# MIT Lie Groups Seminar

2021 - 2022

Meetings: 4:00pm on Wednesdays

This seminar will take place either in-person or online. For in-person seminars, it will be held at 2-142. You are welcome to join in-person seminars by Zoom. For remote participation, the Zoom link is the same as last year's. You can email Andre Dixon or Pavel Etingof for the Zoom meeting Link. To access videos of talks, please email Andre Dixon for the password.

## Fall 2021

• Sept 8

David Vogan
(MIT)

2-142

Constructing unipotent representations

Abstract: In the 1950s, Mackey began a systematic analysis of unitary representations of groups in terms of "induction" from normal subgroups. Ultimately this led to a fairly good reduction of unitary representation theory to the case of simple groups, which lack interesting normal subgroups. At about the same time, Gelfand and Harish-Chandra understood that many representations of simple groups could be constructed using induction from parabolic subgroups. After many refinements and extensions of this work, there still remain a number of interesting representations of simple groups that are often not obtained by parabolic induction.

For the case of real reductive groups, I will discuss a certain (finite) family of representations, called unipotent, whose existence was conjectured by Arthur in the 1980s. Some unipotent representations can in fact be obtained by parabolic induction; I will talk about when this ought to happen, and about the (rather rare) cases in which Arthur's unipotent representations are not induced. (A lot of what I will say is meaningful and interesting over local or finite fields, but I know almost nothing about those cases.)

Slides

Video

• Sep 15

Wicher Malten
(Oxford)

Zoom

From braids to transverse slices in reductive groups

Abstract: We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties. Also building upon recent work of He-Nie, our perspective furnishes a common generalisation and a simple geometric criterion for Weyl group elements to yield strictly transverse slices.

Slides

Video

• Sep 22

George Lusztig
(MIT)

2-142

Total positivity in symmetric spaces

Abstract: The theory of total positive matrices in GL_n(R) was initiated by Schoenberg (1930) and Gantmacher-Krein (1935) and extended to reductive groups in my 1994 paper. It turns out that much of the theory makes sense also for symmetric spaces although some new features arise.

Video

• Sep 29

Ivan Losev
(Yale)

2-142

Harish-Chandra modules over quantizations of nilpotent orbits

Abstract: Let O be a nilpotent orbit in a semisimple Lie algebra over the complex numbers. Then it makes sense to talk about filtered quantizations of O, these are certain associative algebras that necessarily come with a preferred homomorphism from the universal enveloping algebra. Assume that the codimension of the boundary of O is at least 4, this is the case for all birationally rigid orbits (but six in the exceptional type), for example. In my talk I will explain a geometric classification of faithful irreducible Harish-Chandra modules over quantizations of O, concentrating on the case of canonical quantizations -- this gives rise to modules that could be called unipotent. The talk is based on a joint paper with Shilin Yu (in preparation).

Slides

Video

• Oct 6
10AM

Xuhua He
(Chinese U.
Hong Kong)

Zoom

Frobenius-twisted conjugacy classes of loop groups and Demazure product of Iwahaori-Weyl groups

Abstract: The affine Deligne-Lusztig varieties, roughly speaking, describe the intersection of Iwahori-double cosets and Frobenius-twisted conjugacy classes in a loop group. For each fixed Iwahori-double coset $I w I$, there exists a unique Frobenius-twisted conjugacy class whose intersection with $I w I$ is open dense in $I w I$. Such Frobenius-twisted conjugacy class $[b_w]$ is called the generic Frobenius-twisted conjugacy class with respect to the element $w$. Understanding $[b_w]$ leads to some important consequences in the study of affine Deligne-Lusztig varieties. In this talk, I will give an explicit description of $[b_w]$ in terms of Demazure product of the Iwahori-Weyl groups. It is worth pointing out that a priori, $[b_w]$ is related to the conjugation action on $I w I$, and it is interesting that $[b_w]$ can be described using Demazure product instead of conjugation action. This is based on my preprint arXiv:2107.14461.

If time allows, I will also discuss an interesting application. Lusztig and Vogan recently introduced a map from the set of translations to the set of dominant translations in the Iwahori-Weyl group. As an application of the connection between $[b_w]$ and Demazure product, we will give an explicit formula for the map of Lusztig and Vogan.

Slides

Video

• Oct 13

Tony Feng
(MIT)

2-142

Derived Chevalley isomorphisms

Abstract: For a reductive group G, the classical Chevalley isomorphism identifies conjugation-invariant functions on G with Weyl-invariant functions on its maximal torus. Berest-Ramadoss-Yeung have conjectured a derived upgrade of this statement, which predicts that the conjugation-invariant functions on the derived commuting variety of G identify with the Weyl-invariant functions on the derived commuting variety of its maximal torus. In joint work with Dennis Gaitsgory we deduce this conjecture for G = GL_n from investigations into derived aspects of the local Langlands correspondence. I’ll explain this story, assuming no background in derived algebraic geometry.

Video

• Oct 20

Yaping Yang
(U. Melbourne)

Zoom

Frobenii on Morava E-theoretical quantum groups

Abstract: In this talk, I will explain a connection between stable homotopy theory and representation theory. I will focus on one application of this idea to a problem arising from the modular representation theory. More explicitly, we study a family of new quantum groups labelled by a prime number and a positive integer constructed using the Morava E-theories. Those quantum groups are related to Lusztig's 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic. This talk is based on my joint work with Gufang Zhao.

• Oct 27

Yuri Berest
(Cornell)

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• Nov 3

Peter Crooks
(Northeastern)

Abstract:

• Nov 10

Milen Yakimov
(Northeastern)

Abstract:

• Nov 17

Alexander Braverman

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• Nov 24

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• Dec 1

Tasho Kaletha

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• Dec 8

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## Archive

Contact: Andre Dixon

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