The spherical part of the local and global Springer actions

Abstract

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky–Kottwitz–MacPherson and Bezrukavnikov–Varshavsky. We first prove this result for the global Springer action, and then deduce from it the result for the local Springer action. This gives an application of global Springer theory to more classical problems.

Appendix A: Sheaves and correspondences on spaces locally of finite type

In this appendix, all algebraic spaces are locally of finite type over $k$.

1.1 A.1 The category of sheaves

Let $X$ be an algebraic space over $k$ which is locally of finite type. Let $\mathrm{Ft}(X)$ be the set of open subsets $U\subset X$ which are of finite type over $k$. We define

$$\begin{aligned} \underleftarrow{D}^{b}(X):=\varprojlim _{U\in \mathrm{Ft}(X)}D^{b}(U) \end{aligned}$$

When $X$ itself is of finite type over $k,\,\mathrm{Ft}(X)$ has a final object $X$, so obviously $\underleftarrow{D}^{b}(X)=D^{b}(X)$.

Concretely, an object in $\underleftarrow{D}^{b}(X)$ is a system of complexes ${\mathcal {F}}_{U}\in D^{b}(U)$ for each open subset $U\subset X$ of finite type over $k$, together with isomorphisms $\varphi ^{U}_{V}:j^{*}{\mathcal {F}}_{U}\mathop {\rightarrow }\limits ^{\sim }{\mathcal {F}}_{V}$ for each open embedding $j:V\hookrightarrow U$ satisfying obvious transitivity conditions. A morphism $\alpha :\{{\mathcal {F}}_{U}\}\rightarrow \{{\mathcal {G}}_{U}\}$ is a system of maps $\alpha _{U}:{\mathcal {F}}_{U}\rightarrow {\mathcal {G}}_{U}$ in $D^{b}(U)$ such that $\alpha _{U}$ restricts to $\alpha _{V}$ on $V$.

Examples of objects in $\underleftarrow{D}^{b}(X)$ include the constant sheaf ${\mathbb {Q}}_{\ell ,X}:=\{{\mathbb {Q}}_{\ell ,U}\}$ and the dualizing complex ${\mathbb {D}}_{X}:=\{{\mathbb {D}}_{U}\}$.

1.2 A.2 Functors

Let $f:X\rightarrow Y$ be a morphism which is locally of finite type. We have the following functors

• 1. $f^{*}:\underleftarrow{D}^{b}(X)\rightarrow \underleftarrow{D}^{b}(Y)$. For $U\in \mathrm{Ft}(X),\,f(U)$ is contained in some $V\in \mathrm{Ft}(Y)$. Denote by $f_{U,V}:U\rightarrow V$ the restriction of $f$. We define $(f^{*}{\mathcal {G}})_{U}=f_{U,V}^{*}{\mathcal {G}}_{V}$.

• 2. $f^{!}:\underleftarrow{D}^{b}(X)\rightarrow \underleftarrow{D}^{b}(Y)$, defined in a similarly way as $f^{*}$: $(f^{!}{\mathcal {G}})_{U}:=f_{U,V}^{!}{\mathcal {G}}_{V}$.

• 3. If $f$ is of finite type, we have

  $$\begin{aligned} f_{!}:\underleftarrow{D}^{b}(X)\rightarrow \underleftarrow{D}^{b}(Y) \end{aligned}$$

  For $V\in \mathrm{Ft}(Y),\,f^{-1}(V)\in \mathrm{Ft}(X)$. Let $f_{V}:f^{-1}(V)\rightarrow V$ be the restriction of $f$. We define $(f_{!}{\mathcal {F}})_{V}:=f_{V,!}{\mathcal {F}}_{f^{-1}(V)}$. In general, if $f$ is only locally of finite type, we have

  $$\begin{aligned} f_{!}:\underleftarrow{D}^{b}(X)\rightarrow \mathrm{ind}\,\underleftarrow{D}^{b}(Y) \end{aligned}$$

  where $\mathrm{ind}\,\underleftarrow{D}^{b}(Y)$ denotes the category of ind-objects in ${\underleftarrow{D}}^{b}(Y)$. We define $f_{!}{\mathcal {F}}$ as the ind-object ${\mathrm{lim}_\rightarrow }_{U\in \mathrm{Ft}(X)}f_{U,!}{{\mathcal {F}}}_{U}$, where $f_{U}:U\rightarrow Y$, the restriction of $f$, is of finite type, and $f_{U,!}$ is defined above.

• 4. If $f$ is of finite type, we have

  $$\begin{aligned} f_{*}:\underleftarrow{D}^{b}(X)\rightarrow \underleftarrow{D}^{b}(Y) \end{aligned}$$

  defined in a similar way as $f_{!}$: $(f_{*}{\mathcal {F}})_{V}:=f_{V,*}{\mathcal {F}}_{f^{-1}(V)}$. In general, if $f$ is only locally of finite type, we have

  $$\begin{aligned} f_{*}:\underleftarrow{D}^{b}(X)\rightarrow \mathrm{pro}\,\underleftarrow{D}^{b}(Y) \end{aligned}$$

  where $\mathrm{pro}\,\underleftarrow{D}^{b}(Y)$ denotes the category of pro-objects in $\underleftarrow{D}^{b}(Y)$. We define $f_{*}{\mathcal {F}}$ as the pro-object $\varprojlim _{U\in \mathrm{Ft}(X)}f_{U,*}{\mathcal {F}}_{U}$.

In particular, we can still define

$$\begin{aligned} \mathbf {H}_{*}({X/Y}):=f_{!}{\mathbb {D}}_{X}\in \mathrm{ind}\,\underleftarrow{D}^{b}(Y). \end{aligned}$$

When $Y=Speck$, we have

$$\begin{aligned}&\mathrm{H}_{c}^{*}({X})=f_{!}{\mathbb {Q}}_{\ell ,X}, \quad \mathrm{H}_{*}({X})=f_{!}{\mathbb {D}}_{X}\in \mathrm{ind}\,D^{b}({{\mathbb {Q}}}_{\ell }\hbox {-vector spaces});\\&\mathrm{H}^{*}({X})=f_{*}{\mathbb {Q}}_{\ell ,X}, \quad \mathrm{H}^{\mathrm{BM}}_{*}({X})=f_{*}{\mathbb {D}}_{X}\in \mathrm{pro}\,D^{b}({{\mathbb {Q}}}_{\ell }\hbox {-vector spaces}). \end{aligned}$$

1.3 A.3 Cohomological correspondences

In this appendix, we extend the formalism of cohomological correspondences (see [9] and [20, Appendix A]) to situations where the relevant algebraic spaces are locally of finite type.

Consider a correspondence diagram

(46)

where

• $S$ is locally of finite type over a field $k$;

• $f,g$ are locally of finite type;

• $\overrightarrow{c}$ is proper and $\overleftarrow{c}$ is of finite type.

For ${\mathcal {F}}\in D^{b}(X)$ and ${\mathcal {G}}\in D^{b}(Y)$, we define as in [20, Definition A.1.1]

$$\begin{aligned} \mathrm{Corr}(C;{\mathcal {F}},{\mathcal {G}}):=Hom_{\underleftarrow{D}^{b}(C)}\left( \overrightarrow{c}^*{\mathcal {G}},\overleftarrow{c}^!{\mathcal {F}}\right) . \end{aligned}$$

We call an element $\zeta \in \mathrm{Corr}(C;{\mathcal {F}},{\mathcal {G}})$ a cohomological correspondence between ${\mathcal {F}}$ and ${\mathcal {G}}$ with support on $C$.

Given $\zeta \in \mathrm{Corr}(C;{\mathcal {F}},{\mathcal {G}})$, we define

$$\begin{aligned} \zeta _{\#}:g_{!}{\mathcal {G}}\xrightarrow {g_!(\mathrm{ad.})}g_{!}\overrightarrow{c}_{*}\overrightarrow{c}^{*}{\mathcal {G}}\xrightarrow {g_{!}\overrightarrow{c}_{*}\zeta }g_{!}\overrightarrow{c}_{*}\overleftarrow{c}^{!}{\mathcal {F}}=g_{!}\overrightarrow{c}_{!}\overleftarrow{c}^{!}{\mathcal {F}}\\ =f_{!}\overleftarrow{c}_{!}\overleftarrow{c}^{!}{\mathcal {F}}\xrightarrow {f_{!}(\mathrm{ad.})}f_{!}{\mathcal {F}}. \end{aligned}$$

In the equality above, we used $\overrightarrow{c}_{!}=\overrightarrow{c}_{*}$ since it is proper. Arrows indexed by “$\mathrm{ad.}$” all come from the relevant adjunction for the morphisms $\overleftarrow{c}$ and $\overrightarrow{c}$, which are of finite type. Note that $\zeta _{\#}$ is a morphism in $\mathrm{ind}\,D^{b}(S)$.

Most of the results in [20, Appendix A] are still valid in this extended situation. In particular, the results on pull-backs of cohomological correspondences in [20, Appendix A.4] extends verbatim.

1.4 A.4 Composition

Suppose we have the following diagram

(47)

where $C=C_{1}\times _{Y} C_{2}$ and $C_{1}$ and $C_{2}$ satisfy the conditions in beginning of Appendix A.3. Since $\overrightarrow{c_{1}},\overrightarrow{c_{2}}$ are proper, so are $\overrightarrow{d}$ and $\overrightarrow{c}$. Similarly, $\overleftarrow{c}$ is of finite type. Hence $C$, as a correspondence between $X$ and $Z$, also satisfies the conditions in the beginning of Appendix A.3.

Let ${\mathcal {F}}\in \underleftarrow{D}^{b}(X),{\mathcal {G}}\in \underleftarrow{D}^{b}(Y)$ and ${\mathcal {H}}\in \underleftarrow{D}^{b}(Z)$. The convolution product defined in [20, Appendix A.2] extends to the current situation, giving a bilinear map

$$\begin{aligned} \circ :\mathrm{Corr}(C_{1};{\mathcal {F}},{\mathcal {G}})\otimes \mathrm{Corr}(C_{2},{\mathcal {G}},{\mathcal {H}})\rightarrow \mathrm{Corr}(C;{\mathcal {F}},{\mathcal {H}}). \end{aligned}$$

The following statement is a variant of [20, Lemma A.2.1], and is proved by a diagram-chasing:

Lemma 8

Let $\zeta _{1}\in \mathrm{Corr}(C_{1};{\mathcal {F}},{\mathcal {G}})$ and $\zeta _{2}\in \mathrm{Corr}(C_{2};{\mathcal {G}},{\mathcal {H}})$. Then

$$\begin{aligned} (\zeta _{1}\circ \zeta _{2})_{\#}=\zeta _{1,\#}\circ \zeta _{2,\#}:h_{!}{\mathcal {H}}\rightarrow f_{!}{\mathcal {F}}. \end{aligned}$$

The associativity of the convolution $\circ$ also holds, see [20, Lemma A.2.2].

1.5 A.5 Property (G-2)

From now on we assume both $X$ and $Y$ are smooth of equidimension $d$. Recall from [20, Appendix A.6] that we say $C$ has Property (G-2) with respect to an open subset $U\subset S$ if $\dim C_U\le d$ and the image of $C-C_U\rightarrow X\times _{S}Y$ has dimension $<d$.

[20, Lemma A.6.2] now reads

Lemma 9

Suppose $C$ satisfies (G-2) with respect to $U\subset S$. Let $\zeta ,\zeta '\in \mathrm{Corr}(C;{\mathbb {Q}}_{\ell ,X},{\mathbb {Q}}_{\ell ,Y})$. If $\zeta |_U=\zeta '|_U\in \mathrm{Corr}(C_U;{\mathbb {Q}}_{\ell ,X_U},{\mathbb {Q}}_{\ell ,Y_U})$, then $\zeta _\#=\zeta '_\#\in Hom_S(g_!{\mathbb {Q}}_{\ell ,Y},f_!{\mathbb {Q}}_{\ell ,X})$.