$ splits. Considering the indempotent

(2.9) $$ \begin{align} e=e_K:=\frac{1}{|W|}\sum_{w\in W}\theta(w)\in\mathrm{End}(K),\\[-15pt]\nonumber \end{align} $$

we define $K^W\rightarrow K$ as the kernel of $1-e$ .

A W-torsor is a morphism of stacks $\pi :\mathcal {X}\rightarrow \overline {\mathcal {X}}$ that fits to a cartesian diagram

(2.10)

Remark 2.9. (1) A W-torsor $\pi :\mathcal {X}\rightarrow \overline {\mathcal {X}}$ is thus finite étale representable and for any scheme S and morphism $S\rightarrow {\mathcal {Y}}$ , the projection

$$ \begin{align*}S\times_{\overline{\mathcal{X}}}\mathcal{X}\rightarrow S\\[-15pt] \end{align*} $$

has a natural structure of W-torsor (between schemes).

(2) If $\pi :\mathcal {X}\rightarrow \overline {\mathcal {X}}$ is a W-torsor, then $\mathcal {X}$ is equipped with a right action of W in the sense of [Reference Romagny18] and conversely, from right action of W on $\mathcal {X}$ we get a W-torsor by taking the quotient morphism of $\mathcal {X}$ by W. However, although this is implicit, we will not use the definition of group actions on stacks or the notion of quotient of stacks by group action.

An $\ell $ -adic sheaf on $\mathrm {B}(W)$ is a vector space equipped with a right action of W and

$$ \begin{align*}(\pi_o)_*{\overline{\mathbb{Q}}_{\ell}}={\overline{\mathbb{Q}}_{\ell}}[W]\\[-15pt] \end{align*} $$

for right multiplication of W on itself. The left multiplication corresponds to the Galois action. We thus have a decomposition

$$ \begin{align*}(\pi_o)_*{\overline{\mathbb{Q}}_{\ell}}=\bigoplus_{\chi} V_{\chi}\otimes \mathcal{L}_{o,\chi}\\[-15pt] \end{align*} $$

where the sum is over the irreducible ${\overline {\mathbb {Q}}_{\ell }}$ -characters of W, $V_{\chi }$ is a W-module affording the character $\chi $ and $\mathcal {L}_{o,\chi }$ is the irreducible smooth $\ell $ -adic sheaf on $\mathrm {B}(W)$ corresponding to $V_{\chi }^*$ .

By base change from Diagram (2.10) we get the analogous decomposition

(2.11) $$ \begin{align} \pi_*{\overline{\mathbb{Q}}_{\ell}}=\bigoplus_{\chi} V_{\chi}\otimes \mathcal{L}_{\chi} \end{align} $$

Notice that $\mathcal {L}_{\chi }$ may not be irreducible and that $(\pi _*{\overline {\mathbb {Q}}_{\ell }})^W={\overline {\mathbb {Q}}_{\ell }}$ .

Given a W-torsor $\pi :\mathcal {X}\rightarrow \overline {\mathcal {X}}$ , we denote by $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X},W)$ the subcategory of $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ whose objects are isomorphic to objects of the form $\pi ^*(\overline {K})$ with $\overline {K}\in D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})$ and morphisms are given by

$$ \begin{align*}\mathrm{Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{X},W)}(\pi^*\overline{A},\pi^*\overline{B}):=\pi^*\mathrm{ Hom}(\overline{A},\overline{B})\simeq \mathrm{Hom}(\overline{A},\overline{B}). \end{align*} $$

We call $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X},W)$ the category of W-equivariant complexes on $\mathcal {X}$ (with respect to $\pi $ ).

Remark 2.10. The inverse functor of $\pi ^*:D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})\rightarrow D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X},W)$ is given by

$$ \begin{align*}D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{X},W)\rightarrow D_{\mathrm{c}}^{\mathrm{b}}(\overline{\mathcal{X}}),\hspace{.5cm}K\mapsto (\pi_*K)^W. \end{align*} $$

Indeed this follows from the above discussion and

$$ \begin{align*}\pi_*\pi^*\overline{K}=(\pi_*{\overline{\mathbb{Q}}_{\ell}})\otimes\overline{K} \end{align*} $$

which is a consequence of the projection formula.

Given two W-equivariant complexes $A=\pi ^*\overline {A}$ and $B=\pi ^*\overline {B}$ in $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ , we define an action of W on $\mathrm { Hom}_{D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})}(A,B)$ as follows.

We have

$$ \begin{align*} \mathrm{Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{X})}(A,B)&=\mathrm{ Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{X})}(\pi^*\overline{A},\pi^*\overline{B})\\ &=\mathrm{Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\overline{\mathcal{X}})}(\overline{A},\pi_*\pi^*\overline{B})\\ &= \bigoplus_{\chi} V_{\chi}\otimes\mathrm{Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\overline{\mathcal{X}})}\left(\overline{A}, \mathcal{L}_{\chi}\otimes\overline{B}\right). \end{align*} $$

The action of W on the $V_{\chi }$ defines thus an action of W on $\mathrm { Hom}_{D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})}(A,B)$ , and we have

$$ \begin{align*}\mathrm{Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{X})}(A,B)^W=\mathrm{ Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{X},W)}(A,B). \end{align*} $$

Remark 2.11. Our definition of W-equivariant complexes is consistent with the usual one. If W acts (on the right) on a scheme X, then following [Reference Kiehl and Weissauer8, III, 15], a W-equivariant complex on X is a pair $(K,\theta )$ with $K\in D_{\mathrm {c}}^{\mathrm {b}}(X)$ and $\theta =(\theta _w)_{w\in W}$ a collection of isomorphisms

$$ \begin{align*}\theta_w:w^*(K)\rightarrow K \end{align*} $$

such that

1. (i) $\theta _{ww'}=\theta _w\circ w^*(\theta _{w'})$ for all $w,w'\in W$ , and

2. (ii) $\theta _1=1_K$ ,

where $1_K:K\rightarrow K$ denotes the identity morphism.

Moreover, if $(K,\theta ),(K',\theta ')\in D_{\mathrm {c}}^{\mathrm {b}}(X,W)$ , then the group W acts (on the left) on $\mathrm {Hom}_{D_{\mathrm {c}}^{\mathrm {b}}(X)}(K,K')$ as

$$ \begin{align*}w\cdot f=\theta^{\prime}_w\circ w^*(f)\circ(\theta_w)^{-1} \end{align*} $$

for all $w\in W$ and $f\in \mathrm {Hom}(K,K')$ , and

$$ \begin{align*}\mathrm{Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(X,W)}(K,K'):=\mathrm{ Hom}_{D_{\mathrm{c}}^{\mathrm{b}}(X)}(K,K')^W. \end{align*} $$

A commutative diagram

where $\pi $ and $\pi '$ are W-torsors, induces a functor $f^*:D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X}',W)\rightarrow D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X},W)$ , and, moreover, if the above diagram is cartesian and f is representable, then we also get a functor $f_!:D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X},W)\rightarrow D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X}',W)$ . Both are compatible with the usual functors $f^*$ and $f_!$ where we forget the actions of W.

In the next two following lemmas we assume given a W-torsor

$$ \begin{align*}\pi:\mathcal{X}\rightarrow\overline{\mathcal{X}}. \end{align*} $$

Lemma 2.13. Assume that we have a distinguished triangle

in $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ and that d is W-equivariant. Then there exists a distinguished triangle

(2.12)

in $D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})$ and an isomorphism of triangles

If, moreover, we assume that

$$ \begin{align*}\mathrm{Hom}(A,C[-1])=0 \hspace{.5cm}\text{ and }\hspace{.5cm}\mathrm{Hom}(C,A)=0, \end{align*} $$

then the triangle (2.12) is unique (up to a unique isomorphism), and the morphism s is unique.

Proof. As $d:\pi ^*\overline {A}\rightarrow \pi ^*\overline {C}$ is W-equivariant, by definition it descends to a unique morphism $\overline {d}:\overline {A}\rightarrow \overline {C}$ in $D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})$ . We complete this morphism into a distinguished triangle

in $D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})$ .

There exists an isomorphism $s:C'\rightarrow \pi ^*(\overline {C}')$ , such that the following diagram commutes:

The second statement follows from Lemma 2.1.

Proposition 2.14. Assume that we have a Postnikov diagram $\Lambda $ of the form (2.1) in $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ , such that the complex $\Lambda _b$ is the image by $\pi ^*$ of a complex

(2.13)

in $D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})$ . Suppose also that

(2.14) $$ \begin{align} \mathrm{Hom}(A_j,A_i)=0,\hspace{.5cm}\text{ for all } j<i, \end{align} $$

and

(2.15) $$ \begin{align} \mathrm{Hom}(A_j,A_i[-k])=0,\hspace{.5cm}\text{ for all }j>i\text{ and } k\geq 1. \end{align} $$

Then the complex (2.13) can be completed in a unique way (up to a unique isomorphism) into a Postnikov diagram $\overline {\Lambda }$ in $D_{\mathrm {c}}^{\mathrm {b}}(\overline {\mathcal {X}})$ , such that

$$ \begin{align*}\pi^*(\overline{\Lambda})\simeq\Lambda. \end{align*} $$

2.5 Cohomological correspondences

By a cohomological correspondence $\Gamma =({\mathcal {C}},N,p,q)$ we shall mean a correspondence of S-algebraic stacks

(2.16)

together with a kernel $N\in D_{\mathrm {c}}^-({\mathcal {C}})$ .

The correspondence $\Gamma $ comes with a functor

$$ \begin{align*}\mathrm{Res}=\mathrm{Res}_{\Gamma}:D_{\mathrm{c}}^-(\mathcal{X})\rightarrow D_{\mathrm{c}}^-({\mathcal{Y}}),\hspace{.5cm}A\mapsto q_!(N\otimes p^*A) \end{align*} $$

which we call the restriction functor associated to $\Gamma $ , whose right adjoint is the induction functor

$$ \begin{align*}\mathrm{Ind}=\mathrm{Ind}_{\Gamma}:D_{\mathrm{c}}^+({\mathcal{Y}})\rightarrow D_{\mathrm{c}}^+(\mathcal{X}),\hspace{.5cm}B\mapsto p_*\underline{\mathrm{Hom}}(N,q^!B). \end{align*} $$

Lemma 2.15. If we have a morphism of correspondences $f:({\mathcal {C}},p,q)\rightarrow ({\mathcal {C}}',p',q')$ (i.e., a commutative diagram)

(2.17)

then we have natural isomorphisms

$$ \begin{align*}\mathrm{Res}_{({\mathcal{C}},N,p,q)}=\mathrm{Res}_{({\mathcal{C}}',f_! N,p',q')},\hspace{1cm}\mathrm{Ind}_{({\mathcal{C}},N,p,q)}=\mathrm{Ind}_{({\mathcal{C}}',f_!N,p',q')}.\end{align*} $$

Lemma 2.16 (Composition).

Assume that we have two cohomological correspondences $\Gamma =({\mathcal {C}},N,p,q)$ and $\Gamma '=({\mathcal {C}}',N',p',q')$ . Consider the following diagram:

Then

$$ \begin{align*}\mathrm{Res}_{\Gamma'}\circ\mathrm{Res}_{\Gamma}=\mathrm{Res}_{\Gamma'\circ\Gamma} \end{align*} $$

where

$$ \begin{align*}\Gamma'\circ\Gamma:=\left(\mathcal{Z}\times_S\mathcal{X},(q',p)_!(N'\boxtimes_{\mathcal{Y}} N),\overline{\mathrm{pr}}_2,\overline{\mathrm{pr}}_1\right). \end{align*} $$

We keep the correspondence (2.16), and we assume that it can be completed into a diagram

where $\pi $ and $\rho $ are W-torsors and where the square is cartesian.

If N is W-equivariant (i.e., descends to a complex $\overline {N}$ on $\overline {{\mathcal {C}}}$ ), then we have the following factorization

(2.18) $$ \begin{align} \mathrm{Ind}=\mathrm{I}\circ\pi_*,\hspace{1cm}\mathrm{Res}=\pi^*\circ\mathrm{R} \end{align} $$

where $\mathrm {I}:D_{\mathrm {c}}^+(\overline {{\mathcal {Y}}})\rightarrow D_{\mathrm {c}}^+(\mathcal {X})$ and $\mathrm {R}:D_{\mathrm {c}}^-(\mathcal {X})\rightarrow D_{\mathrm {c}}^-(\overline {{\mathcal {Y}}})$ are respectively the induction and restriction functors defined from the correspondence $(\overline {{\mathcal {C}}},\overline {N},\overline {p},\overline {q})$ .

Remark 2.17. Notice that $\mathrm {I}$ can be computed as

$$ \begin{align*}\mathrm{I}(\overline{K})=\mathrm{Ind}(\pi^*\overline{K})^W \end{align*} $$

since

$$ \begin{align*} \mathrm{Ind}(\pi^*\overline{K})&=\mathrm{I}\circ\pi_*\circ\pi^*(\overline{K})\\ &=\bigoplus_{\chi} V_{\chi}\otimes \mathrm{I}(\mathcal{L}_{\chi}\otimes \overline{K}), \end{align*} $$

where $\pi _*{\overline {\mathbb {Q}}_{\ell }}=\bigoplus _{\chi } V_{\chi }\otimes \mathcal {L}_{\chi }$ is the decomposition indexed by the irreducible characters of W (see Remark 2.10).

2.6 Reductive groups

For an affine connected algebraic group H over k, we denote by $\mathfrak {h}$ the Lie algebra of H. We denote by $Z(H)$ the center of H and by $z(\mathfrak {h})$ the center of $\mathfrak {h}$ . We will also denote by ${\mathcal {H}}:=[\mathfrak {h}/H]$ the quotient stack of $\mathfrak {h}$ by H for the adjoint action $\mathrm {Ad}:H\rightarrow \mathrm {GL}(\mathfrak {h})$ .

If, moreover, H is reductive, we denote by

$$ \begin{align*}\mathfrak{car}_{\mathfrak{h}}=\mathfrak{h}/\!/H:=\mathrm{Spec}(k[\mathfrak{h}]^H) \end{align*} $$

the variety of characteristic polynomials of the elements of $\mathfrak {h}$ , and we have a canonical map $\chi _{\mathfrak {h}}:\mathcal {H}\rightarrow \mathfrak {car}_{\mathfrak {h}}$ .

If H is commutative, then ${\mathcal {H}}\simeq \mathfrak {h}\times \mathrm {B}(H)$ , $\mathfrak {car}_{\mathfrak {h}}\simeq \mathfrak {h}$ and $\chi _{\mathfrak {h}}$ is the first projection.

From now, G is a connected reductive group over k, T is a maximal torus of G, B a Borel subgroup of G containing T, U the unipotent radical of B, N the normalizer $\mathrm {N}_G(T)$ of T in G and W the Weyl group $N/T$ . Through this paper we put

$$ \begin{align*}n:=\mathrm{dim}\, T. \end{align*} $$

We will simply use the notation $\mathfrak {car}$ instead of $\mathfrak {car}_{\mathfrak {g}}$ , and we recall that

$$ \begin{align*}\mathfrak{car}={\mathfrak{t}}/\!/W. \end{align*} $$

We let

$$ \begin{align*}\pi:\mathcal{T}\longrightarrow{\overline{\mathcal{T}}}=[{\mathfrak{t}}/N] \end{align*} $$

be the canonical map.

The character group is denoted by $X^*(T)$ and the cocharacter group by $X_*(T)$ .

When it makes sense, we use freely the subscripts ${\mathrm {reg}}$ and ${\mathrm {rss}}$ for restriction to G-regular or G-regular semisimple elements. Similarly we will use the subscript $\mathrm {nil}$ for restriction to nilpotent elements.

We assume throughout this paper that the characteristic of k is not too small so that ${\mathfrak {t}}_{{\mathrm {reg}}}\neq \emptyset $ and centralizers of semisimple elements of ${\mathfrak {g}}$ are Levi subgroups (of parabolic subgroups) of G (see [Reference Letellier11, §2.6] for an explicit bound on p).

2.7 Lusztig correspondence

Consider the correspondence (which we call Lusztig correspondence)

(2.19)

where the arrows are induced by the inclusion ${\mathfrak {b}}\subset {\mathfrak {g}}$ and by the projection ${\mathfrak {b}}\rightarrow {\mathfrak {t}}$ .

Consider

$$ \begin{align*}X:=\{(x,gB)\in{\mathfrak{g}}\times G/B\,|\, \mathrm{Ad}(g)(x)\in{\mathfrak{b}}\}\\[-9pt] \end{align*} $$

The group G acts on X by $g\cdot (x,hB)=( \mathrm {Ad}(g)(x),ghB)$ , and the map

$$ \begin{align*}B\rightarrow X,\,\, x\mapsto(x,B)\\[-9pt] \end{align*} $$

induces an isomorphism $\mathcal {B}\rightarrow \mathcal {X}:=[X/G]$ .

Under the identification $\mathcal {B}\simeq \mathcal {X}$ , the morphism p is the quotient by G of the Grothendieck-Springer resolution

$$ \begin{align*}\mathrm{pr}_1:X\rightarrow {\mathfrak{g}},\,\, (x,gB)\mapsto x \end{align*} $$

from which we deduce the first assertion of the theorem below.

2.8 Lusztig induction and restriction

The morphism p is representable, proper and q is smooth of pure dimension $0$ . We thus have $p_*=p_!$ and $q^*=q^!$ . The Lusztig induction and restriction functors [Reference Lusztig14, (7.1.7)] are the induction and restriction defined from the correspondence (2.19) with the constant sheaf as a kernel:

$$ \begin{align*}\mathrm{Ind}:D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{T})\rightarrow D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{G}),\hspace{.5cm}K\mapsto p_*q^!(K), \end{align*} $$

$$ \begin{align*}\mathrm{Res}:D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{G})\rightarrow D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{T}),\hspace{.5cm}K\mapsto q_!p^*(K). \end{align*} $$

Remark 2.19. Notice that $\mathrm {Res}$ is well-defined as $q_!$ maps bounded complexes to bounded complexes. To see that, we consider the commutative triangle

Since h is an affine fibration, the adjunction morphism

$$ \begin{align*}h_!h^*\rightarrow 1 \end{align*} $$

is an isomorphism, and so we conclude by noticing that a is representable.

By the main result of [Reference Bezrukavnikov and Yom Din3], the functors $\mathrm {Ind}$ and $\mathrm {Res}$ map perverse sheaves to perverse sheaves.

Consider the factorization

(2.20)

where the map $\mathcal {T}\rightarrow \mathfrak {car}$ is either the composition $\mathcal {T}\rightarrow \mathcal {G}\rightarrow \mathfrak {car}$ or the composition $\mathcal {T}\rightarrow {\mathfrak {t}}\rightarrow \mathfrak {car}$ .

By §2.5, we have

$$ \begin{align*} \mathrm{Res}(K)&={\mathrm{pr}_{\mathcal{T}}}_!\left(\mathrm{pr}^*_{\mathcal{G}}(K)\otimes (q,p)_!{\overline{\mathbb{Q}}_{\ell}}\right),\\ \mathrm{Ind}(K)&={\mathrm{pr}_{\mathcal{G}}}_*\,\underline{\mathrm{ Hom}}\left((q,p)_!{\overline{\mathbb{Q}}_{\ell}},\mathrm{pr}_{\mathcal{T}}^!(K)\right)\\ &={\mathrm{pr}_{\mathcal{G}}}_!\left(\mathrm{pr}^*_{\mathcal{T}}(K)\otimes (q,p)_!{\overline{\mathbb{Q}}_{\ell}}\right). \end{align*} $$

The last identity follows from the fact that $q^!=q^*$ and $p_!=p_*$ since we can then regard $\mathrm {Ind}$ as the restriction functor associated to the correspondence $(\mathcal {B},q,p)$ .

Notice also that $(q,p)_!{\overline {\mathbb {Q}}_{\ell }}\in D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {T}\times _{\mathfrak {car}}\mathcal {G})$ .

2.9 Induction and restriction for perverse sheaves

As we say in the previous section, the two functors $\mathrm {Ind}$ and $\mathrm {Res}$ are t-exact but, as we will see later, the kernel $(q,p)_!{\overline {\mathbb {Q}}_{\ell }}$ is not W-equivariant. In this section we introduce slightly modified functors ${^p}\underline {\mathrm {Ind}}$ and ${^p}\underline {\mathrm {Res}}$ defined from a kernel which is naturally W-equivariant.

We consider the following commutative diagram

(2.21)

where $s:\mathcal {T}\simeq {\mathfrak {t}}\times \mathrm {B}(T)\rightarrow {\mathfrak {t}}$ is the projection on ${\mathfrak {t}}$ . Notice that s is not representable.

Consider the induction functor

$$ \begin{align*}\underline{\mathrm{Ind}}:=\mathrm{Ind}\circ s^:D_{\mathrm{c}}^+({\mathfrak{t}})\rightarrow D_{\mathrm{c}}^+(\mathcal{G}),\,\,\,\,K\mapsto p_*q'{^!}(K)[n](n). \end{align*} $$

Since the functor $s^:\mathcal {M}({\mathfrak {t}})\rightarrow \mathcal {M}(\mathcal {T})$ is an equivalence of categories with inverse functor ${^p}{\mathcal {H}}^0\circ \left (s_\right )$ and since $\mathrm {Ind}$ maps perverse sheaves to perverse sheaves (see §2.8), the functor $\underline {\mathrm {Ind}}$ preserves perversity and we denote by ${^p}\underline {\mathrm {Ind}}$ the induced functor on perverse sheaves:

$$ \begin{align*}{^p}\underline{\mathrm{Ind}}:=\mathrm{Ind}\circ s^:\mathcal{M}({\mathfrak{t}})\rightarrow\mathcal{M}(\mathcal{G}),\,\,\,\,K\mapsto p_*q'{^!}(K)[n](n). \end{align*} $$

The functor $\underline {\mathrm {Ind}}$ admits a left adjoint

$$ \begin{align*}\underline{\mathrm{Res}}:=s_!\circ\mathrm{Res}[-n](-n):D_{\mathrm{c}}^-(\mathcal{G})\rightarrow D_{\mathrm{c}}^-({\mathfrak{t}}),\,\,\, K\mapsto q^{\prime}_!p^*(K)[-n](-n) \end{align*} $$

which is right t-exact but not left t-exact (unlike $\underline {\mathrm {Ind}}$ ).

Since $\mathrm {Res}$ preserves perversity, for $K\in \mathcal {M}(\mathcal {G})$ , we have

$$ \begin{align*}\underline{\mathrm{Res}}(K)=s_!s^!(K')=K'\otimes R\Gamma_c(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})[-2n](-n) \end{align*} $$

where $K'\in \mathcal {M}({\mathfrak {t}})$ is defined as $s^!(K')[n](n)=\mathrm {Res}(K)$ .

Define the restriction

$$ \begin{align*}{^p}\underline{\mathrm{Res}}:={^p}{\mathcal{H}}^0\circ\underline{\mathrm{Res}}:\mathcal{M}(\mathcal{G})\rightarrow\mathcal{M}({\mathfrak{t}}),\,\,\,K\mapsto K'. \end{align*} $$

Proof. We have

$$ \begin{align*} \mathrm{Hom}_{\mathcal{M}({\mathfrak{t}})}\left({^p}\underline{\mathrm{Res}}(A),B\right)&=\mathrm{ Hom}_{\mathcal{M}(\mathcal{T})}\left(s^!({^p}\underline{\mathrm{Res}}(A))[n](n),s^!B[n](n)\right)\\ &=\mathrm{Hom}_{\mathcal{M}(\mathcal{T})}\left(\mathrm{Res}(A),s^!B[n](n)\right)\\ &=\mathrm{Hom}_{\mathcal{M}(\mathcal{G})}\left(A,\mathrm{Ind}(s^!B[n](n))\right)\\ &=\mathrm{Hom}_{\mathcal{M}(\mathcal{G})}\left(A,{^p}\underline{\mathrm{Ind}}(B)\right). \\[-36pt] \end{align*} $$

Put

$$ \begin{align*}\overline{{\mathfrak{t}}}:=[{\mathfrak{t}}/W] \end{align*} $$

and let $\pi _{\mathfrak {t}}:{\mathfrak {t}}\rightarrow \overline {{\mathfrak {t}}}$ be the quotient map.

Proposition 2.21. The functors ${{}^p}\underline {\mathrm {Ind}}$ and ${{}^p}\underline {\mathrm {Res}}$ factorize as

$$ \begin{align*}{{}^p}\underline{\mathrm{Ind}}={{}^p}\mathrm{I}\circ\pi_{{\mathfrak{t}}\, *},\hspace{1cm}{{}^p}\underline{\mathrm{Res}}=\pi_{\mathfrak{t}}^*\circ{{}^p}\mathrm{R} \end{align*} $$

where ${{}^p}\mathrm {I}:\mathcal {M}(\overline {{\mathfrak {t}}})\rightarrow \mathcal {M}(\mathcal {G})$ and ${{}^p}\mathrm {R}:\mathcal {M}(\mathcal {G})\rightarrow \mathcal {M}(\overline {{\mathfrak {t}}})$ are defined as

$$ \begin{align*} &{{}^p}\mathrm{I}(K):=\mathrm{pr}_{2\, *}\,\underline{\mathrm{ Hom}}\left({\mathrm{IC}}_{\overline{{\mathfrak{t}}}\times_{\mathfrak{car}}\mathcal{G}},\mathrm{pr}_1^!K\right)[n](n),\\ &{{}^p}\mathrm{R}(K):={{}^p}{\mathcal{H}}^0\left(\mathrm{pr}_{1\, !}\left({\mathrm{IC}}_{\overline{{\mathfrak{t}}}\times_{\mathfrak{car}}\mathcal{G}}\otimes\mathrm{pr}_2^*(K)\right)[-n](-n)\right). \end{align*} $$

We have the following proposition.

Proof. From Proposition 2.21 we find that

$$ \begin{align*}{{}^p}\underline{\mathrm{Ind}}(w\cdot f)=w\cdot{{}^p}\underline{\mathrm{Ind}}(f) \end{align*} $$

for any $f\in \mathrm {Hom}({{}^p}\underline {\mathrm {Res}}(L),K)$ and $w\in W$ , from which we deduce the assertion (1). The second assertion is a consequence of (1) together with following identities:

$$ \begin{align*} &\mathrm{Hom}_{\mathcal{M}(\overline{{\mathfrak{t}}})}\left({{}^p}\mathrm{R}(L),\overline{K}\right)=\mathrm{ Hom}_{\mathcal{M}({\mathfrak{t}})}\left({{}^p}\underline{\mathrm{Res}}(L),K\right)^W,\\ &\mathrm{Hom}_{\mathcal{M}(\mathcal{G})}\left(L,{{}^p}\mathrm{I}(\overline{K})\right)=\mathrm{ Hom}_{\mathcal{M}(\mathcal{G})}\left(L,{{}^p}\underline{\mathrm{Ind}}(K)\right)^W \end{align*} $$

where $K=\pi _{\mathfrak {t}}^*(\overline {K})$ .

2.10 Springer action

From Borho-MacPherson construction of the Springer action [Reference Borho and MacPherson4], the complex $p_!{\overline {\mathbb {Q}}_{\ell }}$ is endowed with an action of the Weyl group W (i.e., a group homomorphism $W\rightarrow \mathrm {Aut}(p_!({\overline {\mathbb {Q}}_{\ell }}))$ ). Their strategy was to prove that $p_*{\overline {\mathbb {Q}}_{\ell }}$ is the intermediate extension of some smooth $\ell $ -adic sheaf on $\mathcal {G}_{\mathrm {rss}}$ on which W acts. From the diagram (2.21), we see that $p_!{\overline {\mathbb {Q}}_{\ell }}=\mathrm {pr}_{2\, !}{\mathrm {IC}}_{{\mathfrak {t}}\times _{\mathfrak {car}}\mathcal {G}}$ , and so the action of W follows from the fact that ${\mathrm {IC}}_{{\mathfrak {t}}\times _{\mathfrak {car}}\mathcal {G}}$ is naturally W-equivariant.

This defines an action of W on $H^i(\mathcal {B},{\overline {\mathbb {Q}}_{\ell }})$ as well as an action on the cohomology of the fibres. These actions are compatible with the restriction maps from the cohomology of $\mathcal {B}$ to the cohomology of the fibres.

On the other hand, the natural map $\mathrm {B}(T)\rightarrow \mathrm {B}(B)$ is a U-fibration, and so $H^i(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})=H^i(\mathrm {B}(B),{\overline {\mathbb {Q}}_{\ell }})$ . The action of W on $\mathrm {B}(T)$ induces thus an action of W on $H^i(\mathrm {B}(B),{\overline {\mathbb {Q}}_{\ell }})$ . This action coincides with the Springer action (regarding $\mathrm {B}(B)$ as the zero fibre of p).

Proof. We know that the restriction map

$$ \begin{align*}i^*:H^i(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}})\rightarrow H^i(\mathrm{B}(B),{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

of the inclusion $i:\mathrm {B}(B)\hookrightarrow \mathcal {B}$ is W-equivariant for the Springer actions.

The morphism $b:\mathcal {B}\rightarrow \mathrm {B}(B)$ is a vector bundle with fibre ${\mathfrak {b}}$ . Hence $b^*$ is an isomorphism with inverse $i^*$ and so is W-equivariant.

3.1 Geometry of Steinberg stacks

We consider the following stacks

$$ \begin{align*}\mathcal{Z}:=\mathcal{B}\times_{\mathcal{G}}\mathcal{B},\hspace{1cm}{\mathcal{Y}}:=\mathcal{B}\times_{\mathcal{S}}\mathcal{B}. \end{align*} $$

We have the following cartesian diagrams

where the top horizontal arrow is the natural map, and the last two arrows are the diagonal morphisms. Since the diagonal morphism ${\mathfrak {t}}\rightarrow {\mathfrak {t}}\times _{\mathfrak {car}}{\mathfrak {t}}$ is closed, the stack ${\mathcal {Y}}$ is a closed substack of $\mathcal {Z}$ .

Remark 3.1. Consider

$$ \begin{align*}Z=\{(x,gB,g'B)\in {\mathfrak{g}}\times G/B\times G/B\,|\, \mathrm{Ad}(g^{-1})(x)\in {\mathfrak{b}},\, \mathrm{Ad}({g'}^{-1})(x)\in {\mathfrak{b}}\} \end{align*} $$

so that $\mathcal {Z}=[Z/G]$ where G acts on Z by

$$ \begin{align*}(x,gB,g'B)\cdot h=(\mathrm{Ad}(h^{-1})x,h^{-1}gB,h^{-1}g'B). \end{align*} $$

The canonical morphisms $\mathcal {B}\rightarrow \mathrm {B}(B)$ and $\mathcal {G}\rightarrow \mathrm {B}(G)$ induce a morphism

and the canonical map $[B\backslash G/B]\rightarrow \mathrm {B}(B)\times _{\mathrm {B}(G)}\mathrm {B}(B)$ is an isomorphism.

Consider the stratification

$$ \begin{align*}[B\backslash G/B]=\coprod_{w\in W}\mathcal{O}_w, \end{align*} $$

where $\mathcal {O}_w=[B\backslash BwB/B]$ . Notice that the natural map $\mathrm {B}(B_w)\rightarrow \mathcal {O}_w$ is an isomorphism.

This induces partitions into locally closed substacks

$$ \begin{align*}\mathcal{Z}=\coprod_{w\in W}\mathcal{Z}_w,\hspace{1cm}{\mathcal{Y}}=\coprod_{w\in W}{\mathcal{Y}}_w. \end{align*} $$

For $w\in W$ , put

$$ \begin{align*}B_w:=B\cap wBw^{-1},\hspace{.5cm}{\mathfrak{b}}_w:=\mathrm{Lie}(B_w),\hspace{.5cm}U_w:=U\cap wUw^{-1}, \hspace{.5cm}{\mathfrak{u}}_w:=\mathrm{Lie}(U_w). \end{align*} $$

Then

$$ \begin{align*}\mathcal{Z}_w=[{\mathfrak{b}}_w/B_w],\hspace{1cm}{\mathcal{Y}}_w=[({\mathfrak{t}}^w+{\mathfrak{u}}_w)/B_w] \end{align*} $$

where $B_w$ acts by the adjoint action and ${\mathfrak {t}}^w$ are the points of ${\mathfrak {t}}$ fixed by w.

Since $B_w=TU_w$ , we proved the following proposition.

3.2 Purity of cohomology

Proof. We prove it for ${\mathcal {Y}}$ , but the proof is completely similar for $\mathcal {Z}$ . Let us choose $k_o$ , a finite subfield of k such that G, B and T are defined over $k_o$ and T is split. Then $\mathrm {Gal}(k/k_o)$ acts trivially on W.

Recall (see §3) that we have a partition into locally closed substacks

(3.1) $$ \begin{align} {\mathcal{Y}}=\coprod_{w\in W}{\mathcal{Y}}_w. \end{align} $$

By our choice of $k_o$ , each stratum is defined over $k_o$ .

We choose a total order $\{w_0,w_1,\dots \}$ on W so that we have a decreasing filtration of closed substacks

$$ \begin{align*}{\mathcal{Y}}={\mathcal{Y}}_0\supset{\mathcal{Y}}_1\supset\cdots\supset{\mathcal{Y}}_{|W|-1}\supset\mathcal{Z}_{|W|}=\emptyset \end{align*} $$

satisfying ${\mathcal {Y}}_i\backslash {\mathcal {Y}}_{i+1}={\mathcal {Y}}_{w_i}$ for all i. This defines a spectral sequence [Reference Deligne5, Chapter 6, Formula (2.5.2)] from which we can reduce the proof of the purity of the cohomology of ${\mathcal {Y}}$ and the vanishing in odd degrees to the analogous statement for ${\mathcal {Y}}_w$ .

The fibres of the projection ${\mathcal {Y}}_w\rightarrow \mathcal {O}_w$ are affine spaces all of same dimension $n(w)=\mathrm {dim}\,{\mathfrak {t}}^w+\mathrm {dim}\,{\mathfrak {u}}_w$ by Proposition 3, and so

$$ \begin{align*}H_c^i({\mathcal{Y}}_w,{\overline{\mathbb{Q}}_{\ell}})=H_c^{i-2n(w)}(\mathcal{O}_w,{\overline{\mathbb{Q}}_{\ell}})(-n(w)). \end{align*} $$

Recall also that the map $\mathrm {B}(B_w)\rightarrow \mathcal {O}_w$ is an isomorphism and that

$$ \begin{align*}H_c^{k+2\mathrm{dim}\, U_w}(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})(\mathrm{dim}\, U_w)=H_c^k(\mathrm{B}(B_w),{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

Therefore,

$$ \begin{align*}H_c^i({\mathcal{Y}}_w,{\overline{\mathbb{Q}}_{\ell}})=H_c^{i-2\mathrm{dim}\,{\mathfrak{t}}^w}(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})(-\mathrm{dim}\,{\mathfrak{t}}^w). \end{align*} $$

Since the cohomology of $\mathrm {B}(T)$ is pure and vanishes in odd degree, the same is true for $H_c^i({\mathcal {Y}}_w,{\overline {\mathbb {Q}}_{\ell }})$ .

3.3 Restriction to the diagonal

Notice that

$$ \begin{align*}H_c^i(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}})=H_c^i(\mathcal{G},p_!{\overline{\mathbb{Q}}_{\ell}}\otimes p_!{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

and so $H_c^i(\mathcal {Z},{\overline {\mathbb {Q}}_{\ell }})$ is naturally equipped with an action of $W\times W$ .

The aim of this section is to prove the following theorem.

Theorem 3.4. The cohomological restriction

$$ \begin{align*}H_c^{2i}(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}})\longrightarrow H_c^{2i}(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

of the diagonal morphism $\mathcal {B}\rightarrow \mathcal {Z}$ is W-equivariant for the Springer actions (where we consider the diagonal action of W on the source).

Let $q':\mathcal {B}\rightarrow {\mathfrak {t}}$ be induced by the canonical projection ${\mathfrak {b}}\rightarrow {\mathfrak {t}}$ . Then

$$ \begin{align*}H^i_c(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}})=H_c^i({\mathfrak{t}}\times_{\mathfrak{car}}{\mathfrak{t}},(q',q')_!{\overline{\mathbb{Q}}_{\ell}}),\hspace{1cm}H_c^i(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}})=H_c^i({\mathfrak{t}},q^{\prime}_!{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

Choose a total ordering $\{w_0,w_1,\dots \}$ on W so that we have a decreasing filtration of closed substacks

$$ \begin{align*}\mathcal{Z}=\mathcal{Z}_0\supset\mathcal{Z}_1\supset\cdots\supset\mathcal{Z}_{|W|-1}\supset\mathcal{Z}_{|W|}=\emptyset \end{align*} $$

satisfying $\mathcal {Z}_i\backslash \mathcal {Z}_{i+1}=\mathcal {Z}_{w_i}$ for all i.

By [Reference Deligne5, Chapter 6, Formula (2.5.2)], we have a spectral sequence

$$ \begin{align*}E_1^{ij}={\mathcal{H}}^{i+j} \left(\left((q',q')|_{\mathcal{Z}_{w_i}}\right)_!{\overline{\mathbb{Q}}_{\ell}}\right)\Rightarrow {\mathcal{H}}^{i+j}(q',q')_!{\overline{\mathbb{Q}}_{\ell}}. \end{align*} $$

For $w\in W$ , consider the morphism

$$ \begin{align*}\Delta_{{\mathfrak{t}},w}:{\mathfrak{t}}\longrightarrow {\mathfrak{t}}\times_{\mathfrak{car}}{\mathfrak{t}},\hspace{1cm}t\mapsto (w(t),t) \end{align*} $$

and the following commutative diagram:

Then

$$ \begin{align*}((q',q')|_{\mathcal{Z}_w})_!{\overline{\mathbb{Q}}_{\ell}}=\Delta_{{{\mathfrak{t}},w}\,*}q^{\prime}_{w\,*}{\overline{\mathbb{Q}}_{\ell}}=\Delta_{{\mathfrak{t}},w\,*}{\overline{\mathbb{Q}}_{\ell}}\otimes R\Gamma_c(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

As $H^{\mathrm {odd}}_c(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})$ vanishes, the spectral sequence degenerates at $E_1$ .

Proof of Theorem 3.4.

Since the above spectral sequence degenerates at $E_1$ , we have ${\mathcal {H}}_c^{\mathrm {odd}}(q',q')_!{\overline {\mathbb {Q}}_{\ell }}=0$ , and ${\mathcal {H}}^{\mathrm {even}}(q',q')_!{\overline {\mathbb {Q}}_{\ell }}$ is a successive extension of the $\Delta _{{\mathfrak {t}},w\,*}{\overline {\mathbb {Q}}_{\ell }}\otimes H^{\mathrm { even}}_c(\mathrm {B}(T),{\overline {\mathbb {Q}}_{\ell }})$ . By Proposition 2.8, ${\mathcal {H}}^{2i}(q',q')_!{\overline {\mathbb {Q}}_{\ell }}$ is thus a perverse sheaf (up to a shift) that is the intermediate extension of its restriction to ${\mathfrak {t}}_{\mathrm {reg}}\times _{\mathfrak {car}}{\mathfrak {t}}_{\mathrm {reg}}$ .

We thus have

$$ \begin{align*}{\mathcal{H}}^{2i}(q',q')_!{\overline{\mathbb{Q}}_{\ell}}=\Delta_*{\overline{\mathbb{Q}}_{\ell}}\otimes H^{2i}_c(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

where $\Delta :W\times {\mathfrak {t}}\rightarrow {\mathfrak {t}}\times _{\mathfrak {car}}{\mathfrak {t}}$ , $(w,t)\mapsto (w(t),t)$ is the normalization morphism.

Using the spectral sequence

$$ \begin{align*}H_c^i\left({\mathfrak{t}}\times_{\mathfrak{car}}{\mathfrak{t}},{\mathcal{H}}^j(q',q')_!{\overline{\mathbb{Q}}_{\ell}}\right)\Rightarrow H_c^{i+j}(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

together with the fact that $H_c^i({\mathfrak {t}}\times _{\mathfrak {car}}{\mathfrak {t}},\Delta _*{\overline {\mathbb {Q}}_{\ell }})=0$ unless $i=2n$ , we deduce that

$$ \begin{align*}H_c^{-2i}(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}})=H_c^{2n}({\mathfrak{t}}\times_{\mathfrak{car}}{\mathfrak{t}},\Delta_*{\overline{\mathbb{Q}}_{\ell}})\otimes H_c^{-2n-2i}(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

The restriction morphism $H_c^i(\mathcal {Z},{\overline {\mathbb {Q}}_{\ell }})\rightarrow H_c^i(\mathcal {B},{\overline {\mathbb {Q}}_{\ell }})$ is induced by the restriction morphism

$$ \begin{align*}\Delta_{{\mathfrak{t}},1}^*:H_c^{2n}({\mathfrak{t}}\times_{\mathfrak{car}}{\mathfrak{t}},\Delta_*{\overline{\mathbb{Q}}_{\ell}})\rightarrow H_c^{2n}({\mathfrak{t}},{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

which is W-equivariant, hence the theorem.

4.1 Postnikov diagrams associated to T-torsors

Let ${\mathcal {V}}$ be an algebraic stack and $\dot {{\mathcal {V}}}$ a T-torsor $\rho :\dot {{\mathcal {V}}}\rightarrow {\mathcal {V}}$ . We want to compute $\rho _!{\overline {\mathbb {Q}}_{\ell }}$ .

This torsor corresponds to a morphism

$$ \begin{align*}\sigma:{\mathcal{V}}\rightarrow\mathrm{B}(T). \end{align*} $$

We have a Chern class morphism

$$ \begin{align*}c_{\mathrm{can}}:X^*(T)\rightarrow H^2(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})(1) \end{align*} $$

associated to the canonical T-torsor $\mathrm {Spec}(k)\rightarrow \mathrm {B}(T)$ , which induces an isomorphism

$$ \begin{align*}X^*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}\simeq H^2(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})(1). \end{align*} $$

Composed with $\sigma ^*$ , it induces a Chern class morphism

$$ \begin{align*}c_{\rho}:X^*(T)\rightarrow H^2({\mathcal{V}},{\overline{\mathbb{Q}}_{\ell}})(1). \end{align*} $$

Concretely, if $\chi \in X^*(T)$ , then $c_{\rho }(\chi )$ is the Chern class of the line bundle

$$ \begin{align*}(\dot{{\mathcal{V}}}\times\mathbb{A}^1)/T\rightarrow{\mathcal{V}}, \end{align*} $$

where T acts on $\mathbb {A}^1$ as $x\cdot t=\chi (t^{-1})x$ , for $(x,t)\in \mathbb {A}^1\times T$ .

Since $H^2({\mathcal {V}},{\overline {\mathbb {Q}}_{\ell }})={\mathrm {Ext}}^2_{\mathcal {V}}({\overline {\mathbb {Q}}_{\ell }},{\overline {\mathbb {Q}}_{\ell }})=\mathrm {Hom}_{\mathcal {V}}({\overline {\mathbb {Q}}_{\ell }},{\overline {\mathbb {Q}}_{\ell }}[2])$ , we can regard c as a map

$$ \begin{align*}c_{\rho}:{\overline{\mathbb{Q}}_{\ell}}\rightarrow X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[2](1) \end{align*} $$

in $D_{\mathrm {c}}^{\mathrm {b}}({\mathcal {V}})$ .

This defines a complex of objects of $D_{\mathrm {c}}^{\mathrm {b}}({\mathcal {V}})$

$$ \begin{align*}{\overline{\mathbb{Q}}_{\ell}}[-2n](-n)\overset{\partial_n}{\longrightarrow} X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[2-2n](1-n)\overset{\partial_{n-1}}{\longrightarrow}\bigwedge^2X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[4-2n](2-n)\overset{\partial_{n-2}}{\longrightarrow}\cdots \end{align*} $$

(4.2) $$ \begin{align} \cdots\rightarrow\bigwedge^{n-1}X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[-2](-1)\overset{\partial_1}{\longrightarrow}\bigwedge^nX_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}, \end{align} $$

where $\partial _{n-i+1}[2n-2i+2](n-i+1)$ is induced by the morphism

$$ \begin{align*}\bigwedge^{i-1}X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}\longrightarrow\bigwedge^iX_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[2](1) \end{align*} $$

that maps v to $v\wedge c_{\rho }(1)$ .

4.2 Postnikov diagram of ${\mathcal {N}}$

Since the map $\hat {f}:\hat {\mathcal {B}}\rightarrow \hat {{\mathcal {S}}}$ is a small resolution of singularities, we have

$$ \begin{align*}\hat{f}_!{\overline{\mathbb{Q}}_{\ell}}={\mathrm{IC}}_{\hat{{\mathcal{S}}}}. \end{align*} $$

Following §4.1 we now regard the Chern class morphism

$$ \begin{align*}c_{\delta}:X^*(T)\rightarrow H^2(\hat{\mathcal{B}},{\overline{\mathbb{Q}}_{\ell}})(1) \end{align*} $$

associated to the T-torsor $\delta :\mathcal {B}\rightarrow \hat {\mathcal {B}}$ as a morphism

$$ \begin{align*}c_{\delta}:{\overline{\mathbb{Q}}_{\ell}}\rightarrow X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[2](1). \end{align*} $$

Applying $\hat {f}_!$ we get

$$ \begin{align*}\hat{f}_!(c_{\delta}):{\mathrm{IC}}_{\hat{{\mathcal{S}}}}\rightarrow X_*(T)\otimes{\mathrm{IC}}_{\hat{{\mathcal{S}}}}[2](1). \end{align*} $$

Applying the functor $\hat {f}_!$ to the complex $\Lambda _b(\delta _!{\overline {\mathbb {Q}}_{\ell }})$ we obtain a complex

(4.3) $$ \begin{align} &{\mathrm{IC}}_{\hat{{\mathcal{S}}}}[-2n](-n)\overset{\partial_n}{\longrightarrow} X_*(T)\otimes{\mathrm{IC}}_{\hat{{\mathcal{S}}}}[2-2n](1-n)\overset{\partial_{n-1}}{\longrightarrow}\cdots \end{align} $$

$$\begin{align*} &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \cdots \overset{\partial_1}{\longrightarrow}\left(\bigwedge^{n}X_*(T)\right)\otimes{\mathrm{IC}}_{\hat{{\mathcal{S}}}}, \notag \end{align*}$$

where $\partial _{n-i+1}[2n-2i+2](n-i+1)$ is induced by the morphism

$$ \begin{align*}\left(\bigwedge^{i-1}X_*(T)\right)\otimes{\mathrm{IC}}_{\hat{{\mathcal{S}}}}\longrightarrow\left(\bigwedge^iX_*(T)\right)\otimes{\mathrm{IC}}_{\hat{{\mathcal{S}}}}[2](1) \end{align*} $$

given by $v\otimes s\mapsto v\wedge f_!(c_{\delta })(s)$ .

Proof. The assertion (1) follows from the fact that the morphism $\hat {f}$ is small and that the cohomology groups of $\Lambda (\delta _!{\overline {\mathbb {Q}}_{\ell }})$ are constant (see Proposition 4.1(1)). The assertion (2) follows from Lemma 2.3.

To prove assertion (3), we choose a finite subfield $k_o$ of k on which G, T and B are defined, and we let $T_o$ and $\hat {{\mathcal {S}}}_o$ be the induced $k_o$ -structure on T and $\hat {{\mathcal {S}}}$ . We want to prove that $\hat {f}_{o!}\left (\Lambda (\delta _{o!}{\overline {\mathbb {Q}}_{\ell }})\right )$ is the unique Postnikov diagram that completes the complex (4.3) with T and $\hat {{\mathcal {S}}}$ replaced by $T_o$ and $\hat {{\mathcal {S}}}_o$ .

We need to prove that the hypothesis of Lemma 2.5 is satisfied. The condition (2.5) is verified as

$$ \begin{align*}\mathrm{Hom}({\mathrm{IC}}_{\hat{{\mathcal{S}}}_o}[-2j](-j),{\mathrm{IC}}_{\hat{{\mathcal{S}}}_o}[-2i](-i))=\mathrm{Ext}^{2(j-i)}({\mathrm{IC}}_{\hat{{\mathcal{S}}}_o},{\mathrm{IC}}_{\hat{{\mathcal{S}}}_o}(j-i))=0 \end{align*} $$

for all $j<i$ .

The proof of the condition (2.6) reduces to prove that

$$ \begin{align*}\mathrm{Hom}({\mathrm{IC}}_{\hat{{\mathcal{S}}}_o}[-2j](-j),{\mathrm{IC}}_{\hat{{\mathcal{S}}}_o}[-2i-k](-i))=\mathrm{Ext}^{2(j-i)-k}({\mathrm{IC}}_{\hat{{\mathcal{S}}}_o},{\mathrm{IC}}_{\hat{{\mathcal{S}}}_o}(j-i))=0 \end{align*} $$

for all $j>i$ and $k\geq 1$ . But this is clear from the next proposition using the exact sequence (2.8).

Proof. We need to compute

$$ \begin{align*}\mathrm{Ext}^i_{\hat{{\mathcal{S}}}}({\mathrm{IC}}_{\hat{{\mathcal{S}}}},{\mathrm{IC}}_{\hat{{\mathcal{S}}}})=\mathrm{Hom}({\mathrm{IC}}_{\hat{{\mathcal{S}}}},{\mathrm{IC}}_{\hat{{\mathcal{S}}}}[i]). \end{align*} $$

Since $\hat {{\mathcal {S}}}=\mathrm {B}(T)\times {\mathcal {S}}$ and since the statement of the proposition is true if we replace $\hat {S}$ by $\mathrm {B}(T)$ (noticing that ${\mathrm {IC}}_{\mathrm {B}(T)}={\overline {\mathbb {Q}}_{\ell }}$ ), by Künneth formula we are reduced to prove the proposition with ${\mathcal {S}}$ instead of $\hat {{\mathcal {S}}}$ .

Consider the cartesian diagram

Since f is representable, the morphism $\mathrm {pr}_1$ is also representable. We have

$$ \begin{align*} \mathrm{Hom}({\mathrm{IC}}_{\mathcal{S}},{\mathrm{IC}}_{\mathcal{S}}[i])&=\mathrm{Hom}(f_!{\overline{\mathbb{Q}}_{\ell}},f_!{\overline{\mathbb{Q}}_{\ell}}[i])\\ &=\mathrm{Hom}(f^*f_!{\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}}[i])\\ &=\mathrm{Hom}(\mathrm{pr}_{1!}{\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}}[i])\\ &=\mathrm{Hom}({\overline{\mathbb{Q}}_{\ell}},\mathrm{pr}_1^!{\overline{\mathbb{Q}}_{\ell}}[i])\\ &=H^i({\mathcal{Y}},\mathrm{pr}_1^!{\overline{\mathbb{Q}}_{\ell}})\\ &=H_c^{-i}({\mathcal{Y}},{\overline{\mathbb{Q}}_{\ell}}), \end{align*} $$

where the last identity is by Poincaré duality (notice that $\mathrm {dim}\,{\mathcal {Y}}=0$ ). We thus conclude from Proposition 3.3.

4.3 Restriction to nilpotent elements

We let $\mathfrak {z}$ be a locally closed subset of $z(\mathfrak {g})$ . We consider the Lusztig correspondence

over $\mathfrak {z}=\mathfrak {z}/\!/W\subset \mathfrak {car}$ .

The T-torsor $\delta _{\mathfrak {z}}:\mathcal {B}_{\mathfrak {z}}\rightarrow \hat {\mathcal {B}}_{\mathfrak {z}}=\mathrm {B}(T)\times \mathcal {B}_{\mathfrak {z}}$ induced from $\delta $ gives rise to a Chern class morphism

$$ \begin{align*}c_{\delta_{\mathfrak{z}}}:{\overline{\mathbb{Q}}_{\ell}}\rightarrow X_*(T)\otimes{\overline{\mathbb{Q}}_{\ell}}[2](1) \end{align*} $$

and so following the same lines as in §4.2, we end up with a complex

(4.4) $$ \begin{align} &\left(\overline{\mathbb{Q}}_{\ell,\,\mathrm{B}(T)}\boxtimes p_{\mathfrak{z}\,!}{\overline{\mathbb{Q}}_{\ell}}\right)[-2n](-n)\rightarrow X_*(T)\otimes \left(\overline{\mathbb{Q}}_{\ell,\,\mathrm{B}(T)}\boxtimes p_{\mathfrak{z}\,!}{\overline{\mathbb{Q}}_{\ell}}\right)[2-2n](1-n)\rightarrow\cdots \end{align} $$

$$\begin{align*} &\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \cdots\longrightarrow\left(\bigwedge^nX_*(T)\right)\otimes \left(\overline{\mathbb{Q}}_{\ell,\,\mathrm{B}(T)}\boxtimes p_{\mathfrak{z}\,!}{\overline{\mathbb{Q}}_{\ell}}\right).\notag \end{align*}$$

which is the restriction of the complex (4.3) to $\hat {{\mathcal {S}}}_{\mathfrak {z}}$ .

If $\mathfrak {z}=0$ , we will replace the subscript $\mathfrak {z}$ by nil.

Let $\mathrm {pr}_{\mathrm {nil}}:\hat {{\mathcal {S}}}_{\mathfrak {z}}\rightarrow \hat {{\mathcal {S}}}_{\mathrm { nil}}=\mathrm {B}(T)\times \mathcal {G}_{\mathrm {nil}}$ be the morphism induced by the projection $\mathfrak {z}+{\mathfrak {g}}_{\mathrm {nil}}\rightarrow {\mathfrak {g}}_{\mathrm {nil}}$ .

5.1 The main theorem

We have a natural stratification of $\mathfrak {car}$ into smooth locally closed subsets indexed by the set $\mathfrak {L}$ of G-conjugacy classes of Levi subgroups (of parabolic subgroups) of G

$$ \begin{align*}\mathfrak{car}=\coprod_{\lambda\in\mathfrak{L}}\mathfrak{car}_{\lambda}. \end{align*} $$

Concretely, let L be a representative of $\lambda \in \mathfrak {L}$ containing T. Then $\mathfrak {car}_{\lambda }$ is the image of

$$ \begin{align*}z(\mathfrak{l})^o:=\{x\in \mathfrak{l}\,|\,C_G(x)=L\}\subset z(\mathfrak{l})\subset {\mathfrak{t}} \end{align*} $$

by the quotient map ${\mathfrak {t}}\rightarrow \mathfrak {car}$ .

If $\mathcal {X}$ is either $\mathcal {T},\mathcal {B},\hat {\mathcal {B}}, {\mathcal {S}}, \hat {{\mathcal {S}}}, {\overline {\mathcal {T}}} $ or $\mathcal {G}$ we put

$$ \begin{align*}\mathcal{X}_{\lambda}:=\mathcal{X}\times_{\mathfrak{car}}\mathfrak{car}_{\lambda} \end{align*} $$

for any $\lambda \in \mathfrak {L}$ .

We consider the Lusztig correspondence over $\lambda \in \mathfrak {L}$

Notice that $\mathcal {T}_{\lambda }=[{\mathfrak {t}}_{\lambda }/T]\simeq {\mathfrak {t}}_{\lambda }\times \mathrm {B}(T)$ with

$$ \begin{align*}{\mathfrak{t}}_{\lambda}=\coprod_{g\in N_G(T)/N_G(L,T)}g\,z(\mathfrak{l})^og^{-1}. \end{align*} $$

Notice that if for some $g\in G$ and $s\in z(\mathfrak {l})^o$ we have $\mathrm {Ad}(g)(s)\in z(\mathfrak {l})^o$ , then $g\in N_G(L)$ [Reference Letellier11, Lemma 2.6.16].

We thus have isomorphisms of stacks

$$ \begin{align*}\mathcal{G}_{\lambda}\simeq [(z(\mathfrak{l})^o+\mathfrak{l}_{\mathrm{nil}})/N_G(L)],\hspace{1cm}\overline{\mathcal{T}}_{\lambda}=[{\mathfrak{t}}_{\lambda}/N]\simeq [z(\mathfrak{l})^o/N_G(L,T)]. \end{align*} $$

Put $B^L:=B\cap L$ , and let ${\mathfrak {u}}^L$ be the Lie algebra of the unipotent radical of $B^L$ . We consider the following commutative diagram (as in §4.3 with $G=L$ and $\mathfrak {z}=z(\mathfrak {l})^o$ ):

(5.1)

Put

$$ \begin{align*}W^L:=N_L(T)/T,\hspace{1cm}\widehat{W}^L:=N_G(L,T)/T,\hspace{1cm}W_L:=N_G(L)/L \end{align*} $$

We have an exact sequence

$$ \begin{align*}1\rightarrow W^L\rightarrow \widehat{W}^L\rightarrow W_L\rightarrow 1 \end{align*} $$

and the group $\widehat {W}^L$ acts on

$$ \begin{align*}[z(\mathfrak{l})^o/T]\times_{z(\mathfrak{l})^o} [(z(\mathfrak{l})^o+\mathfrak{l}_{\mathrm{nil}})/L] \end{align*} $$

in the natural way on the first factor and via $W_L$ on the second factor.

The first two vertical arrows of the diagram (5.1) are $\widehat {W}^L$ -torsors while the right vertical arrow is $W_L$ -torsor.

Remark 5.2. We have the following commutative diagram

where the top square is cartesian. From this diagram we see that

$$ \begin{align*}{\overline{{\mathcal{N}}}}_{\lambda}|_{\mathcal{T}_{\lambda}\times_{\mathfrak{car}}\mathcal{G}_{\lambda}}\simeq{\mathcal{N}}|_{\mathcal{T}_{\lambda}\times_{\mathfrak{car}}\mathcal{G}_{\lambda}}=(q_{\lambda},p_{\lambda})_!{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

where ${\mathcal {N}}$ is as in §4. In other words, the complex ${\mathcal {N}}_{\lambda }:={\mathcal {N}}|_{\mathcal {T}_{\lambda }\times _{\mathfrak {car}}\mathcal {G}_{\lambda }}$ is W-equivariant.

5.2 Proof of Theorem 5.1

The essential case is when $L=G$ which case reduces to the nilpotent elements (see §4.3). We thus consider the following diagram:

and we want to prove that the complex $(q_{\mathrm {nil}},p_{\mathrm {nil}})_!{\overline {\mathbb {Q}}_{\ell }}$ is W-equivariant and so descends to $\mathrm {B}(N)\times \mathcal {G}_{\mathrm {nil}}$ .

Consider now the diagram

Notice that the complex $(q_{\mathrm {nil}},p_{\mathrm {nil}})_!{\overline {\mathbb {Q}}_{\ell }}$ is the restriction of $(\tau ,p)_!{\overline {\mathbb {Q}}_{\ell }}$ to $\mathrm {B}(T)\times \mathcal {G}_{\mathrm {nil}}$ .

It is thus enough to prove that $(\tau ,p)_!{\overline {\mathbb {Q}}_{\ell }}$ is W-equivariant.

The map $(\tau ,p)$ decomposes as

Following the strategy of §4.1, we see that the complex $(\tau ,p)_!{\overline {\mathbb {Q}}_{\ell }}$ is the outcome of the Postnikov diagram $\hat {p}_!\Lambda (\delta _!{\overline {\mathbb {Q}}_{\ell }})$ . The base of this Postnikov diagram is the complex

(5.2) $$ \begin{align} &{\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}})[-2n](-n)\overset{\partial_n}{\longrightarrow} X_*(T)\otimes{\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}})[2-2n](1-n)\overset{\partial_{n-1}}{\longrightarrow}\cdots \end{align} $$

$$\begin{align*} &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \cdots\overset{\partial_1}{\longrightarrow} \left(\bigwedge^nX_*(T)\right)\otimes{\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}}), \notag \end{align*}$$

where $\hat {\mathcal {L}}$ is the semisimple local system ${\overline {\mathbb {Q}}_{\ell }}\boxtimes \mathcal {L}$ with $\mathcal {L}=p_{{\mathrm {rss}}\, !}{\overline {\mathbb {Q}}_{\ell }}$ where $p_{\mathrm {rss}}:\mathcal {B}_{\mathrm {rss}}\rightarrow \mathcal {G}_{\mathrm {rss}}$ is the restriction of p to semisimple regular elements.

We consider on vertices of (5.2) the W-action given by the Springer action on ${\mathrm {IC}}_{\mathcal {G}}(\mathcal {L})=p_!{\overline {\mathbb {Q}}_{\ell }}$ and the obvious one on $X_*(T)$ .

Theorem 5.3 will be a consequence of Theorem 5.6 below.

Corollary 5.4. The complex (5.2) descends to a unique complex $\overline {\Lambda }_b$ in $D_{\mathrm {c}}^{\mathrm {b}}(\hat {\mathcal {G}})$ whose vertices are

$$ \begin{align*}\left(\bigwedge^{2n-i}X_*(T)\right)\otimes\left(\overline{\mathbb{Q}}_{\ell,\, \mathrm{B}(N)}\boxtimes {\mathrm{IC}}_{\mathcal{G}}(\mathcal{L})\right)[2n-2i](n-i). \end{align*} $$

If $k_o$ is a finite subfield of k on which $G, T$ and B are defined, then $\overline {\Lambda }_b$ is naturally defined over $k_o$ .

Theorem 5.5. The complex $\overline {\Lambda }_b$ of Corollary 5.4 can be completed into a unique Postnikov diagram $\overline {\Lambda }$ (up to a unique isomorphism) such that for any finite subfield $k_o$ of k on which $G,T$ and B are defined, the diagram $\overline {\Lambda }$ is also defined over $k_o$ with $\overline {\Lambda }_b$ equipped with its natural $k_o$ -structure.

We have

$$ \begin{align*}\overline{\Lambda}|_{\hat{\mathcal{G}}}\simeq\hat{p}_!\Lambda(\delta_!{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

In particular, the outcome $(\tau ,p)_!{\overline {\mathbb {Q}}_{\ell }}$ of $\hat {p}_!\Lambda (\delta _!{\overline {\mathbb {Q}}_{\ell }})$ descends naturally to $\mathrm {B}(N)\times \mathcal {G}$ .

As explained in §2.4, the group W acts on

$$ \begin{align*}{\mathrm{Ext}}^i_{\hat{\mathcal{G}}} \left({\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}}),{\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}})\right)=\mathrm{Hom}\left({\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}}), {\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}})[i]\right). \\[-15pt]\end{align*} $$

Theorem 5.6. The map

$$ \begin{align*}X^*(T)\overset{c_{\delta}}{\longrightarrow} H^2(\hat{\mathcal{B}},{\overline{\mathbb{Q}}_{\ell}})(1)={\mathrm{Ext}}^2_{\hat{\mathcal{B}}}({\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}})(1)\longrightarrow{\mathrm{Ext}}^2_{\hat{\mathcal{G}}}\left({\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}}),{\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}})\right)(1),\\[-15pt] \end{align*} $$

where the first arrow is the Chern class morphism of $\delta $ and the second arrow is given by $\hat {p}_!$ , is W-equivariant.

This is apriori not obvious as the natural action of W on $\hat {\mathcal {G}}$ does not lift to $\hat {\mathcal {B}}$ .

We have the following proposition.

Proposition 5.7. The Chern class morphism

$$ \begin{align*}c_{\delta}:X^*(T)\rightarrow H^2(\hat{\mathcal{B}},{\overline{\mathbb{Q}}_{\ell}})(1)\\[-15pt] \end{align*} $$

of the T-torsor $\delta $ is W-equivariant.

Proof. As $\hat {\mathcal {B}}=\mathrm {B}(T)\times \mathcal {B}$ , by Künneth formula we have

$$ \begin{align*}H^2(\hat{\mathcal{B}},{\overline{\mathbb{Q}}_{\ell}})(1)= H^2(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}})(1)\oplus H^2(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})(1). \end{align*} $$

Then

$$ \begin{align*}c_{\delta}=c_{\pi}\oplus -c_{\mathrm{can}},\\[-15pt] \end{align*} $$

where $\pi $ is the T-torsor $\dot {\mathcal {B}}\rightarrow \mathcal {B}$ . The Chern class morphism $c_{\mathrm {can}}$ of the trivial T-torsor $\mathrm {Spec}(k)\rightarrow \mathrm {B}(T)$ is W-equivariant.

We thus need to prove that $c_{\pi }$ is W-equivariant.

The Chern class morphism $c_{\pi }$ decomposes as

$$ \begin{align*}X^*(T)\rightarrow H^2(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})\simeq H^2(\mathrm{B}(B),{\overline{\mathbb{Q}}_{\ell}})\rightarrow H^2(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}}),\\[-15pt] \end{align*} $$

where the first map is $c_{\mathrm {can}}$ and the second map is the cohomological restriction of the canonical map $\mathcal {B}\rightarrow \mathrm {B}(B)$ . It is W-equivariant by Lemma 2.23.

Theorem 5.6 is a consequence of Proposition 5.7 together with the following one.

Proposition 5.8. The map

(5.3) $$ \begin{align} H^2(\hat{\mathcal{B}},{\overline{\mathbb{Q}}_{\ell}})(1)={\mathrm{Ext}}^2_{\hat{\mathcal{B}}}({\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}})(1)\rightarrow{\mathrm{Ext}}^2_{\hat{\mathcal{G}}}\left({\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}}),{\mathrm{IC}}_{\hat{\mathcal{G}}}(\hat{\mathcal{L}})\right)(1)\\[-15pt]\nonumber \end{align} $$

induced by $\hat {p}_!$ is W-equivariant.

Proof. By Künneth formula, we are reduced to prove that the same statement is true with $\hat {\mathcal {B}}$ and $\hat {\mathcal {G}}$ replaced by $\mathcal {B}$ and $\mathcal {G}$ . That is, we need to see that the map

(5.4) $$ \begin{align} H^2(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}})(1)={\mathrm{Ext}}^2_{\mathcal{B}}({\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}})(1)\rightarrow{\mathrm{Ext}}^2_{\mathcal{G}}\left({\mathrm{IC}}_{\mathcal{G}}(\mathcal{L}),{\mathrm{IC}}_{\mathcal{G}}(\mathcal{L})\right)(1) \end{align} $$

given by the functor $p_!$ is W-equivariant.

Consider the cartesian diagram

We have

$$ \begin{align*} {\mathrm{Ext}}^i_{\mathcal{G}}({\mathrm{IC}}_{\mathcal{G}}(\mathcal{L}),{\mathrm{IC}}_{\mathcal{G}}(\mathcal{L}))&=\mathrm{ Hom}(p_!{\overline{\mathbb{Q}}_{\ell}},p_!{\overline{\mathbb{Q}}_{\ell}}[i])\\ &=\mathrm{Hom}(p^*p_!{\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}}[i])\\ &=H^i(\mathcal{Z},\mathrm{pr}_1^!{\overline{\mathbb{Q}}_{\ell}})\\ &=H^{-i}_c(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

The dual of the map (5.4) is the cohomological restriction

$$ \begin{align*}H_c^{-2}(\mathcal{Z},{\overline{\mathbb{Q}}_{\ell}})\longrightarrow H_c^{-2}(\mathcal{B},{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

of the diagonal morphism $\mathcal {B}\rightarrow \mathcal {Z}$ . The proposition is thus a consequence of Theorem 3.4.

5.3 Descent over regular elements

We consider the diagram (4.1) over regular elements

and put

$$ \begin{align*}{\mathcal{N}}_{\mathrm{reg}}:=(q,p)_{{\mathrm{reg}}\, !}{\overline{\mathbb{Q}}_{\ell}}={\mathcal{N}}|_{\hat{{\mathcal{S}}}_{\mathrm{reg}}}. \end{align*} $$

Proof. Following the strategy of §5.2, we need to prove that the base $\Lambda _b({\mathcal {N}}_{\mathrm {reg}})$ of the Postnikov diagram $\Lambda ({\mathcal {N}}_{\mathrm {reg}})$ is W-equivariant. With the map $\hat {f}_{\mathrm {reg}}$ being an isomorphism, the vertices of $\Lambda _b({\mathcal {N}}_{\mathrm {reg}})$ are constant sheaves (up to a shift) and so are clearly W-equivariant. To prove that the arrows are W-equivariant, we need to prove (analogously to Theorem 5.6) that the map

$$ \begin{align*}X^*(T)\longrightarrow H^2(\hat{\mathcal{B}}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}})(1)\longrightarrow{\mathrm{Ext}}^2(\hat{f}_{{\mathrm{reg}}\, !}{\overline{\mathbb{Q}}_{\ell}},\hat{f}_{{\mathrm{reg}}\, !}{\overline{\mathbb{Q}}_{\ell}})(1) \end{align*} $$

is W-equivariant. The first map being W-equivariant (see Proposition 5.7), we need to see that the second map too. However, notice that $\hat {f}_{\mathrm {reg}}$ is an isomorphism, and so the second map is an isomorphism

$$ \begin{align*}H^2(\hat{\mathcal{B}}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}})\longrightarrow H^2(\hat{{\mathcal{S}}}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

which is dual to the restriction morphism

$$ \begin{align*}H^{-2}_c(\hat{{\mathcal{S}}}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}})\longrightarrow H^{-2}_c(\hat{\mathcal{B}}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}}). \end{align*} $$

By Künneth formula, we are reduced to prove that the restriction map (which is an isomorphism)

$$ \begin{align*}H_c^{-2}({\mathcal{S}}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}})\longrightarrow H^{-2}_c(\mathcal{B}_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}})\\[-15pt] \end{align*} $$

is W-equivariant. Notice that $H_c^{-2}({\mathcal {S}}_{\mathrm {reg}},{\overline {\mathbb {Q}}_{\ell }})$ and $H^{-2}_c(\mathcal {B}_{\mathrm {reg}},{\overline {\mathbb {Q}}_{\ell }})$ are both the hypercohomology of the proper pushforwards of ${\overline {\mathbb {Q}}_{\ell }}$ along the maps $\mathrm {pr}_2:{\mathcal {S}}_{\mathrm {reg}}\rightarrow \mathcal {G}_{\mathrm {reg}}$ and $p_{\mathrm {reg}}:\mathcal {B}_{\mathrm {reg}}\rightarrow \mathcal {G}_{\mathrm {reg}}$ , respectively, and these pushforwards are the intermediate extensions of the pushforwards over semisimple regular elements. The result follows from the fact that W acts on $\mathcal {B}_{\mathrm {rss}}$ and that the isomorphism $\mathcal {B}_{\mathrm {rss}}\simeq {\mathcal {S}}_{\mathrm {rss}}$ is W-equivariant.

5.4 “Un calcul triste”

In this section we give an example where the kernel ${\mathcal {N}}$ is not W-equivariant.

For $w\in W$ , put $\mathcal {B}_w:=[\mathrm {Ad}(w){\mathfrak {b}}/wBw^{-1}]$ . We then have a cartesian diagram

from which we get a commutative diagram (base change)

where the horizontal arrows are given by the functors $(q^{\prime }_w,p_w)_!$ and $(q,p)_!$ , respectively. Identifying $H^2(\mathcal {B},{\overline {\mathbb {Q}}_{\ell }})$ and $H^2(\mathcal {B}_w,{\overline {\mathbb {Q}}_{\ell }})$ with $IH^2({\mathcal {S}},{\overline {\mathbb {Q}}_{\ell }})$ on one hand, and $(q',p)_!{\overline {\mathbb {Q}}_{\ell }}$ and $(q^{\prime }_w,p_w)_!{\overline {\mathbb {Q}}_{\ell }}$ with ${\mathrm {IC}}_{\mathcal {S}}$ on the other hand, we end up with a commutative diagram

Similarly to §5.2, we see that that the arrows of the diagram $\Lambda _b({\mathcal {N}})$ are W-equivariant if and only if $a=a_w$ for all $w\in W$ .

Note that

$$ \begin{align*} &S:={\mathfrak{t}}\times_{\mathfrak{car}}{\mathfrak{g}}=\{(x,t)\in {\mathfrak{sl}}_2\times\mathbb{A}^1\,|\, \det(x)=-t^2\}\\ &X_+=\{(x,t,D)\in{\mathfrak{sl}}_2\times\mathbb{A}^1\times\mathbb{P}^1\,|\, (x,t)\in S, D\subset\mathrm{ Ker}(x-t\mathrm{Id})\}\\[-15pt] \end{align*} $$

and denote by $\rho _+:X_+\rightarrow S$ the map $(x,t,D)\mapsto (x,t)$ . We consider also the analogous map $\rho _-:X_-\rightarrow S$ , where we replace $\mathrm {Ker}(x-t\mathrm {Id})$ in the definition of $X_+$ by $\mathrm { Ker}(x+t\mathrm {Id})$ .

Note that if B denotes the Borel subgroup of upper triangular matrices, then

$$ \begin{align*}X_+\simeq \{(x,t,gB)\in{\mathfrak{sl}}_2\times\mathbb{A}^1\times \mathrm{SL}_2/B\,|\, g^{-1}xg\in(t,-t)+{\mathfrak{u}}\} \end{align*} $$

and $X_-$ corresponds to choosing the opposite Borel subgroup of lower triangular matrices.

We have compactifications

$$ \begin{align*} &\overline{S}:=\{(X;T;Z)\in\mathbb{P}^4\,|\, \det(X)=-T^2\}\\ &\overline{X}_+:=\{((X;T;Z),D)\in\overline{S}\times\mathbb{P}^1\,|\, D\subset\mathrm{Ker}(X-T\mathrm{Id})\}\\ &\overline{X}_-:=\{((X;T;Z),D)\in\overline{S}\times\mathbb{P}^1\,|\, D\subset\mathrm{Ker}(X+T\mathrm{Id})\}\ \end{align*} $$

where we identify ${\mathfrak {sl}}_2$ with $\mathbb {A}^3$ in the obvious way, and the small resolutions $\overline {\rho }_+:\overline {X}_+\rightarrow \overline {S}$ and $\overline {\rho }_-:\overline {X}_-\rightarrow \overline {S}$ .

Identifying $IH^2(\overline {S},{\overline {\mathbb {Q}}_{\ell }})$ with $H^2(\overline {X}_+,{\overline {\mathbb {Q}}_{\ell }})$ (resp. with $H^2(\overline {X}_-,{\overline {\mathbb {Q}}_{\ell }})$ ) and ${\mathrm {IC}}_{\overline {S}}$ with $\overline {\rho }_{+*}{\overline {\mathbb {Q}}_{\ell }}$ (resp. $\overline {\rho }_{-\, *}{\overline {\mathbb {Q}}_{\ell }}$ ) we get a map

$$ \begin{align*}\overline{a}_+:IH^2(\overline{S},{\overline{\mathbb{Q}}_{\ell}})\rightarrow{\mathrm{Ext}}^2\left({\mathrm{IC}}_{\overline{S}},{\mathrm{IC}}_{\overline{S}}\right)\end{align*} $$

(resp. we get a map $\overline {a}_-:IH^2(\overline {S},{\overline {\mathbb {Q}}_{\ell }})\rightarrow {\mathrm {Ext}}^2\left ({\mathrm {IC}}_{\overline {S}},{\mathrm {IC}}_{\overline {S}}\right )$ ).

If we compose $\overline {a}_+$ (resp. $\overline {a}_-$ ) with the natural map

$$ \begin{align*}{\mathrm{Ext}}^2\left({\mathrm{IC}}_{\overline{S}},{\mathrm{IC}}_{\overline{S}}\right)\rightarrow\mathrm{Hom}\left(IH^2(\overline{S},{\overline{\mathbb{Q}}_{\ell}}),IH^4(\overline{S},{\overline{\mathbb{Q}}_{\ell}})\right) \end{align*} $$

we get the cup-product

$$ \begin{align*}IH^2(\overline{S},{\overline{\mathbb{Q}}_{\ell}})\otimes IH^2(\overline{S},{\overline{\mathbb{Q}}_{\ell}})\overset{\smile_+}{\longrightarrow} IH^4(\overline{S},{\overline{\mathbb{Q}}_{\ell}}) \end{align*} $$

induced by the one on $H^*(\overline {X}_+,{\overline {\mathbb {Q}}_{\ell }})$ (resp. we get the cup-product $\smile _-$ induced by the one on $H^*(\overline {X}_-,{\overline {\mathbb {Q}}_{\ell }})$ ).

We have the following result [Reference Verdier20].

Proposition 5.12 (Verdier).

The two cup-products

are not the same.

Proof. For the convenience of the reader, we outline the proof.

Consider the following commutative diagram

Consider the divisors

$$ \begin{align*} &D_{1,+}:=\mathrm{pr}_+\left(f^{-1}(\{\infty\}\times\mathbb{P}^1)\right)=f_+^{-1}(\infty),\hspace{.5cm}D_{2,+}:=\mathrm{pr}_+\left(f^{-1}(\mathbb{P}^1\times\{\infty\})\right)\\ &D_{1,-}:=\mathrm{pr}_-\left(f^{-1}(\mathbb{P}^1\times\{\infty\})\right)=f_-^{-1}(\infty),\hspace{.5cm}D_{2,-}:=\mathrm{pr}_-\left(f^{-1}(\{\infty\}\times\mathbb{P}^1)\right)\\[-15pt] \end{align*} $$

Under the natural identifications

$$ \begin{align*}H^2(\overline{X}_+,{\overline{\mathbb{Q}}_{\ell}})\simeq IH^2(\overline{S},{\overline{\mathbb{Q}}_{\ell}})\simeq H^2(\overline{X}_-,{\overline{\mathbb{Q}}_{\ell}})\\[-15pt] \end{align*} $$

the class $\mathrm {cl}(D_{1,+/-})\in H^2(\overline {X}_{+/-},{\overline {\mathbb {Q}}_{\ell }})$ corresponds to $\mathrm { cl}(D_{2,-/+})\in H^2(\overline {X}_{-/+},{\overline {\mathbb {Q}}_{\ell }})$ , and so the proposition follows from the calculations

$$ \begin{align*}\mathrm{cl}(D_{1,+/-})\smile_{+/-}\mathrm{cl}(D_{1,+/-})=0,\hspace{.5cm}\mathrm{ cl}(D_{2,+/-})\smile_{+/-}\mathrm{cl}(D_{2,+/-})\neq 0.\\[-36pt] \end{align*} $$

As a consequence of the proposition, we get that the two maps $\overline {a}_+$ and $\overline {a}_-$ are different.

Let us now deduce from this that $a_+$ and $a_-$ are also different.

Put

$$ \begin{align*}K:=\underline{\mathrm{Hom}}({\mathrm{IC}}_{\overline{S}},{\mathrm{IC}}_{\overline{S}}).\\[-15pt] \end{align*} $$

Notice that

$$ \begin{align*}K|_S=\underline{\mathrm{Hom}}({\mathrm{IC}}_S,{\mathrm{IC}}_S).\\[-15pt] \end{align*} $$

Put

$$ \begin{align*} &\overline{\sigma}:=\overline{a}_+-\overline{a}_-:IH^2(\overline{S},{\overline{\mathbb{Q}}_{\ell}})\rightarrow {\mathrm{Ext}}^2({\mathrm{IC}}_{\overline{S}},{\mathrm{IC}}_{\overline{S}})=H^2(\overline{S},K)\\ &\sigma:=a_+-a_-:IH^2(S,{\overline{\mathbb{Q}}_{\ell}})\rightarrow {\mathrm{Ext}}^2({\mathrm{IC}}_S,{\mathrm{IC}}_S)=H^2(S,K|_S).\\[-15pt] \end{align*} $$

We have the following commutative diagram:

where the vertical arrows are the restriction maps.

Let $\overline {\alpha }\in IH^2(\overline {S},{\overline {\mathbb {Q}}_{\ell }})$ be such that $\overline {\sigma }(\overline {\alpha })\neq 0$ , and denote by $\alpha \in IH^2(S,{\overline {\mathbb {Q}}_{\ell }})$ the image of $\overline {\alpha }$ .

Proposition 5.13. We have

$$ \begin{align*}\sigma(\alpha)\neq 0\\[-15pt] \end{align*} $$

and so $a_+\neq a_-$ .

Proof. We need to prove that

(5.5) $$ \begin{align} \varphi_S\left(\overline{\sigma}(\overline{\alpha})\right)\neq 0. \end{align} $$

We have the open covering

$$ \begin{align*}\overline{S}=\overline{S}_{\mathrm{reg}}\cup S \end{align*} $$

where $\overline {S}_{\mathrm {reg}}$ is the smooth open subset of elements $(X;T;Z)$ in $\overline {S}$ such that $X\neq 0$ .

Notice that $S_{\mathrm {reg}}=\overline {S}_{\mathrm {reg}}\cap S$ , and so we have the exact sequence (Mayer-Vietoris)

$$ \begin{align*}\cdots\rightarrow H^1(S_{\mathrm{reg}},K|_{S_{\mathrm{reg}}})\rightarrow H^2(\overline{S},K)\overset{\varphi}{\longrightarrow} H^2(\overline{S}_{\mathrm{reg}},K|_{\overline{S}_{\mathrm{reg}}})\oplus H^2(S,K|_S)\rightarrow\cdots \end{align*} $$

Since $S_{\mathrm {reg}}$ and $\overline {S}_{\mathrm {reg}}$ are smooth open subsets of S and $\overline {S}$ , respectively, we have

$$ \begin{align*}K|_{S_{\mathrm{reg}}}=\underline{\mathrm{Hom}}_{S_{\mathrm{reg}}}({\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}})={\overline{\mathbb{Q}}_{\ell}},\hspace{.5cm}\text{ and }\hspace{.5cm} K|_{\overline{S}_{\mathrm{reg}}}={\overline{\mathbb{Q}}_{\ell}}. \end{align*} $$

Notice also that

$$ \begin{align*}H^1(S_{\mathrm{reg}},{\overline{\mathbb{Q}}_{\ell}})=0 \end{align*} $$

as $\rho |_{X_{\mathrm {reg}}}:X_{\mathrm {reg}}\simeq S_{\mathrm {reg}}$ and $H^1(X_{\mathrm {reg}},{\overline {\mathbb {Q}}_{\ell }})\simeq H^1(X,{\overline {\mathbb {Q}}_{\ell }})=0$ .

The map $\varphi $ is thus injective, and so to prove (5.5), we are reduced to prove that

$$ \begin{align*}\varphi_{\overline{S}_{\mathrm{reg}}}(\overline{\sigma}(\overline{\alpha}))= 0 \end{align*} $$

where $\varphi _{\overline {S}_{\mathrm {reg}}}:H^2(\overline {S},K)\rightarrow H^2(\overline {S}_{\mathrm {reg}},{\overline {\mathbb {Q}}_{\ell }})$ is the restriction map.

As the restrictions of $\overline {\rho }_+$ and $\overline {\rho }_-$ over regular elements induce isomorphisms $\overline {X}_{+,{\mathrm {reg}}}\simeq \overline {S}_{\mathrm {reg}}$ and $\overline {X}_{-,{\mathrm {reg}}}\simeq \overline {S}_{\mathrm {reg}}$ we have the following commutative diagram:

and the analogous diagram with $\overline {X}_-$ instead of $\overline {X}_+$ . From these two commutative diagrams we deduce that

$$ \begin{align*}\varphi_{\overline{S}_{\mathrm{reg}}}\circ \overline{\sigma}=0.\\[-36pt] \end{align*} $$

6.1 Definition

We consider the cohomological correspondence where $\mathrm {pr}_i$ is the i-th projection

with kernel ${\overline {{\mathcal {N}}}}_{\lambda }$ .

We define the associated restriction and induction functors

$$ \begin{align*} &\mathrm{R}_{\lambda}:D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{G}_{\lambda})\rightarrow D_{\mathrm{c}}^{\mathrm{b}}({\overline{\mathcal{T}}}_{\lambda}),\hspace{.5cm}K\mapsto\mathrm{pr}_{1!}\left(\mathrm{pr}_2^*(K)\otimes{\overline{{\mathcal{N}}}}_{\lambda}\right),\\ &\mathrm{I}_{\lambda}:D_{\mathrm{c}}^{\mathrm{b}}({\overline{\mathcal{T}}}_{\lambda})\rightarrow D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{G}_{\lambda}),\hspace{.5cm}K\mapsto \mathrm{pr}_{2*}\,\underline{\mathrm{ Hom}}\left({\overline{{\mathcal{N}}}}_{\lambda},\mathrm{pr}_1^!(K)\right) \end{align*} $$

as in §2.5.

If we denote $\mathrm {Ind}_{\lambda }$ and $\mathrm {Res}_{\lambda }$ the induction and restriction functors associated to the correspondence

with kernel ${\mathcal {N}}_{\lambda }={\mathcal {N}}|_{\mathcal {T}_{\lambda }\times _{\mathfrak {car}}\mathcal {G}_{\lambda }}$ , then by Remark 5.2,

$$ \begin{align*}\mathrm{Ind}_{\lambda}=\mathrm{I}_{\lambda}\circ\pi_{\lambda\, *},\hspace{1cm}\mathrm{Res}_{\lambda}=\pi_{\lambda}^*\circ\mathrm{R}_{\lambda}.\\[-15pt] \end{align*} $$

Since $q_{\lambda }^!=q_{\lambda }^*$ and $p_{\lambda \,!}=p_{\lambda \, *}$ , we see that $\mathrm {Ind}_{\lambda }$ decomposes also as the composition of $\pi _{\lambda \, !}$ followed by the functor

$$ \begin{align*}D_{\mathrm{c}}^{\mathrm{b}}({\overline{\mathcal{T}}}_{\lambda})\rightarrow D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{G}_{\lambda}),\hspace{.5cm}K\mapsto \mathrm{pr}_{2!}\left(\mathrm{pr}_1^*(K)\otimes{\overline{{\mathcal{N}}}}_{\lambda}\right)\\[-15pt] \end{align*} $$

and so by Remark 2.17, we have

(6.1) $$ \begin{align} \mathrm{I}_{\lambda}(K)=\mathrm{pr}_{2!}\left(\mathrm{pr}_1^*(K)\otimes{\overline{{\mathcal{N}}}}_{\lambda}\right).\\[-15pt]\nonumber \end{align} $$

As in §2.9, we define a pair of adjoint functors $({^p}\mathrm {R}_{\lambda },{^p}\mathrm {I}_{\lambda })$

where $\overline {{\mathfrak {t}}}_{\lambda }:=[{\mathfrak {t}}_{\lambda }/W]$ . We let $\overline {s}_{\lambda }:{\overline {\mathcal {T}}}_{\lambda }\rightarrow \overline {{\mathfrak {t}}}_{\lambda }$ be the quotient map of the identity map ${\mathfrak {t}}_{\lambda }\rightarrow {\mathfrak {t}}_{\lambda }$ by $N\rightarrow W$ .

Lemma 6.1. We have

$$ \begin{align*}{^p}\mathrm{I}_{\lambda}=\mathrm{I}_{\lambda}\circ \overline{s}_{\lambda}^,\hspace{.5cm}{^p}\mathrm{R}_{\lambda}={^p}{\mathcal{H}}^0\circ \overline{s}_{\lambda\, !}\circ \mathrm{R}_{\lambda} [-n](-n). \end{align*} $$

Proof. Since the functors $\overline {s}_{\lambda }^$ and ${^p}{\mathcal {H}}^0\circ \overline {s}_{\lambda \, !}[-n](-n)$ between $\mathcal {M}(\overline {{\mathfrak {t}}}_{\lambda })$ and $\mathcal {M}({\overline {\mathcal {T}}}_{\lambda })$ are inverse to each other, by adjunction we deduce the second equality from the first one.

Analogously to the definition ${^p}\underline {\mathrm {Ind}}$ (see §2.9) we define ${^p}\underline {\mathrm {Ind}}_{\lambda }$ by

$$ \begin{align*}{^p}\underline{\mathrm{Ind}}_{\lambda}:=\mathrm{Ind}_{\lambda}\circ s_{\lambda}^={^p}\mathrm{I}_{\lambda}\circ \pi_{{\mathfrak{t}}_{\lambda}\, !}.\\[-15pt] \end{align*} $$

where

By Remark 2.17, to prove the first equality, we need to see that

$$ \begin{align*}{^p}\underline{\mathrm{Ind}}_{\lambda}=\mathrm{I}_{\lambda}\circ \overline{\mathrm{pr}}_{{\mathfrak{t}}_{\lambda}}^\circ \pi_{{\mathfrak{t}}_{\lambda}\, !} \end{align*} $$

Using that $\mathrm {Ind}_{\lambda }=\mathrm {I}_{\lambda }\circ \pi _{\lambda \, !}$ , we are reduced to prove that

$$ \begin{align*}\overline{s}_{\lambda}^\circ \pi_{{\mathfrak{t}}_{\lambda}\, !}=\pi_{\lambda\, !}\circ s_{\lambda}^. \end{align*} $$

But this follows from the base change theorem as $\overline {s}_{\lambda }$ and $s_{\lambda }$ are smooth and of the same relative dimension.

6.2 Semisimple regular elements

Let us see that $\mathrm {I}_{\lambda }$ and $\mathrm {R}_{\lambda }$ are the identity functors when $\lambda $ represents semisimple regular elements (i.e., when $\lambda $ is the conjugacy class of maximal tori), in which case we use the subscript ${\mathrm {rss}}$ instead of $\lambda $ in the notation.

Recall that

$$ \begin{align*}U\times{\mathfrak{t}}_{{\mathrm{reg}}}\longrightarrow{\mathfrak{b}}_{{\mathrm{rss}}},\hspace{.5cm}(u,t)\mapsto \mathrm{Ad}(u)(t) \end{align*} $$

is an isomorphism and so, taking the quotient by B on both sides, we end up with an isomorphism

$$ \begin{align*}\mathcal{T}_{{\mathrm{reg}}}\overset{\sim}{\longrightarrow}\mathcal{B}_{{\mathrm{rss}}}. \end{align*} $$

Moreover, the natural injection ${\mathfrak {t}}_{{\mathrm {reg}}}\rightarrow {\mathfrak {g}}_{{\mathrm {rss}}}$ induces an isomorphism

$$ \begin{align*}{\overline{\mathcal{T}}}_{\!{\mathrm{reg}}}\simeq\mathcal{G}_{{\mathrm{rss}}}. \end{align*} $$

Therefore, the morphism $(q_{\mathrm {rss}},p_{\mathrm {rss}})$ is the diagonal morphism

$$ \begin{align*}\mathcal{T}_{{\mathrm{reg}}}\longrightarrow\mathcal{T}_{{\mathrm{reg}}}\times_{\mathfrak{car}}\mathcal{G}_{{\mathrm{rss}}}=\mathcal{T}_{{\mathrm{reg}}}\times_{\mathfrak{car}}{\overline{\mathcal{T}}}_{\!{\mathrm{reg}}}. \end{align*} $$

We thus have the following cartesian diagram:

where the horizontal arrows are the quotient maps by W, and the vertical arrows are the diagonal morphisms (which are T-torsors). Therefore,

$$ \begin{align*}{\mathcal{N}}_{\mathrm{rss}}=(q_{\mathrm{rss}},p_{\mathrm{rss}})_!{\overline{\mathbb{Q}}_{\ell}}\simeq (\pi_{\mathrm{reg}}\times 1)^*\Delta_{{\overline{\mathcal{T}}}_{\!{\mathrm{reg}}}!}{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

(i.e., ${\overline {{\mathcal {N}}}}_{{\mathrm {rss}}}=\Delta _{{\overline {\mathcal {T}}}_{\!{\mathrm {reg}}}!}{\overline {\mathbb {Q}}_{\ell }}$ ). The functors $\mathrm {I}_{{\mathrm {rss}}}:D_{\mathrm {c}}^{\mathrm {b}}({\overline {\mathcal {T}}}_{\!{\mathrm {reg}}})\rightarrow D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{{\mathrm {rss}}})\simeq D_{\mathrm {c}}^{\mathrm {b}}({\overline {\mathcal {T}}}_{\!{\mathrm {reg}}})$ and $\mathrm {R}_{{\mathrm {rss}}}:D_{\mathrm {c}}^{\mathrm {b}}({\overline {\mathcal {T}}}_{\!{\mathrm {reg}}})\simeq D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{{\mathrm {rss}}})\rightarrow D_{\mathrm {c}}^{\mathrm {b}}({\overline {\mathcal {T}}}_{\!{\mathrm {reg}}})$ are thus the identity functors.

6.3 The isomorphism $\mathrm {R}_{\lambda }\circ \mathrm {I}_{\lambda }\simeq 1$

By (6.1) we can regard $\mathrm {I}_{\lambda }$ as a restriction functor. We thus consider the composition of correspondences

where the arrows are the obvious projections.

We put

$$ \begin{align*}{\overline{{\mathcal{N}}}}_{\lambda}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\lambda}:=\mathrm{pr}_{{\overline{\mathcal{T}}}_{\lambda},{\overline{\mathcal{T}}}_{\lambda} !}(\mathrm{pr}_{12}^*{\overline{{\mathcal{N}}}}_{\lambda}\otimes\mathrm{pr}_{23}^*{\overline{{\mathcal{N}}}}_{\lambda})=\mathrm{pr}_{{\overline{\mathcal{T}}}_{\lambda},{\overline{\mathcal{T}}}_{\lambda}!}({\overline{{\mathcal{N}}}}_{\lambda}\boxtimes_{\mathcal{G}_{\lambda}}{\overline{{\mathcal{N}}}}_{\lambda}). \end{align*} $$

Then it follows from §2.5 that the functor $\mathrm {R}_{\lambda }\circ \mathrm {I}_{\lambda }$ is the restriction functor of the cohomological correspondence

$$ \begin{align*}({\overline{\mathcal{T}}}_{\lambda}\times_{\mathfrak{car}_{\lambda}}{\overline{\mathcal{T}}}_{\lambda},{\overline{{\mathcal{N}}}}_{\lambda}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\lambda},a,b). \end{align*} $$

Theorem 6.2. There is a natural isomorphism

$$ \begin{align*}{\overline{{\mathcal{N}}}}_{\lambda}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\lambda}\overset{\simeq}{\longrightarrow}\Delta_{{\overline{\mathcal{T}}}_{\lambda}!}{\overline{\mathbb{Q}}_{\ell}}, \end{align*} $$

where

$$ \begin{align*}\Delta_{{\overline{\mathcal{T}}}_{\lambda}}:{\overline{\mathcal{T}}}_{\lambda}\rightarrow{\overline{\mathcal{T}}}_{\lambda}\times_{\mathfrak{car}_{\lambda}}{\overline{\mathcal{T}}}_{\lambda} \end{align*} $$

is the diagonal morphism. In particular we have $\mathrm {R}_{\lambda }\circ \mathrm {I}_{\lambda }\simeq 1$ .

As we did in §5.2, we explain the proof in the nilpotent case, which is the essential case. We thus consider the first diagram of §5.2, and we put

$$ \begin{align*}{\mathcal{N}}_{\mathrm{nil}}:=(q_{\mathrm{nil}},p_{\mathrm{nil}})_!{\overline{\mathbb{Q}}_{\ell}} \in D_{\mathrm{c}}^{\mathrm{b}}(\mathrm{B}(T)\times\mathcal{G}_{\mathrm{nil}}). \end{align*} $$

We have

$$ \begin{align*}({\overline{{\mathcal{N}}}}_{\mathrm{nil}}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\mathrm{nil}})|_{\mathrm{B}(T)\times\mathrm{B}(T)}=\mathrm{pr}_{\mathrm{B}(T),\mathrm{B}(T)\,!}({\mathcal{N}}_{\mathrm{nil}}\boxtimes_{\mathcal{G}_{\mathrm{nil}}}{\mathcal{N}}_{\mathrm{nil}}) \end{align*} $$

where

$$ \begin{align*}\mathrm{pr}_{\mathrm{B}(T),\mathrm{B}(T)}:\hat{{\mathcal{S}}}_{\mathrm{nil}}\times_{\mathcal{G}}\hat{{\mathcal{S}}}_{\mathrm{nil}}=\mathrm{B}(T)\times\mathcal{G}_{\mathrm{nil}}\times\mathrm{B}(T)\longrightarrow\mathrm{B}(T)\times\mathrm{B}(T) \end{align*} $$

is the obvious projection.

Consider the decomposition of $(q_{\mathrm {nil}},q_{\mathrm {nil}})$ as

By Künneth formula, we have

(6.2) $$ \begin{align} ({\overline{{\mathcal{N}}}}_{\mathrm{nil}}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\mathrm{nil}})|_{\mathrm{B}(T)\times\mathrm{B}(T)}=(q_{\mathrm{nil}},q_{\mathrm{nil}})_!{\overline{\mathbb{Q}}_{\ell}}. \end{align} $$

We consider the following commutative diagram

(6.3)

with diagonal morphisms $\Delta _{\mathcal {B}_{\mathrm {nil}}}$ and $\Delta _{\mathrm {B}(T)}$ .

The morphism $\Delta _{\mathcal {B}_{\mathrm {nil}}}:\mathcal {B}_{\mathrm {nil}}\rightarrow \mathcal {Z}_{\mathrm {nil}}$ is schematic closed, and so by adjunction we have a morphism

$$ \begin{align*}{\overline{\mathbb{Q}}_{\ell}}\rightarrow\Delta_{\mathcal{B}_{\mathrm{nil}}*}{\overline{\mathbb{Q}}_{\ell}}=\Delta_{\mathcal{B}_{\mathrm{nil}}!}{\overline{\mathbb{Q}}_{\ell}}, \end{align*} $$

and by applying the functor $(q_{\mathrm {nil}},q_{\mathrm {nil}})_!$ , we get a morphism

(6.4)

On the other hand, the morphism $q_{\mathrm {nil}}$ being smooth with fibres isomorphic to $[{\mathfrak {u}}/U]$ , we have by adjunction a morphism

(6.5)

which is an isomorphism, and so we obtain an isomorphism

(6.6)

Composing (6.4) with (6.6), we end up with a morphism

(6.7)

From (6.2) together with (6.7) we get a morphism

$$ \begin{align*}{\overline{{\mathcal{N}}}}_{\mathrm{nil}}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\mathrm{nil}}\longrightarrow(\pi_{\mathrm{nil}}\times\pi_{\mathrm{nil}})_!\Delta_{\mathrm{B}(T) !}{\overline{\mathbb{Q}}_{\ell}}=\Delta_{\mathrm{B}(N) !}\pi_{\mathrm{nil}\,!}{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

where $\pi _{\mathrm {nil}}$ is the canonical map $\mathrm {B}(T)\rightarrow \mathrm {B}(N)$ .

Composing with the adjunction morphism

$$ \begin{align*}\pi_{\mathrm{nil}\,!}(\pi_{\mathrm{nil}})^*=\pi_{\mathrm{nil}\,!}(\pi_{\mathrm{nil}})^!\longrightarrow 1 \end{align*} $$

we find

(6.8)

Consider the cartesian diagram

where the top arrow is given by $(w,t)\mapsto (w(t),t)$ and the bottom arrow is the diagonal morphism.

If we denote by $\Delta _{\mathrm {B}(T),w}:\mathrm {B}(T)\rightarrow \mathrm {B}(T)\times \mathrm {B}(T)$ the w-twisted diagonal morphism induced by $t\mapsto (w(t),t)$ , then

$$ \begin{align*}f_!{\overline{\mathbb{Q}}_{\ell}}=\bigoplus_{w\in W}\Delta_{\mathrm{B}(T),w\, !}{\overline{\mathbb{Q}}_{\ell}}=(\pi_{\mathrm{nil}}\times\pi_{\mathrm{nil}})^*\Delta_{\mathrm{B}(N)!}{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

and so the pullback of (6.8) along the map $\pi _{\mathrm {nil}}\times \pi _{\mathrm {nil}}$ provides a morphism

(6.9) $$ \begin{align} (q_{\mathrm{nil}},q_{\mathrm{nil}})_!{\overline{\mathbb{Q}}_{\ell}}\rightarrow\bigoplus_{w\in W}\Delta_{\mathrm{B}(T),w\,!}{\overline{\mathbb{Q}}_{\ell}}. \end{align} $$

of $W\times W$ -equivariant complexes.

Theorem 6.2 is thus a consequence of the following result.

It is enough to show that the morphism

$$ \begin{align*}{\mathcal{H}}^i(q_{\mathrm{nil}},q_{\mathrm{nil}})_!{\overline{\mathbb{Q}}_{\ell}}\rightarrow {\mathcal{H}}^i\left(\bigoplus_{w\in W}\Delta_{\mathrm{B}(T),w\,!}{\overline{\mathbb{Q}}_{\ell}}\right) \end{align*} $$

induced by the morphism (6.9) is an isomorphism for all i.

To prove this, we prove that we have an isomorphism

$$ \begin{align*}{\mathcal{H}}^i(q,q)_!{\overline{\mathbb{Q}}_{\ell}}\rightarrow {\mathcal{H}}^i\left(\bigoplus_{w\in W}\Delta_{\mathcal{T},w\,!}{\overline{\mathbb{Q}}_{\ell}}\right) \end{align*} $$

where $\Delta _{\mathcal {T},w}:\mathcal {T}\rightarrow \mathcal {T}\times _{\mathfrak {car}}\mathcal {T}$ is induced by $t\mapsto (w(t),t)$ .

To prove the proposition, we will use a weight argument. Therefore, we choose a finite subfield $k_o$ of k such that $G, T,$ and B are defined over $k_o$ , and T is split over $k_o$ .

Recall that we have a stratification (see §3)

$$ \begin{align*}\mathcal{Z}=\coprod_{w\in W}\mathcal{Z}_w. \end{align*} $$

The restriction of $(q,q)$ to $\mathcal {Z}_w$ is the composition of the projection $q_w:\mathcal {Z}_w\rightarrow \mathcal {T}$ followed by the morphism $ \Delta _{\mathcal {T},w}:\mathcal {T}\rightarrow \mathcal {T}\times _{\mathfrak {car}}\mathcal {T}$ .

As the cohomology of $[{\mathfrak {u}}_w/U_w]$ is trivial, we deduce that

(6.10) $$ \begin{align} ((q,q)|_{\mathcal{Z}_w})_!{\overline{\mathbb{Q}}_{\ell}}=\Delta_{\mathcal{T},w\,!}{\overline{\mathbb{Q}}_{\ell}}. \end{align} $$

We choose a total order $\{w_0,w_1,\dots \}$ on W so that we have a decreasing filtration of closed substacks

$$ \begin{align*}\mathcal{Z}=\mathcal{Z}_0\supset\mathcal{Z}_1\supset\cdots\supset\mathcal{Z}_{|W|-1}\supset\mathcal{Z}_{|W|}=\emptyset \end{align*} $$

satisfying $\mathcal {Z}_i\backslash \mathcal {Z}_{i+1}=\mathcal {Z}_{w_i}$ for all i.

By [Reference Deligne5, Chapitre 6, 2.5], we have a spectral sequence

$$ \begin{align*}E_1^{ij}={\mathcal{H}}^{i+j}\left((q,q)|_{\mathcal{Z}_i\backslash\mathcal{Z}_{i+1}}\right)_!{\overline{\mathbb{Q}}_{\ell}}\Rightarrow {\mathcal{H}}^{i+j}(q,q)_!{\overline{\mathbb{Q}}_{\ell}}. \end{align*} $$

Proof. As $\mathcal {Z}_i\backslash \mathcal {Z}_{i+1}=\mathcal {Z}_{w_i}$ and $\mathcal {T}=T\times \mathrm {B}(T)$ , by (6.10) we have

(6.11) $$ \begin{align} E_1^{ij}={\mathcal{H}}^{i+j}\Delta_{\mathcal{T},w_i\,!}{\overline{\mathbb{Q}}_{\ell}}=\bigwedge^{2n-i-j}X_*(T)\otimes\left(\Delta_{{\mathfrak{t}},w_i\, !}{\overline{\mathbb{Q}}_{\ell}}\boxtimes {{\overline{\mathbb{Q}}_{\ell}}}_{,\mathrm{B}(T)\times\mathrm{B}(T)}\right)(n-i-j) \end{align} $$

where $\Delta _{{\mathfrak {t}},w}:{\mathfrak {t}}\rightarrow {\mathfrak {t}}\times _{\mathfrak {car}} {\mathfrak {t}}$ is given by $t\mapsto (w(t),t)$ . Notice that

$$ \begin{align*}\Delta_{{\mathfrak{t}},w_i\,!}{\overline{\mathbb{Q}}_{\ell}}\boxtimes {{\overline{\mathbb{Q}}_{\ell}}}_{,\mathrm{B}(T)\times\mathrm{B}(T)} \end{align*} $$

is (up to a shift) perverse and pure of weight $0$ . Hence $E_1^{ij}$ is (up to a shift) pure of weight $2(i+j)$ .

Therefore, the sources and the targets of the differentials $d_1$ are (up to a shift) perverse of different weight, and so $d_1$ vanishes. We keep applying the same argument for the other differentials and we deduce that the spectral sequence degenerates at $E_1$ , i.e. $E_1=E_{\infty }$ .

Following the proof of (6.4), we have also a morphism

$$ \begin{align*}(q,q)_!{\overline{\mathbb{Q}}_{\ell}}\longrightarrow \Delta_{\mathcal{T}\, !}{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

where $\Delta _{\mathcal {T}}:\mathcal {T}\rightarrow \mathcal {T}\times _{\mathfrak {car}}\mathcal {T}$ is the diagonal morphism. We thus get a morphism

$$ \begin{align*}{\mathcal{H}}^i(q,q)_!{\overline{\mathbb{Q}}_{\ell}}\longrightarrow{\mathcal{H}}^i\Delta_{\mathcal{T}\, !}{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

and since, by Proposition 6.4, the sheaf ${\mathcal {H}}^i(q,q)_!{\overline {\mathbb {Q}}_{\ell }}$ is $W\times W$ -equivariant, we get a morphism

$$ \begin{align*}{\mathcal{H}}^i(q,q)_!{\overline{\mathbb{Q}}_{\ell}}\longrightarrow\bigoplus_{w\in W}{\mathcal{H}}^i\Delta_{\mathcal{T},w\, !}{\overline{\mathbb{Q}}_{\ell}}. \end{align*} $$

The source and the target of this morphism are intermediate extensions of their restriction to regular elements (see Proposition 6.4). It is thus an isomorphism, as its restriction to regular element is an isomorphism.

7.1 Derived categories

Fix $\lambda \in \mathfrak {L}$ . In this section we choose a geometric point $c:\mathrm {Spec}(k)\rightarrow \mathfrak {car}_{\lambda }$ . For any stack $\mathcal {X}$ above $\mathfrak {car}_{\lambda }$ we put

$$ \begin{align*}\mathcal{X}_c:=\mathrm{Spec}(k)\times_{\mathfrak{car}}\mathcal{X}, \end{align*} $$

and for any complex $K\in D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X})$ we denote by $K_c\in D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {X}_c)$ the pullback of K along the base change morphism $\mathcal {X}_c\rightarrow \mathcal {X}$ .

We have a commutative diagram with Cartesian squares

(7.1)

We then define the pair $(\mathrm {R}_c,\mathrm {I}_c)$ of adjoint functors

by

$$ \begin{align*}\mathrm{I}_c: K\mapsto\mathrm{pr}_{2!}({\overline{{\mathcal{N}}}}_c\otimes\mathrm{pr}_1^*(K)),\hspace{1cm}\mathrm{R}_c : K\mapsto\mathrm{pr}_{1!}({\overline{{\mathcal{N}}}}_c\otimes\mathrm{pr}_2^*(K)). \end{align*} $$

The following lemma follows from the base change theorem and the projection formula.

Theorem 7.2. We have

$$ \begin{align*}\mathrm{R}_c\circ\mathrm{I}_c\simeq 1.\hspace{1cm} \end{align*} $$

Proof. The isomorphism of functors $\mathrm {R}_c\circ \mathrm {I}_c\simeq 1$ is obtained by taking the pullback along $\overline {\mathcal {T}_c}\times \overline {\mathcal {T}_c}\rightarrow {\overline {\mathcal {T}}}_{\lambda }\times _{\mathfrak {car}_{\lambda }}{\overline {\mathcal {T}}}_{\lambda }$ of the isomorphism of complexes

$$ \begin{align*}{\overline{{\mathcal{N}}}}_{\lambda}\circ_{\mathcal{G}}{\overline{{\mathcal{N}}}}_{\lambda}\overset{\simeq}{\longrightarrow}\Delta_{{\overline{\mathcal{T}}}_{\lambda}\, !}{\overline{\mathbb{Q}}_{\ell}} \end{align*} $$

of Theorem 6.2.

We have also the functor

$$ \begin{align*}\mathrm{Ind}_c:D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{T}_c)\longrightarrow D_{\mathrm{c}}^{\mathrm{b}}(\mathcal{G}_c) \end{align*} $$

defined by base change from the diagram (2.20). It satisfies

$$ \begin{align*}\mathrm{Ind}_c=\mathrm{I}_c\circ\pi_{c\, !} \end{align*} $$

where $\pi _c$ is the W-torsor $\mathcal {T}_c\rightarrow {\overline {\mathcal {T}}}_c$ . When $c=0$ , we will use the notation $\pi _{\mathrm {nil}}$ instead of $\pi _c$ .

We define $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_c)^{\mathrm {Spr}}$ as the triangulated subcategory of $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_c)$ generated by the simple direct factors of $\mathrm {Ind}_c({\overline {\mathbb {Q}}_{\ell }})$ which, by Springer theory, are indexed by the irreducible characters of the Weyl group $W_c$ of the centralizer of c.

Proof. The essential case is $c=0$ in which case we have $\mathcal {T}_c=\mathrm {B}(T)$ , ${\overline {\mathcal {T}}}_c=\mathrm {B}(N)$ and $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_c)^{\mathrm { Spr}}=D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{\mathrm {nil}})^{\mathrm {Spr}}$ . The simple objects of $D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$ and $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{\mathrm {nil}})^{\mathrm {Spr}}$ are both parametrized by the irreducible characters of W. This is Springer theory in the second case and, in the first case, the simple objects are the simple direct summands of $\pi _{\mathrm { nil}\,*}{\overline {\mathbb {Q}}_{\ell }}$ . The pullback functor $\overline {s}_{\mathrm {nil}}^$ along the obvious morphism $\overline {s}_{\mathrm {nil}}:\mathrm {B}(N)\rightarrow \mathrm {B}(W)$ induces an equivalence between the categories of perverse sheaves that is compatible with the indexation of the simple perverse sheaves by the irreducible characters of W. By Lemma 6.1 applied to nilpotents, we have

$$ \begin{align*}{^p}\mathrm{I}_{\mathrm{nil}}=\mathrm{I}_{\mathrm{nil}}\circ\overline{s}_{\mathrm{nil}}^. \end{align*} $$

We are thus reduced to prove the statement of the proposition for ${^p}\mathrm {I}_{\mathrm { nil}}:\mathcal {M}(\mathrm {B}(W))\rightarrow \mathcal {M}(\mathcal {G}_{\mathrm {nil}})^{\mathrm {Spr}}$ . Unlike $\mathrm {I}_{\mathrm {nil}}$ , the construction of ${^p}\mathrm {I}_{\mathrm {nil}}$ extends to a global functor ${^p}\mathrm {I}$ . A standard argument using the functor ${^p}\mathrm {I}$ and its restriction to semisimple regular elements proves the proposition for ${^p}\mathrm {I}_{\mathrm {nil}}$ .

Consider the decomposition

$$ \begin{align*}\pi_{\mathrm{nil}\, !}{\overline{\mathbb{Q}}_{\ell}}=\bigoplus_{\varphi} V_{\varphi}\otimes \mathcal{L}_{\varphi} \end{align*} $$

where the sum runs over the irreducible ${\overline {\mathbb {Q}}_{\ell }}$ -characters of W, $V_{\varphi }$ is an irreducible representation affording the character $\varphi $ and the $\mathcal {L}_{\varphi }$ are the irreducible smooth $\ell $ -adic sheaves on $\mathrm {B}(N)$ .

Proposition 7.5. For any irreducible characters $\varphi $ and $\varphi '$ of W, we have

$$ \begin{align*}\mathrm{Ext}^i(\mathcal{L}_{\varphi},\mathcal{L}_{\varphi'})=\left(V_{\varphi}\otimes H^i(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})\otimes V_{\varphi'}^*\right)^W. \end{align*} $$

Proof. The proof is similar to that of [Reference Achar1, Theorem 4.6]. We give it for the convenience of the reader. We have

$$ \begin{align*} \mathrm{Ext}^i(\mathcal{L}_{\varphi},\mathcal{L}_{\varphi'})&=\mathrm{Hom}_W\left(V_{\varphi}^*,\mathrm{ Ext}^i(\pi_{\mathrm{nil}\,!}{\overline{\mathbb{Q}}_{\ell}},\mathcal{L}_{\varphi'})\right)\\ &=\mathrm{Hom}_W\left(V_{\varphi}^*,\mathrm{Ext}^i({\overline{\mathbb{Q}}_{\ell}},\pi_{\mathrm{nil}}^*(\mathcal{L}_{\varphi'}))\right)\\ &=\mathrm{Hom}_W\left(V_{\varphi}^*,\mathrm{ Ext}^i_{\mathrm{B}(T)}({\overline{\mathbb{Q}}_{\ell}},{\overline{\mathbb{Q}}_{\ell}})\otimes\mathrm{ Hom}({\overline{\mathbb{Q}}_{\ell}},\pi_{\mathrm{nil}}^*(\mathcal{L}_{\varphi'}))\right)\\ &=\mathrm{Hom}_W\left(V_{\varphi}^*,H^i(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})\otimes\mathrm{Hom}(\pi_{\mathrm{ nil}\,!}{\overline{\mathbb{Q}}_{\ell}},\mathcal{L}_{\varphi'})\right)\\ &=\left(V_{\varphi}\otimes H^i(\mathrm{B}(T),{\overline{\mathbb{Q}}_{\ell}})\otimes V_{\varphi'}^*\right)^W. \end{align*} $$

The third equality follows from the canonical isomorphism

$$ \begin{align*}\pi_{\mathrm{nil}}^*(\mathcal{L}_{\varphi'})\simeq {\overline{\mathbb{Q}}_{\ell}}\otimes\mathrm{ Hom}({\overline{\mathbb{Q}}_{\ell}},\pi_{\mathrm{nil}}^*(\mathcal{L}_{\varphi'})) \end{align*} $$

as the category $\mathcal {M}(\mathrm {B}(T))$ is semisimple linear with unique simple object ${\overline {\mathbb {Q}}_{\ell }}[-n]$ .

Define $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{\lambda })^{\mathrm {Spr}}$ as the full subcategory of $D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{\lambda })$ of complexes K such that $K_c\in D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_c)^{\mathrm {Spr}}$ for all geometric point c of $\mathfrak {car}_{\lambda }$ .

Proof. Since $\mathrm {R}_{\lambda }\circ \mathrm {I}_{\lambda }\simeq 1$ , it suffices to see that the morphism

$$ \begin{align*}K\longrightarrow \mathrm{I}_{\lambda}\circ\mathrm{R}_{\lambda}(K) \end{align*} $$

is an isomorphism for all $K\in D_{\mathrm {c}}^{\mathrm {b}}(\mathcal {G}_{\lambda })^{\mathrm {Spr}}$ . But it follows from the fact that it is true over $\mathcal {G}_c$ for any geometrical point c of $\mathfrak {car}_{\lambda }$ by the above proposition.