Lectures on Springer theories and orbital integrals

The cotangent bundle $\widetilde{\mathcal{N}}:=T^{*}\mathcal{B}$ classifies pairs $(e,B)$ where $e\in\mathcal{N}$ and $B$ is a Borel subgroup of $G$ such that $e\in\mathfrak{n}$, where $\mathfrak{n}$ is the nilpotent radical of $\textup{Lie}\ B$. The Springer resolution is the forgetful map

sending $(e,B)$ to $e$. This map is projective.

For $e\in\mathcal{N}$, the fiber $\mathcal{B}_{e}:=\pi^{-1}(e)$ is called the Springer fiber of $e$. By definition, $\mathcal{B}_{e}$ is the closed subscheme of $\mathcal{B}$ consisting of those Borel subgroups $B\subset G$ such that $e$ is contained in the nilpotent radical of $\textup{Lie}\ B$.

Consider the variety $\widetilde{\mathfrak{g}}$ of pairs $(X,B)$ where $X\in\mathfrak{g}$ and $B\in\mathcal{B}$ such that $X\in\textup{Lie}\ B$. The forgetful map $(X,B)\mapsto X$

is called the Grothendieck alteration ¹¹1The term “alteration” refers to a proper, generically finite map whose source is smooth over $k$., also known as the Grothendieck simultaneous resolution.

Let $\mathfrak{c}=\mathfrak{g}\sslash G\cong\mathfrak{t}\sslash W$ be the categorical quotient of $\mathfrak{g}$ by the adjoint action of $G$. Then we have a commutative diagram

Here $\chi:\mathfrak{g}\to\mathfrak{c}$ is the natural quotient map and $\widetilde{\chi}:\widetilde{\mathfrak{g}}\to\mathfrak{t}$ sends $(X,B)\in\widetilde{\mathfrak{g}}$ to the image of $X$ in $\mathfrak{b}/\mathfrak{n}$ (where $\mathfrak{b}=\textup{Lie}\ B$ with nilpotent radical $\mathfrak{n}$), which can be canonically identified with $\mathfrak{t}$ (upon choosing a Borel containing $\mathfrak{t}$). The diagram (1) is Cartesian when restricting the left column to regular elements ²²2An element $X\in\mathfrak{g}$ is called regular if its centralizer in $G$ has dimension $r$, the rank of $G$. in $\mathfrak{g}$. In particular, if we restrict the diagram (1) to the regular semisimple locus $\mathfrak{c}^{\textup{rs}}\subset\mathfrak{c}$ ³³3An element $X\in\mathfrak{t}$ is regular semisimple if $\alpha(X)\neq 0$ for any root $\alpha$. Denote by $\mathfrak{t}^{\textup{rs}}\subset\mathfrak{t}$ the open subscheme of regular semisimple elements in $\mathfrak{t}$. Since $\mathfrak{t}^{\textup{rs}}$ is stable under $W$, it is the full preimage of an open subset $\mathfrak{c}^{\textup{rs}}\subset\mathfrak{c}$. The open subset $\mathfrak{g}^{\textup{rs}}=\chi^{-1}(\mathfrak{c}^{\textup{rs}})$ is by definition the locus of regular semisimple elements of $\mathfrak{g}$., we see that $\pi_{\mathfrak{g}}|_{\widetilde{\mathfrak{g}}^{\textup{rs}}}:\widetilde{\mathfrak{g}}^{\textup{rs}}\to\mathfrak{g}^{\textup{rs}}$ is a $W$-torsor; i.e., there is an action of $W$ on $\pi_{\mathfrak{g}}^{-1}(\mathfrak{g}^{\textup{rs}})$ preserving the projection to $\mathfrak{g}^{\textup{rs}}$ making the fibers of $\pi_{\mathfrak{g}}$ into principal homogeneous spaces for $W$. The map $\pi_{\mathfrak{g}}$ becomes branched but still finite over the regular locus $\mathfrak{g}^{\textup{reg}}\subset\mathfrak{g}$.

For $X\in\mathfrak{g}$, we denote the fiber $\pi_{\mathfrak{g}}^{-1}(X)$ simply by $\widetilde{\mathcal{B}}_{X}$. Restricting $\pi_{\mathfrak{g}}$ to $\mathcal{N}$, the fibers $\widetilde{\mathcal{B}}_{e}$ for a nilpotent element $e\in\mathcal{N}$ is the closed subvariety of $\mathcal{B}$ consisting of those $B$ such that $e\in\textup{Lie}\ B$. Clearly $\mathcal{B}_{e}$ is a subscheme of $\widetilde{\mathcal{B}}_{e}$, and the two schemes have the same reduced structure. However, as schemes, $\widetilde{\mathcal{B}}_{e}$ and $\mathcal{B}_{e}$ are different in general. See §1.3.2 below and Exercise 1.7.2.

When $e=0$, $\mathcal{B}_{e}=\mathcal{B}$.

The unique dense open $G$-orbit consists of regular nilpotent elements, i.e., those $e$ such that $\dim C_{G}(e)=r$. When $e$ is a regular nilpotent element, $\mathcal{B}_{e}$ is a single point: there is a unique Borel subgroup of $G$ whose Lie algebra contains $e$. What is this Borel subgroup?

By the Jacobson-Morosov theorem, we may extend $e$ to an $\mathfrak{sl}_{2}$-triple $(e,h,f)$ in $\mathfrak{g}$. The adjoint action of $h$ on $\mathfrak{g}$ has integer weights, and it decomposes $\mathfrak{g}$ into weight spaces $\mathfrak{g}(n)$, $n\in\mathbb{Z}$. Let $\mathfrak{b}=\oplus_{n\geq 0}\mathfrak{g}(n)$. This is a Borel subalgebra of $\mathfrak{g}$, and the corresponding Borel subgroup is the unique point in $\mathcal{B}_{e}$.

When $e$ is regular, the fiber $\widetilde{\mathcal{B}}_{e}$ of the Grothendieck alteration is a non-reduced scheme whose underlying reduced scheme is a point. The coordinate ring of $\widetilde{\mathcal{B}}_{e}$ is isomorphic to the coinvariant algebra $\textup{Sym}(\mathfrak{t}^{*})/(\textup{Sym}(\mathfrak{t}^{*})^{W}_{+})$, which is, interestingly, also isomorphic to the cohomology ring with $k$-coefficients of the complex flag variety $\mathcal{B}_{\mathbb{C}}$.

When $G=\textup{SL}(V)$ for some vector space $V$ of dimension $n$, $\mathcal{B}$ is the moduli space of full flags $0=V_{0}\subset V_{1}\subset V_{2}\subset\cdots\subset V_{n-1}\subset V_{n}=V$. The Springer fiber $\mathcal{B}_{e}$ consists of those flags such that $eV_{i}\subset V_{i-1}$.

Consider the case $G=\textup{SL}_{3}$ and $e=\left(\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 0&0&0\end{array}\right)$ under the standard basis $\{v_{1},v_{2},v_{3}\}$ of $V$. Then $\mathcal{B}_{e}$ is the union of two $\mathbb{P}^{1}$s: the first $\mathbb{P}^{1}$ consisting of flags $0\subset V_{1}\subset\langle{v_{1},v_{2}}\rangle\subset V$ with varying $V_{1}$ inside the fixed plane $\ker(e)=\langle{v_{1},v_{2}}\rangle$; the second $\mathbb{P}^{1}$ consisting of flags $0\subset\langle{v_{1}}\rangle\subset V_{2}\subset V$ with a varying $V_{2}$ containing $\textup{Im}(e)=\langle{v_{1}}\rangle$.

Consider the case $G=\textup{SL}_{4}$ and the nilpotent element $e:v_{3}\mapsto v_{1}\mapsto 0,v_{4}\mapsto v_{2}\mapsto 0$ under the standard basis $\{v_{1},v_{2},v_{3},v_{4}\}$. If a flag $0\subset V_{1}\subset V_{2}\subset V_{3}\subset V$ is in $\mathcal{B}_{e}$, then $V_{1}\subset\ker(e)=\langle{v_{1},v_{2}}\rangle$, and $V_{3}\supset\textup{Im}(e)=\langle{v_{1},v_{2}}\rangle$. We denote $H:=\langle{v_{1},v_{2}}\rangle$. There are two cases:

1. (1)

   $V_{2}=H$. Then we may choose $V_{1}$ to be any line in $H$ and $V_{3}$ to be any hyperplane containing $V_{2}$. We get a closed subvariety of $\mathcal{B}_{e}$ isomorphic to $\mathbb{P}(H)\times\mathbb{P}(H)\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. We denote this closed subvariety of $\mathcal{B}_{e}$ by $C_{1}$.

2. (2)

   $V_{2}\neq H$. This defines an open subscheme $U$ of $\mathcal{B}_{e}$. Suppose the line $V_{1}\subset H$ is spanned by $av_{1}+bv_{2}$ for some $[a:b]\in\mathbb{P}(H)$, then the image of $V_{2}$ in $V/H=\langle{v_{3},v_{4}}\rangle$ is spanned by $av_{3}+bv_{4}$. Fixing $V_{1}$, the choices of $V_{2}$ are given by $\textup{Hom}(\langle{av_{3}+bv_{4}}\rangle,H/V_{1})\cong\textup{Hom}(V_{1},H/V_{1})$. Once $V_{2}$ is fixed, $V_{3}=V_{2}+H$ is also fixed. Therefore $U$ is isomorphic to the total space of the line bundle $\mathcal{O}(2)$ over $\mathbb{P}(H)\cong\mathbb{P}^{1}$.

From the above discussion we see that $\mathcal{B}_{e}$ has dimension $2$, $C_{1}$ is an irreducible component of $\mathcal{B}_{e}$ and so is the closure of $U$, which we denote by $C_{2}$. We have $C_{1}\cup C_{2}=\mathcal{B}_{e}$ and $C_{1}\cap C_{2}$ is the diagonal inside $C_{1}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

When $G=\textup{SL}_{n}=\textup{SL}(V)$, Spaltenstein [Spa76] and Steinberg [St] gave a description of the irreducible components of $\mathcal{B}_{e}$ using standard Young tableaux of size $n$. This will be relevant to the Springer correspondence that we will discuss later, see §1.5.8. Below we follow the presentation of [Spaltenstein, Ch II, §5].

Fix a nilpotent element $e\in\mathcal{N}$ whose Jordan type is a partition $\lambda$ of $n$. This means, if the partition $\lambda$ is $n=\lambda_{1}+\lambda_{2}+\cdots$, $e$ has Jordan blocks of sizes $\lambda_{1},\lambda_{2},\cdots$. We shall construct a (non-algebraic) map $\mathcal{B}_{e}\to ST(\lambda)$, where $ST(\lambda)$ is the discrete set of standard Young tableau for the partition $\lambda$. For each full flag $0=V_{0}\subset V_{1}\subset\cdots\subset V_{n-1}\subset V_{n}=V$ such that $eV_{i}\subset V_{i-1}$, $e$ induces a nilpotent endomorphism of $V/V_{n-i}$. Let $\mu_{i}$ be the Jordan type of the $e$ on $V/V_{n-i}$, then $\mu_{i}$ is a partition of $i$. The increasing sequence of partitions $\mu_{1},\mu_{2},\cdots,\mu_{n}=\lambda$ satisfies that $\mu_{i}$ is obtained from $\mu_{i-1}$ by increasing one part of $\mu_{i-1}$ by 1 (including creating a part of size $1$). This gives an increasing sequence $Y_{1},\cdots,Y_{n}=Y(\lambda)$ of subdiagrams of the Young diagram $Y(\lambda)$ of $\lambda$. We label the unique box in $Y_{i}-Y_{i-1}$ by $i$ to get a standard Young tableau.

Spaltenstein [Spaltenstein, Ch II, Prop 5.5] showed that the closure of the preimage of each standard Young tableaux in $\mathcal{B}_{e}$ is an irreducible component. Moreover, all irreducible components of $\mathcal{B}_{e}$ arise in this way and they all have the same dimension $d_{e}=\frac{1}{2}\sum_{i}\lambda^{*}_{i}(\lambda^{*}_{i}-1)$, where $\lambda^{*}$ is the conjugate partition of $\lambda$. In particular, the top dimensional cohomology $\textup{H}^{2d_{e}}({\mathcal{B}_{e}})$ has dimension equal to $\#ST(\lambda)$, which is also the dimension of an irreducible representation of the symmetric group $S_{n}$. This statement is a numerical shadow of the Springer correspondence, which says that $\textup{H}^{2d_{e}}({\mathcal{B}_{e}})$ is naturally an irreducible representation of $S_{n}$.

Spaltenstein [Spaltenstein, Ch II, Prop 5.9] also showed that there exists a stratification of $\mathcal{B}$ into affine spaces such that $\mathcal{B}_{e}$ is a union of strata. This implies that the restriction map on cohomology $\textup{H}^{*}({\mathcal{B}})\to\textup{H}^{*}({\mathcal{B}_{e}})$ is surjective.

Consider the case $G=\textup{Sp}(V)$ for some symplectic vector space $V$ of dimension $2n$, then $\mathcal{B}$ is the moduli space of full flags

such that $V^{\bot}_{i}=V_{2n-i}$ for $i=1,\cdots,n$. The Springer fiber $\mathcal{B}_{e}$ consists of those flags such that $eV_{i}\subset V_{i-1}$ for all $i$.

Consider the case where $\dim V=4$. We choose a basis $\{v_{1},v_{2},v_{3},v_{4}\}$ for $V$ such that the symplectic form $\omega$ on $V$ satisfies $\omega(v_{i},v_{5-i})=1$ if $i=1$ and $2$, and $\omega(v_{i},v_{j})=0$ for $i+j\neq 5$. Let $e$ be the nilpotent element in $\mathfrak{g}=\mathfrak{sp}(V)$ given by $e:v_{4}\mapsto v_{1}\mapsto 0,v_{3}\mapsto 0,v_{2}\mapsto 0$. Then a flag $0\subset V_{1}\subset V_{2}\subset V_{1}^{\bot}\subset V$ in $\mathcal{B}_{e}$ must satisfy $\langle{v_{1}}\rangle\subset V_{2}\subset\langle{v_{1},v_{2},v_{3}}\rangle$, and this is the only condition for it to lie in $\mathcal{B}_{e}$ (Exercise 1.7.3). Such a $V_{2}$ corresponds to a line $V_{2}/\langle{v_{1}}\rangle\subset\langle{v_{2},v_{3}}\rangle$, hence a point in $\mathbb{P}^{1}=\mathbb{P}(\langle{v_{2},v_{3}}\rangle)$. Over this $\mathbb{P}^{1}$ we have a tautological rank two bundle $\mathcal{V}_{2}$ whose fiber at $V_{2}/\langle{v_{1}}\rangle$ is the two-dimensional vector space $V_{2}$. The further choice of $V_{1}$ gives a point in the projectivization of $\mathcal{V}_{2}$. The exact sequence $0\to\langle{v_{1}}\rangle\to V_{2}\to V_{2}/\langle{v_{1}}\rangle\to 0$ gives an exact sequence of vector bundles $0\to\mathcal{O}\to\mathcal{V}_{2}\to\mathcal{O}(-1)\to 0$ over $\mathbb{P}^{1}$. Therefore $\mathcal{V}_{2}$ is isomorphic to $\mathcal{O}(-1)\oplus\mathcal{O}$, and $\mathcal{B}_{e}\cong\mathbb{P}(\mathcal{O}(-1)\oplus\mathcal{O})$ is a Hirzebruch surface.

The example considered in §1.3.4 is a simplest case of a subregular Springer fiber. There is a unique nilpotent orbit $\mathcal{O}_{\operatorname{subreg}}$ of codimension 2 in $\mathcal{N}$, which is called the subregular nilpotent orbit. For $e\in\mathcal{O}_{\operatorname{subreg}}$, it is known that $\mathcal{B}_{e}$ is a union of $\mathbb{P}^{1}$’s whose configuration we now describe. We may form the dual graph to $\mathcal{B}_{e}$ whose vertices are the irreducible components of $\mathcal{B}_{e}$ and two vertices are joined by an edge if the two corresponding components intersect (it turns out that they intersection at a single point).

For simplicity assume $G$ is of adjoint type. Let $G^{\prime}$ be another adjoint simple group whose type is defined as follows. When $G$ is simply-laced, take $G^{\prime}=G$. When $G$ is of type $B_{n}$, $C_{n}$, $F_{4}$ and $G_{2}$, take $G^{\prime}$ to be of type $A_{2n-1}$, $D_{n+1}$, $E_{6}$ and $D_{4}$ respectively. One can show that $\mathcal{B}_{e}$ is always a union of $\mathbb{P}^{1}$ whose dual graph is the Dynkin diagram of $G^{\prime}$. The rule in the non-simply-laced case is that each long simple root corresponds to 2 or 3 $\mathbb{P}^{1}$’s while each short simple root corresponds to a unique $\mathbb{P}^{1}$. Such a configuration of $\mathbb{P}^{1}$’s is called a Dynkin curve, see [StConj, §3.10, Definition and Prop 2] and [Slodowy, §6.3].

For example, when $G$ is of type $A_{n}$, then $\mathcal{B}_{e}$ is a chain consisting of $n$ $\mathbb{P}^{1}$’s: $\mathcal{B}_{e}=C_{1}\cup C_{2}\cup\cdots\cup C_{n}$ with $C_{i}\cap C_{i+1}$ a point and otherwise disjoint.

Brieskorn [Brieskorn], following suggestions of Grothendieck, related the singularity of the nilpotent cone along the subregular orbits with Kleinian singularities, and he also realized the semi-universal deformation of this singularity inside $\mathfrak{g}$. Assume $G$ is simply-laced. One can construct a transversal slice $S_{e}\subset\mathcal{N}$ through $e$ of dimension two such that $S_{e}$ consists of regular elements except $e$. Then $S_{e}$ is a normal surface with a Kleinian singularity at $e$ of the same type as the Dynkin diagram of $G$. ⁴⁴4A Kleinian singularity is a surface singularity analytically isomorphic to the singularity at $(0,0)$ of the quotient of $\mathbb{A}^{2}$ by a finite subgroup of $\textup{SL}_{2}$.. The preimage $\widetilde{S}_{e}:=\pi^{-1}(S_{e})\subset\widetilde{\mathcal{N}}$ turns out to be the minimal resolution of $S_{e}$, and hence $\mathcal{B}_{e}$ is the union of exceptional divisors. Upon identifying the components of $\mathcal{B}_{e}$ with simple roots of $G$, the intersection matrix of the exceptional divisors is exactly the negative of the Cartan matrix of $G$. Slodowy [Slodowy] extended the above picture to non-simply-laced groups, and we refer to his book [Slodowy] for a beautiful account of the connection between the subregular orbit and Kleinian singularities.

The Springer fibers $\mathcal{B}_{e}$ are connected. See [StConj, p.131, Prop 1], [Spaltenstein, Ch II, Cor 1.7], and Exercise 1.7.11.

Spaltenstein [Spa77], [Spaltenstein, Ch II, Prop 1.12] showed that all irreducible components of $\mathcal{B}_{e}$ have the same dimension.

Let $d_{e}$ be the dimension of $\mathcal{B}_{e}$. Steinberg [St, Thm 4.6] and Springer [Springer] showed that

The group $G\times\mathbb{G}_{m}$ acts on $\mathcal{N}$ (where $\mathbb{G}_{m}$ by dilation). Let $\widetilde{G}_{e}=\textup{Stab}_{G\times\mathbb{G}_{m}}(e)$ be the stabilizer. Then $\widetilde{G}_{e}$ acts on $\mathcal{B}_{e}$. Note that $\widetilde{G}_{e}$ always surjective onto $\mathbb{G}_{m}$ with kernel $C_{G}(e)$.

The action of $\widetilde{G}_{e}$ on $\textup{H}^{*}({\mathcal{B}_{e}})$ factors through the finite group $A_{e}=\pi_{0}(\widetilde{G}_{e})=\pi_{0}(G_{e})$. Note that $A_{e}$ depends not only on the isogeny class of $G$, but on the isomorphism class of $G$. For example, for $e$ regular, $A_{e}=\pi_{0}(ZG)$ where $ZG$ is the center of $G$. The action of $A_{e}$ on $\textup{H}^{*}({\mathcal{B}_{e}})$ further factors through the image of $A_{e}$ in the adjoint group $G^{\textup{ad}}$.

Springer [SpringerPure] proved that the cohomology of $\mathcal{B}_{e}$ is always pure (in the sense of Hodge theory when $k=\mathbb{C}$, or in the sense of Frobenius weights when $k$ is a finite field).

Let $e\in\mathcal{N}$. Consider the restriction map $i^{*}_{e}:\textup{H}^{*}({\mathcal{B}})\to\textup{H}^{*}({\mathcal{B}_{e}})$ induced by the inclusion $\mathcal{B}_{e}\hookrightarrow\mathcal{B}$. Then the image of $i^{*}_{e}$ is exactly the invariants of $\textup{H}^{*}({\mathcal{B}_{e}})$ under $A_{e}$. This is a theorem of Hotta and Springer [HottaSpringer, Theorem 1.1]. In particular, when $G$ is of type $A$, such restriction maps are always surjective.

De Concini, Lusztig and Procesi [DLP] proved that $\textup{H}^{i}({\mathcal{B}_{e}})$ vanishes for all odd $i$ and any $e\in\mathcal{N}$. When $k=\mathbb{C}$, they prove a stronger statement: $\textup{H}^{i}({\mathcal{B}_{e},\mathbb{Z}})$ vanishes for odd $i$ and is torsion-free for even $i$.

In fact there are two natural actions of $W$ on $\textup{H}^{*}({\mathcal{B}_{e}})$ that differ by tensoring with the sign representation of $W$. In these notes we use the action that is normalized by the following properties. The trivial representation of $W$ corresponds to regular nilpotent $e$ and the trivial $\rho$. The sign representation of $W$ corresponds to $e=0$. Note however that Springer’s original paper [Springer] uses the other action.

Taking $e=0$, Springer’s theorem gives a graded action of $W$ on $\textup{H}^{*}({\mathcal{B}})$. What is this action? First, this action can be seen geometrically by considering $G/T$ instead of $\mathcal{B}=G/B$. In fact, since $N_{G}(T)/T=W$, the right action of $N_{G}(T)$ on $G/T$ induces an action of $W$ on $G/T$, which then induces an action of $W$ on $\textup{H}^{*}({G/T})$. Since the projection $G/T\to G/B$ is an affine space bundle, it follows that $\textup{H}^{*}({\mathcal{B}})\cong\textup{H}^{*}({G/T})$. It can be shown that under this isomorphism, the action of $W$ on $\textup{H}^{*}({G/T})$ corresponds exactly to Springer’s action on $\textup{H}^{*}({\mathcal{B}})$.

Let $S=\textup{Sym}(\mathbb{X}^{*}(T)\otimes\mathbb{Q}_{\ell})$ be the graded symmetric algebra where $\mathbb{X}^{*}(T)$ has degree $2$. The reflection representation of $W$ on $\mathbb{X}^{*}(T)$ then induces a graded action of $W$ on $S$. Recall Borel’s presentation of the cohomology ring of the flag variety

where $S_{+}\subset S$ is the ideal spanned by elements of positive degree, and $(S^{W}_{+})$ denotes the ideal of $S$ generated by $W$-invariants on $S_{+}$. Then (5) is in fact an isomorphism of $W$-modules (see [Springer, Prop 7.2]). By a theorem of Chevalley, $S/(S^{W}_{+})$ is isomorphic to the regular representation of $W$, therefore, as a $W$-module, $\textup{H}^{*}({\mathcal{B}})$ is also isomorphic to the regular representation of $W$.

Springer’s original proof of Theorem 3 uses trigonometric sums over $\mathfrak{g}(\mathbb{F}_{q})$ and, when $k$ has characteristic zero, his proof uses reduction to finite fields. The following theorem due to Borho and MacPherson [BM] can be used to give a direct proof of the Springer correspondence for all base fields $k$ of large characteristics or characteristic zero. To state it, we need to use the language of constructible (complexes of) $\mathbb{Q}_{\ell}$-sheaves and perverse sheaves, for which we refer to the standard reference [BBD] and de Cataldo’s lectures [dC] in this volume.

We sketch three constructions of the $W$-action on $\mathbf{R}\pi_{*}\mathbb{Q}_{\ell}[\dim\mathcal{N}]$.

This construction (or rather the version where $\mathfrak{g}$ is replaced by $G$) is due to Lusztig [LuGreen, §3]. The dimension formula for Springer fibers (2) imply that

• •

  The map $\pi:\widetilde{\mathcal{N}}\to\mathcal{N}$ is semismall. ⁵⁵5A proper surjective map $f:X\to Y$ of irreducible varieties is called semismall (resp. small) if for any $d\geq 1$, $\{y\in Y|\dim f^{-1}(y)\geq d\}$ has codimension at least $2d$ (resp. $2d+1$) in $Y$.

There is an extension of the dimension formula (2) for the dimension of $\widetilde{\mathcal{B}}_{X}$ valid for all elements $X\in\mathfrak{g}$. Using this formula one can show that

• •

  The map $\pi_{\mathfrak{g}}:\widetilde{\mathfrak{g}}\to\mathfrak{g}$ is small.

As a well-known fact in the theory of perverse sheaves, the smallness of $\pi_{\mathfrak{g}}$ (together with the fact that $\widetilde{\mathfrak{g}}$ is smooth) implies that $\mathcal{S}_{\mathfrak{g}}:=\mathbf{R}\pi_{\mathfrak{g},*}\mathbb{Q}_{\ell}[\dim\mathfrak{g}]$ is a perverse sheaf which is the middle extension of its restriction to any open dense subset of $\mathfrak{g}$. Over the regular semisimple locus $\mathfrak{g}^{\textup{rs}}\subset\mathfrak{g}$, $\pi_{\mathfrak{g}}$ is a $W$-torsor, therefore $\mathcal{S}_{\mathfrak{g}}|_{\mathfrak{g}^{rs}}$ is a local system shifted in degree $-\dim\mathfrak{g}$ that admits an action of $W$. By the functoriality of middle extension, $\mathcal{S}_{\mathfrak{g}}$ admits an action of $W$. Taking stalks of $\mathcal{S}_{\mathfrak{g}}$, we get an action of $W$ on $\textup{H}^{*}({\widetilde{\mathcal{B}}_{X}})$ for all $X\in\mathfrak{g}$.

In particular, for a nilpotent element $e$, we get an action of $W$ on $\textup{H}^{*}({\mathcal{B}_{e}})=\textup{H}^{*}({\widetilde{\mathcal{B}}_{e}})$, because $\widetilde{\mathcal{B}}_{e}$ and $\mathcal{B}_{e}$ have the same reduced structure. This is the action defined by Springer in his original paper [Springer], which differs from our action by tensoring with the sign character of $W$.

By the semismallness of $\pi$, the complex $\mathcal{S}=\mathbf{R}\pi_{*}\mathbb{Q}_{\ell}[\dim\mathcal{N}]$ is also a perverse sheaf. However, it is not the middle extension from an open subset of $\mathcal{N}$. There is a notion of Fourier transform for $\mathbb{G}_{m}$-equivariant sheaves on affine spaces [Laumon]. One can show that $\mathcal{S}$ is isomorphic to the Fourier transform of $\mathcal{S}_{\mathfrak{g}}$ and vice versa. The $W$-action on $\mathcal{S}_{\mathfrak{g}}$ then induces an action of $W$ on $\mathcal{S}$ by the functoriality of Fourier transform. Taking the stalk of $\mathcal{S}$ at $e$ we get an action of $W$ on $\textup{H}^{*}({\mathcal{B}_{e}})$. This action is normalized according to our convention in §1.5.1.

Consider the Steinberg variety $\textup{St}_{\mathfrak{g}}=\widetilde{\mathfrak{g}}\times_{\mathfrak{g}}\widetilde{\mathfrak{g}}$ which classifies triples $(X,B_{1},B_{2})$, where $B_{1},B_{2}$ are Borel subgroups of $G$ and $X\in\textup{Lie}\ B_{1}\cap\textup{Lie}\ B_{2}$. The irreducible components of $\textup{St}_{\mathfrak{g}}$ are indexed by elements in the Weyl group: for $w\in W$, letting $\textup{St}_{w}$ be the closure of the graph of the $w$-action on $\widetilde{\mathfrak{g}}^{\textup{rs}}$, then $\textup{St}_{w}$ is an irreducible component of St and these exhaust all irreducible components of St. The formalism of cohomological correspondences allows us to get an endomorphism of the complex $\mathcal{S}_{\mathfrak{g}}=\mathbf{R}\pi_{\mathfrak{g},*}\mathbb{Q}_{\ell}[\dim\mathfrak{g}]$ from each $\textup{St}_{w}$. It is nontrivial to show that these endomorphisms together form an action of $W$ on $\mathcal{S}_{\mathfrak{g}}$. The key ingredient in the argument is still the smallness of the map $\pi_{\mathfrak{g}}$. After the $W$-action on $\mathcal{S}_{\mathfrak{g}}$ is defined, one then define the Springer action on $\textup{H}^{*}({\mathcal{B}_{e}})$ by either twisting the action of $W$ on the stalk $\mathcal{S}_{\mathfrak{g},e}$ by the sign representation as in §1.5.3, or by using Fourier transform as in §1.5.4. We refer to [GS, Remark 3.3.4] for some discussion of this construction. See also [CG, §3.4] for a similar but different construction using limits of $\textup{St}_{w}$ in the nilpotent Steinberg variety $\widetilde{\mathcal{N}}\times_{\mathcal{N}}\widetilde{\mathcal{N}}$.

Note that the above three constructions all allow one to show that $\textup{End}(\mathcal{S})\cong\mathbb{Q}_{\ell}[W]$, hence giving a proof of Theorem 7.

We sketch a construction of Slodowy [Slo4, §4] which works for $k=\mathbb{C}$. This construction was conjectured to give the same action of $W$ on $\textup{H}^{*}({\mathcal{B}_{e}})$ as the one in Theorem 3. A similar construction by Rossmann appeared in [Rossmann, §2], in which the author identified his action with that constructed by Kazhdan and Lusztig in [KL80], and the latter was known to be the same as Springer’s action. Thus all these constructions give the same $W$-action as in Theorem 3.

Let $e\in\mathcal{N}$ and let $S_{e}\subset\mathfrak{g}$ be a transversal slice to the orbit of $e$. Upon choosing an $\mathfrak{sl}_{2}$-triple $(e,h,f)$ containing $e$, there is a canonical choice of such a transversal slice $S_{e}=e+\mathfrak{g}_{f}$, where $\mathfrak{g}_{f}$ is the centralizer of $f$ in $\mathfrak{g}$. Now consider the following diagram where the squares are Cartesian except for the rightmost one

Here the rightmost square is (1), $S^{\textup{nil}}_{e}=S_{e}\cap\mathcal{N}$ and $\widetilde{S}_{e}$ and $\widetilde{S}^{\textup{nil}}_{e}$ are the preimages of $S_{e}$ and $S^{\textup{nil}}_{e}$ under $\pi_{\mathfrak{g}}$. Let $V_{e}\subset S_{e}$ be a small ball around $e$ and let $V_{0}\subset\mathfrak{c}$ be an even smaller ball around $0\in\mathfrak{c}$. Let $U_{e}=V_{e}\cap\chi^{-1}(V_{0})\subset S_{e}$. Then the diagram (8) restricts to a diagram

Here $U^{\textup{nil}}_{e}=U_{e}\cap\mathcal{N}$, and $\widetilde{V}_{0},\widetilde{U}_{e}$ and $\widetilde{U}^{\textup{nil}}_{e}$ are the preimages of $V_{0},U_{e}$ and $U^{\textup{nil}}_{e}$ under the vertical maps in (8). The key topological facts here are

• •

  The inclusion $\mathcal{B}_{e}\hookrightarrow\widetilde{U}^{\textup{nil}}_{e}$ admits a deformation retract, hence it is a homotopy equivalence;

• •

  The map $\widetilde{\chi}_{e}:\widetilde{U}_{e}\to\widetilde{V}_{0}$ is a trivializable fiber bundle (in the sense of differential topology).

Now a general fiber of $\widetilde{\chi}_{e}$ admits a homotopy action of $W$ by the second point above because the rightmost square in (9) is Cartesian over $V_{0}\cap\mathfrak{c}^{\textup{rs}}$ and the map $\widetilde{V}_{0}\to V_{0}$ is a $W$-torsor over $V_{0}\cap\mathfrak{c}^{\textup{rs}}$. By the first point above, $\mathcal{B}_{e}$ has the same homotopy type with $\widetilde{U}^{\textup{nil}}_{e}=\widetilde{\chi}^{-1}_{e}(0)$, hence $\mathcal{B}_{e}$ also has the same homotopy type as a general fiber of $\widetilde{\chi}_{e}$ because $\widetilde{\chi}_{e}$ is a fiber bundle. Combining these facts, we get an action of $W$ on the homotopy type of $\mathcal{B}_{e}$, which is a stronger structure than an action of $W$ on the cohomology of $\mathcal{B}_{e}$. A consequence of this construction is that the $W$-action on $\textup{H}^{*}({\mathcal{B}_{e}})$ in Theorem 3 preserves the ring structure.

We decompose the perverse sheaf $\mathcal{S}$ into isotypical components under the $W$-action

where $V_{\chi}$ is the space on which $W$ acts via the irreducible representation $\chi$, and $\mathcal{S}_{\chi}=\textup{Hom}_{W}(V_{\chi},\mathcal{S})$ is a perverse sheaf on $\mathcal{N}$. Since $\textup{End}(\mathcal{S})\cong\mathbb{Q}_{\ell}[W]$, we conclude that each $\mathcal{S}_{\chi}$ is nonzero and that

The decomposition theorem [BBD, Th 6.2.5] implies that each $\mathcal{S}_{\chi}$ is a semisimple perverse sheaf. Therefore (10) implies that $\mathcal{S}_{\chi}$ is simple. Hence $\mathcal{S}_{\chi}$ is of the form $\textup{IC}(\overline{\mathcal{O}},\mathcal{L})$ where $\mathcal{O}\subset\mathcal{N}$ is a nilpotent orbit and $\mathcal{L}$ is an irreducible $G$-equivariant local system on $\mathcal{O}$. Moreover, since $\textup{Hom}(\mathcal{S}_{\chi},\mathcal{S}_{\chi^{\prime}})=0$ for $\chi\neq\chi^{\prime}$, the simple perverse sheaves $\{\mathcal{S}_{\chi}\}_{\chi\in\textup{Irr}(W)}$ are non-isomorphic to each other. This proves part (3) of Theorem 3 by interpreting the right side of (4) as the set of isomorphism classes of irreducible $G$-equivariant local systems on nilpotent orbits. If $\mathcal{S}_{\chi}=\textup{IC}(\overline{\mathcal{O}},\mathcal{L})$ and $e\in\mathcal{O}$, the semismallness of $\pi$ allows us to identify the stalk $\mathcal{L}_{e}$ with an $A_{e}$-isotypic subspace of $\textup{H}^{2d_{e}}({\mathcal{B}_{e}})$. This proves part (2) of Theorem 3. ∎

We give some further examples of the Springer correspondence.

When $G=\textup{SL}_{n}$, all $A_{e}$ are trivial. The Springer correspondence sets a bijection between irreducible representations of $W=S_{n}$ and nilpotent orbits of $\mathfrak{g}=\mathfrak{sl}_{n}$, both parametrized by partitions of $n$. In §1.3.6 we have seen that if $e$ has Jordan type $\lambda$, the top dimensional cohomology $\textup{H}^{2d_{e}}({\mathcal{B}_{e}})$ has a basis indexed by the standard Young tableaux of $\lambda$, the latter also indexing a basis of the irreducible representation of $S_{n}$ corresponding to the partition $\lambda$. For example, for $G=\textup{SL}_{3}$, the Springer correspondences reads

• •

  trivial representation $\leftrightarrow$ regular orbit, partition $3=3$;

• •

  two-dimensional representation $\leftrightarrow$ subregular orbit, partition $3=2+1$;

• •

  sign representation $\leftrightarrow$ {0}, partition $3=1+1+1$.

Consider the case $e$ is a subregular nilpotent element. In this case, the component group $A_{e}$ can be identified with the automorphism group of the Dynkin diagram of $G^{\prime}$ introduced in §1.3.8 (see [Slodowy, §7.5, Proposition]). After identifying the irreducible components of $\mathcal{B}_{e}$ with the vertices of the Dynkin diagram of $G^{\prime}$, the action of $A_{e}$ on $\textup{H}^{2}({\mathcal{B}_{e}})$ is by permuting the basis given by irreducible components in the same way as its action on the Dynkin diagram of $G^{\prime}$. For example, when $G=G_{2}$, we may write $\mathcal{B}_{e}=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}$ with $C_{1}$, $C_{2}$, $C_{3}$ each intersecting $C_{4}$ in a point and otherwise disjoint. The group $A_{e}$ is isomorphic to $S_{3}$, and its action on $\textup{H}^{2}({\mathcal{B}_{e}})$ fixes the fundamental class of $C_{4}$ and permutes the fundamental classes of $C_{1},C_{2}$ and $C_{3}$.

Note that $\textup{H}^{2}({\mathcal{B}_{e}})^{A_{e}}$ always has dimension $r$, the rank of $G$. In fact, as a $W$-module, $\textup{H}^{2}({\mathcal{B}_{e}})^{A_{e}}$ is isomorphic to the reflection representation of $W$ on $\mathfrak{t}^{*}$. In other words, under the Springer correspondence, the pair $(e=\textup{subregular},\rho=1)$ corresponds to the reflection representation of $W$.

The $W$-action on $\textup{H}^{*}({\mathcal{B}_{e}})$ can be extended to an action of a larger algebra in various ways, if we use more sophisticated cohomology theories. On the equivariant cohomology $\textup{H}^{*}_{\widetilde{G}_{e}}(\mathcal{B}_{e})$, there is an action of the graded affine Hecke algebra (see Lusztig [LuGrHk]). On the $\widetilde{G}_{e}$-equivariant $K$-group of $\mathcal{B}_{e}$, there is an action of the affine Hecke algebra (see Kazhdan-Lusztig [KLAffHk] and Chriss-Ginzburg [CG]).

There are obvious analogs of the Springer resolution and the Grothendieck alteration when $\mathcal{N}$ and $\mathfrak{g}$ are replaced with the unipotent variety $\mathcal{U}\subset G$ and $G$ itself. When $\textup{char}(k)$ is large, the exponential map identifies $\mathcal{N}$ with $\mathcal{U}$ in a $G$-equivariant manner, hence the theories of Springer fibers for nilpotent elements and unipotent elements are identical. The group version of the perverse sheaf $\mathcal{S}_{\mathfrak{g}}$ and its irreducible direct summands are precursors of character sheaves, a theory developed by Lusztig ([LuChShI], [LuChShII], [LuChShIV] and [LuChShV]) to study characters of the finite groups $G(\mathbb{F}_{q})$.

We may define analogs of $\mathcal{B}_{e}$ in partial flag varieties. Let $\mathcal{P}$ be a partial flag variety of $G$ classifying parabolic subgroups $P$ of $G$ of a given type. There are two analogs of the map $\pi:\widetilde{\mathcal{N}}\to\mathcal{N}$ one may consider.

First, instead of considering $\widetilde{\mathcal{N}}=T^{*}\mathcal{B}$, we may consider $T^{*}\mathcal{P}$, which classifies pairs $(e,P)\in\mathcal{N}\times\mathcal{P}$ such that $e\in\textup{Lie}\ \mathfrak{n}_{P}$, where $\mathfrak{n}_{P}$ is the nilpotent radical of $\textup{Lie}\ P$. Let $\tau_{\mathcal{P}}:T^{*}\mathcal{P}\to\mathcal{N}$ be the first projection. In general this map is not surjective, its image is the closure of a nilpotent orbit $\mathcal{O}_{\mathcal{P}}$. The orbit $\mathcal{O}_{P}$ is characterized by the property that its intersection with $\mathfrak{n}_{P}$ is dense in $\mathfrak{n}_{P}$, for any $P\in\mathcal{P}$. This is called the Richardson class associated to parabolic subgroups of type $\mathcal{P}$. When $G$ is of type $A$, each nilpotent class $\mathcal{O}$ is the Richardson class associated to parabolic subgroups of some type $\mathcal{P}$ (not unique in general). The map $T^{*}\mathcal{P}\to\overline{\mathcal{O}}$ is a resolution of singularities. For general $G$, not every nilpotent orbit is Richardson.

Second, we may consider the subscheme $\widetilde{\mathcal{N}}_{\mathcal{P}}\subset\mathcal{N}\times\mathcal{P}$ classifying pairs $(e,P)$ such that $e\in\mathcal{N}_{P}$, where $\mathcal{N}_{P}\subset\textup{Lie}\ P$ is the nilpotent cone of $P$. The projection $\pi_{\mathcal{P}}:\widetilde{\mathcal{N}}_{\mathcal{P}}\to\mathcal{N}$ is now surjective, and is a partial resolution of singularities. The Springer resolution $\pi$ can be factored as

We have an embedding $T^{*}\mathcal{P}\hookrightarrow\widetilde{\mathcal{N}}_{\mathcal{P}}$. We may consider the fibers of either $\tau_{\mathcal{P}}$ or $\pi_{\mathcal{P}}$ as parabolic analogs of Springer fibers. We call them partial Springer fibers. The Springer action of $W$ on the cohomology of $\mathcal{B}_{e}$ has an analog for partial Springer fibers. For more information, we refer the readers to [BM].

The Grothendieck alteration $\pi_{\mathfrak{g}}:\widetilde{\mathfrak{g}}\to\mathfrak{g}$ admits a generalization where $\mathfrak{g}$ is replaced with another linear representation of $G$.

Fix a Borel subgroup $B$ of $G$. Let $(V,\rho)$ be a representation of $G$ and $V^{+}\subset V$ be a subspace which is stable under $B$. Now we use the pair $(V,V^{+})$ instead of the pair $(\mathfrak{g},\mathfrak{b})$, we get a generalization of the Grothendieck alteration. More precisely, let $\widetilde{V}\subset V\times\mathcal{B}$ be the subscheme consisting of pairs $(v,gB)\in V\times\mathcal{B}$ such that $v\in\rho(g)V^{+}$. Let $\pi_{V}:\widetilde{V}\to V$ be the first projection. The fibers of $\pi_{V}$ are called Hessenberg varieties.