Section 5. (0N07) · ScalingStacks

5. Springer resolution and affine Hecke algebras

We now apply Theorem 4.9 to the special case of the Springer resolution. Let $G$ be a reductive algebraic group over $\overline{\mathbb{F}}_{p}.$ Denote by $\mathcal{N}_{nil}\subset\operatorname{Lie}(G)$ the nilpotent cone, by $\mu:\widetilde{\mathcal{N}}\to\mathcal{N}_{nil}$ the Springer resolution and by $Z=\widetilde{\mathcal{N}}\times_{\mathcal{N}_{nil}}\widetilde{\mathcal{N}}$ the Steinberg variety. There is an additional dilation action of $\mathbb{G}_{m}$ on $\widetilde{\mathcal{N}}$ and $\mathcal{N}_{nil},$ and we will consider the group $A=G\times\mathbb{G}_{m}$ or $A=G.$

Moreover, assume that

(1)

$p$ is a good prime for every classical group appearing as a constituent in $G$ and
(2)

$p>3(h+1)$ , where $h$ denotes the maximum of all Coxeter numbers of exceptional constituents in $G.$

0MZ3

Lemma 5.1. The conditions (PT) and (FO) hold in this setup.

0MZ4

Proof. Using the conditions on $p$ [Ebe21, Theorem 1.1] shows that the motive $\operatorname{M}(\mu^{-1}(\{x\}))$ of a Springer fiber is pure Tate which implies (PT). Moreover, there are only finitely many $G$ -orbits in $\mathcal{N}_{nil},$ see [Car93], which shows that condition (FO) holds. ∎

The motivic extension algebra $E$ for the action of $A=G\times\mathbb{G}_{m}$ can be identified with Lusztig’s graded affine Hecke algebra associated to $G$ , see [Lus88] and [Lus89],

E\cong(\operatorname{CH}_{A}^{\bullet}(Z)_{\mathbb{Q}},\star)\cong\overline{\mathbb{H}}(G).

We can now use Theorem 4.9 to recover [Ebe21, Corollary 1.4].

0MZ5

Theorem 5.2. There is an equivalence of categories

\operatorname{DM}^{Spr}_{A}(\mathcal{N}_{nil},\mathbb{Q})\cong\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(\overline{\mathbb{H}}(G)).

For $A=G$ a similar result holds, where $E=S\rtimes\mathbb{Q}[W]$ is the semidirect product of the symmetric algebra $S=\operatorname{Sym}(X(T)_{\mathbb{Q}})$ of the rationalized character lattice of a maximal torus $T$ in $G$ and the group algebra of the Weyl group $W=N_{G}(T)/T.$

0MZ6

Remark 5.3. It would be very desirable to obtain a similar result for the affine Hecke algebra $\mathbb{H}(G).$ Let $A=G\times\mathbb{G}_{m}.$ The affine Hecke algebra arises as the $A$ -equivariant $K$ -theory of the Steinberg variety $\mathbb{H}(G)=K_{0}^{A}(Z)_{\mathbb{Q}}.$ Lusztig’s graded affine Hecke algebra arises from a completion process from the affine Hecke algebra. The Atiyah-Segal completion theorem and the Chern character map identifies

\mathbb{H}(G)_{I}^{\wedge}=(K_{0}^{A}(Z)_{\mathbb{Q}})_{I}^{\wedge}\cong K_{0}(EA\times_{A}Z)_{\mathbb{Q}}\cong\prod_{i}\operatorname{CH}^{i}_{A}(Z)_{\mathbb{Q}}\supset\bigoplus_{i}\operatorname{CH}^{i}_{A}(Z)_{\mathbb{Q}}=\overline{\mathbb{H}}(G),

the graded affine Hecke algebra as (a subspace of) the completion of the affine Hecke algebra at the augmentation ideal $I$ of the representation ring of $A.$ So, conjecturally, there should be an equivalence between the category of Springer $K$ -motives on the nilpotent cone

\operatorname{DK}^{Spr}_{A}(\mathcal{N}_{nil},\mathbb{Q})\cong\operatorname{D_{perf}}(\mathbb{H}(G))

and the perfect derived category of the affine Hecke algebra. Here, $\operatorname{DK}_{A}(\mathcal{N}_{nil},\mathbb{Q})$ denotes a (yet to be defined) category of equivariant $K$ -motives which is an equivariant version of the $K$ -motives defined by the first author in [Ebe19].

This conjecture highlights another advantage of motivic sheaves. Namely, one can construct categories of motivic sheaves for generalized cohomology theories such as $K$ -theory.