MIT Lie Groups Seminar

2020 - 2021

Meetings: 4:30pm on Wednesdays

This seminar will take place entirely online. Please email Andre Dixon or Pavel Etingof for the Zoom meeting Link. To access videos of talks, please email Andre Dixon for the password.

Fall 2020

• Sept 9

David Vogan
(MIT)

Structure of Harish-Chandra cells

Abstract: One of the fundamental contributions of Kazhdan and Lusztig's 1979 Inventiones paper was the notion of "cells" in Weyl groups. They gave a decomposition of the left regular representation of W as a direct sum of "left cell" representations, which encode deep and powerful information about group representations. In the case of the symmetric group S_n=W, the left cells are irreducible representations. In all other cases they are not. Lusztig in his 1984 book gave a beautiful description of all left cells in terms of the geometry of a nilpotent orbit.

There is a parallel notion of "Harish-Chandra cells" in the representation theory of a real reductive group G(R). Again each cell is a representation of W, encoding deep information about the G(R) representations. I will formulate a conjecture extending Lusztig's calculation of left cell representations to this case, and explain its connection with Arthur's theory of unipotent representations.

Slides

Video

• Sept 16

Charlotte Chan
(MIT)

A strong Henniart identity for reductive groups over finite fields

Abstract: In 1992, Henniart proved that supercuspidal representations for p-adic GLn are determined by their character on so-called very regular elements. This has been useful in many ways as it allows for convenient comparison between various constructions of supercuspidal representations for GLn. We describe a version of this type of result which holds for (some) representations of reductive groups over finite fields. This is joint work with Masao Oi.

Slides

Video

• Sept 23

Jonathan Wang
(MIT)

Spherical varieties, L-functions, and crystal bases

Abstract:The program of Sakellaridis and Venkatesh proposes a unified framework to study integral representations of L-functions through the lens of spherical varieties. For X an affine spherical variety, the (hypothetical) IC complex of the infinite-dimensional formal arc space of X is conjecturally related to special values of local unramified L-functions. We formulate this relation precisely using a new conjectural geometric construction of the crystal basis of a finite-dimensional representation (determined by X) of the dual group. We prove these conjectures for a large class of spherical varieties. This is joint work with Yiannis Sakellaridis.

Slides

Video

• Sept 30

Olivier Dudas
(CNRS)

Macdonald polynomials and decomposition numbers for finite unitary groups

Abstract:(work in progress with R. Rouquier) In this talk I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. I will first explain how one can produce a "natural" self-equivalence in the case of $\mathrm{GL}_n(q)$ coming from the topology of the Hilbert scheme of $\mathbb{C}^2$. The combinatorial part of this equivalence is related to Macdonald's theory of symmetric functions and gives $(q,t)$-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence.

Slides

• Oct 7

Viktor Ostrik
(U. Oregon, Eugene)

Two dimensional field theories and partial fractions

Abstract:This talk is based on joint work with M.Khovanov and Y.Kononov. By evaluating a topological field theory in dimension 2 on surfaces of genus 0,1,2 etc we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

Slides

Video

• Oct 14

Maarten Solleveld

Abstract:Suppose that one has a supercuspidal representation of a Levi subgroup of some reductive $p$-adic group $G$. Bernstein associated to this a block Rep$(G)^s$ in the category of smooth $G$-representations. We address the question: what does Rep$(G)^s$ look like?

Usually this is investigated with Bushnell--Kutzko types, but these are not always available. Instead, we approach it via the endomorphism algebra of a progenerator of Rep$(G)^s$. We will show that Rep$(G)^s$ is "almost" equivalent with the module category of an affine Hecke algebra -- a statement that will be made precise in several ways.

In the end, this leads to a classification of the irreducible representations in Rep$(G)^s$ in terms of the complex torus and the finite groups that are canonically associated to this Bernstein component.

Slides

Video

• Oct 21

Dima Arinkin
(University of Wisconin)

Compactifying the category of D-modules on the stack of G-bundles

Abstract:

Let X be a projective curve, G a reductive group. Let Bun be the stack of G-bundles over X, and consider the category of D-modules on Bun. (This category appears on the “automorphic” side of the geometric Langlands correspondence.) Drinfeld and Gaitsgory prove that, despite the “unbounded” (non-quasi compact) nature of Bun, the category of D-modules is well-behaved (compactly generated).

In this talk, we will “compactify” this category in a stronger sense; this can be viewed as compactifying the quantized cotangent bundle to Bun. While the basic idea of such compactification goes back to ideas of Deligne and Simpson, its construction relies on non-trivial properties of the geometry of Bun (similar to the Drinfeld-Gaitsgory Theorem).

Slides

Video

• Oct 28

Joel Kamnitzer
(U. Toronto)

Categorical g-actions for modules over truncated shifted Yangians

Abstract:

Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians.

Motivated by the geometric Satake correspondence and the theory of symplectic duality/3d mirror symmetry, we expect a categorical g-action on modules for these truncated shifted Yangians. I will explain three results in this direction. First, we have an indirect realization of this action, using equivalences with KLRW-modules. Second, we have a geometric relation between these generalized slices by Hamiltonian reduction. Finally, we have an algebraic version of this Hamiltonian reduction which we are able to relate to the first realization.

Video

Slides

• Nov 4

Kostiantyn Tolmachov
(U Toronto)

Monodromic model for Khovanov-Rozansky homology

Abstract:

Khovanov-Rozansky homology is a knot invariant which, by the result of Khovanov, can be computed as the Hochschild cohomology functor applied to Rouquier complexes of Soergel bimodules. I will describe a new geometric model for the Hochschild cohomology of Soergel bimodules, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing individual Hochsсhild cohomology groups as images of explicit character sheaves. Based on the joint work with Roman Bezrukavnikov.

Slides

Video

• Nov 10
Tuesday*

Harrison Chen
(Cornell)

Coherent Springer theory and categorical Deligne-Langlands

Abstract: Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of unipotent principal block representations of a p-adic group with certain unipotent Langlands parameters in the Langlands dual group (plus the data of certain representations). We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra (or specializing at q, unipotent principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category. This is a joint work with David Ben-Zvi, David Helm and David Nadler.

Slides

Video

• Nov 18

Yakov Kononov
(Columbia)

Elliptic stable envelopes and 3-dimensional mirror symmetry

Abstract: The action of quantum groups on the K-theory of Nakajima varieties takes the simplest form in the stable bases, invented by D.Maulik and A.Okounkov, and in their most advanced (elliptic) version by M.Aganagic and A.Okounkov. In collaboration with A.Smirnov we discovered and proved the factorization property of elliptic stable envelopes. As a consequence, we proved the conjectures of E.Gorsky and A.Negut. Also it gives a new interesting description of the operators of quantum difference equations, shift operators and other quantities in enumerative geometry. The talk is based on joint works with A.Smirnov.

Slides

Video

• Dec 2
6pm*

Masao Oi
(Kyoto University)

Twisted endoscopic character relation for Kaletha's regular supercuspidal L-packets

Abstract: Recently Kaletha constructed the local Langlands correspondence (i.e., L-packets and their L-parameters) for a wide class of supercuspidal representations. In this talk, I would like to discuss my ongoing work on the twisted endoscopic character relation for Kaletha's supercuspidal L-packets.

The strategy is to imitate Kaletha's proof of the standard endoscopic character relation in the setting of twisted endoscopy. Thus first I am going to review Kaletha's construction of supercuspidal L-packets and his proof of the standard endoscopic character relation. Then I will explain a few key points in the twisting process with an emphasis on Waldspurger's philosophy "l'endoscopic tordue n'est pas si tordue".

• Dec 9

George Lusztig
(MIT)

From families in Weyl groups to unipotent elements

• Dec 16
1pm*

Erez Lapid
(Weizmann Institute)

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