Representation theory RTG Seminar Fall 2024

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.

Lecture notes by Guanjie Huang.

Upcoming talks

See the seminar outline for more details and suggestions for each talk.

Week 9 (November 11): Classical Whittaker models.

Elad Zelingher

Week 10 (November 18): Geometric Whittaker models

Miao (Pam) Gu

Week 11 (November 25): Coherent sheaves

Jialiang Zou

Week 12 (December 2): Construction of the functor

Robert Cass

Week 13 (December 9): Summary plus epsilon

Charlotte Chan

Previous talks

Week 0 (August 26): Introduction and planning

Robert Cass

Notes

Week 1 (September 9): Introduction to the local Langlands correspondence

Elad Zelingher

Abstract:

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.

Notes

Week 2 (September 16): Introduction to Hecke algebras

Elad Zelingher

Abstract:

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.

Notes

Week 3 (September 23): The nilpotent cone, Springer resolution, Springer fibers, and the Steinberg variety

Alexander Hazeltine

Abstract:

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.

Week 4 (September 30): Kazhdan-Lusztig isomorphism I

Alex Bauman

Abstract:

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.

Week 5 (October 7): Kazhdan--Lusztig isomorphism

Calvin Yost-Wolff

Abstract:

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.

Week 6 (October 21): A crash course on derived categories and perverse sheaves

Robert Cass

Abstract:

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.

Notes

Week 7 (October 28): Geometric Satake

Lukas Scheiwiller Robert Cass

Abstract:

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.

Week 8 (November 4): Gaitsgory's central sheaves

Sean Cotner

Abstract:

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.

Resources