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 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.
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.