Learning seminar on Bezrukavnikov's equivalence
This Fall our seminar will run Mondays at 3:00pm in EH 3088, organized jointly by Robert Cass, Charlotte Chan, Kartik Prasanna and Elad Zelingher. Each talk should be 60 minutes long; running over due to too much audience engagement is acceptable but 4:15pm is a strict upper bound.
The aim of this seminar is to casually introduce some ideas in geometric representation theory, with an emphasis on applications to Langlands. The path to doing this will be to understand the content of Bezrukavnikov's equivalence.
Theorem
(Bezrukavnikov):
Let $G$ be a split reductive group over an algebraically closed field of characteristic $p$. Let
$\hat{G}$ be the Langlands dual group over $\overline{\mathbb{Q}}_\ell$, for $\ell \neq p$. Then
we have an equivalence of categories
$$D_{\mathcal{I}}(\mathrm{Fl}, \overline{\mathbb{Q}}_\ell) \cong
D^b\mathrm{Coh}^{\hat{G}}(\hat{\mathcal{N}} \times^L_{\hat{\frak{g}}} \hat{\mathcal{N}}).$$
The left side is the derived category of Iwahori-equivariant etale sheaves on the affine flag
variety for $G$. The right side is the derived category of $\hat{G}$-equivariant coherent
sheaves on a version of the Steinberg variety for $\hat{G}$. Starting from basic algebraic
geometry, we will build our understanding of all of these terms and more throughout the seminar,
roughly following the route paved by Geordie
Williamson's year-long course 2019-2020.
See the seminar outline for more details and suggestions for each talk.
We will discuss coherent sheaves on projective space (P^n), flag varieties (G/B), base affine space (G/U), and its closure (denoted as G/U-bar). Time permitting, we will also touch on the Borel-Weil-Bott theorem.
I will give an introduction to the local Langlands correspondence, focusing on the case of general linear groups and providing examples that are useful to keep in mind.
I will explain the notions of the spherical and the Iwahori Hecke algebras and discuss their properties, with an emphasis on the case of general linear groups.
In this talk, we introduce the nilpotent cone, Springer resolution, Springer fibers, and the Steinberg variety and provide examples. In a broader scope, this is a first step towards geometrizing the Deligne-Langlands correspondence.
We introduce geometric convolution algebras, and equivariant Grothendieck groups, and use them to give a statement of the Kazhdan-Lusztig conjecture, which relates the Iwahori-Matsumoto Hecke algebra to equivariant coherent sheaves on the Steinberg variety.
We will deduce the Kazhdan–Lusztig isomorphism from explicitly working out the $\mathrm{SL}_2$ case and bootstrapping this case to compare the action of $K^{G \times \mathbb{G}_m}(\tilde{N} \otimes_N \tilde{N})$ on $K^{G \times \mathbb{G}_m}(\tilde{N})$ with the anti-spherical module of the Iwahori-Hecke algebra. Along the way we will perform equivariant cohomology calculations on pieces of the Steinberg variety and discuss Borel-Weil-Bott.
I will give an overview of derived categories, $\ell$-adic sheaves and perverse sheaves, with an eye toward topics needed in the geometric Satake equivalence. I will also recall what has been done in previous talks at the level of $K$-groups, and state the categorical upgrades we will pursue in the remainder of semester.
As a first step toward Bezrukavnikov's equivalence we will describe the geometric Satake equivalence, which categorifies the classical Satake isomorphism for the unramified Hecke algebra. It is also a cornerstone of the geometric Langlands program and plays a fundamental role in geometric constructions of Langlands correspondences, such as in the works of V. Lafforgue and Fargues-Scholze. Along the way we will discuss equivariant perverse sheaves, the affine Grassmannian, and the representation theory of reductive algebraic groups.
We will geometrize Bernstein's description of the center of the Iwahori--Hecke algebra, by way of Gaitsgory's central sheaves. To this end, we will define the functor of nearby cycles and apply it to several variations of the affine Grassmannian.
I will explain the notion of generic representations of general linear groups and of their Whittaker models. I will also discuss formulas for special values of special Whittaker functions.
We introduce the categorification of Whittaker models. We construct the Iwahori-Whittaker category, and then define the averaging functor between this category and the antispherical category. Time permitting, we will discuss the Iwahori-Whittaker version of geometric Satake.