Lie Groups and Representation Theory Archives for Academic Year 2021


A mod p geometric Satake isomorphism

When: Mon, September 14, 2020 - 2:00pm
Where: Zoom
Speaker: Robert Cass (Harvard) - http://people.math.harvard.edu/~rcass/

Abstract: We explain a mod p version of the geometric Satake isomorphism which gives a sheaf-theoretic description of the spherical mod p Hecke algebra. In our setup the mod p Satake category is not controlled by the dual group but rather a certain affine monoid scheme. Our proofs rely crucially on the theory of F-singularities, and along the way we prove new results about the singularities of affine Schubert varieties. We also discuss some further results on the resulting monoid in joint work with Cedric Pepin.

An Arthur packet for real split G_2

When: Wed, September 30, 2020 - 2:00pm
Where: https://umd.zoom.us/j/96890967721
Speaker: Sam Mundy (Columbia University) - Abstract: In this talk we will have fun manipulating Arthur parameters for the exceptional group G_2. We will use these parameters to construct certain Arthur packets for G_2 via the work of Adams--Johnson. This is motivated by trying to understand cohomological properties of CAP representations for G_2.

An elliptic Fourier transform for unipotent representations of p-adic groups

When: Mon, October 5, 2020 - 2:00pm
Where: https://umd.zoom.us/j/96890967721
Speaker: Dan Ciubotaru (Oxford) - Abstract: An important ingredient in the character theory of finite groups of Lie type is Lusztig's nonabelian Fourier transform which is the change of basis matrix between the basis of irreducible characters and the basis of "almost characters" (trace functions for character sheaves). The expectation is that a similar picture should exist for admissible representations of reductive p-adic groups, and in the case of representations with unipotent cuspidal support, Lusztig (2014) proposed several conjectures. Independently, Moeglin and Waldspurger defined an involution on the space of tempered unipotent representations of the odd orthogonal groups as part of their proof of the stability of tempered unipotent L-packets. Motivated by a more recent reformulation of this involution by Waldspurger (2017), we define a "nonabelian Fourier transform" on the space of elliptic representations of a semisimple p-adic group and verify, for split exceptional groups, that it commutes, via maximal parahoric restrictions, with Lusztig's Fourier transform for the finite reductive quotients.

Algebraic groups with good reduction

When: Wed, October 21, 2020 - 2:00pm
Where: Zoom
Speaker: Igor Rapinchuk (Michigan State University) - https://sites.google.com/site/irapinchuk1/home
Abstract: Techniques involving reduction are very common in number theory and arithmetic geometry. In particular, elliptic curves and general abelian varieties having good reduction have been the subject of very intensive investigations over the years. The purpose of this talk is to report on recent work that focuses on good reduction in the context of reductive linear algebraic groups over finitely generated fields. In addition, we will highlight some applications to the study of local-global principles and the analysis of algebraic groups having the same maximal tori. (Parts of this work are joint with V. Chernousov and A. Rapinchuk.)

TBA

When: Mon, October 26, 2020 - 2:00pm
Where: https://umd.zoom.us/j/96890967721
Speaker: Laura Rider (University of Georgia) - Modular Perverse Sheaves on the affine Flag Variety
Abstract: There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equivalence. This was achieved by Bezrukavnikov in characteristic 0 about a decade ago. In this talk, I will discuss a first step toward solving this problem in the modular case joint with R. Bezrukavnikov and S. Riche.

Affine Deligne-Lusztig varieties and Generalized affine Springer fibers

When: Wed, November 11, 2020 - 8:00am
Where: Online
Speaker: Xuhua He (CUHK) -
Abstract: The notion of affine Springer fiber was introduced by Kazhdan and Lusztig in 1988. It plays a crucial role in the geometric representation theory and the Langlands program. The generalized affine Springer fibers were first studied by Kottwitz and Viehmann for the hyperspecial level structure in 2012 and by Lusztig for arbitrary parahoric level structure in 2015. Many geometric properties for the hyperspecial level structure were further studied by Bouthier and Chi.

In this talk, I will propose a new approach to study the generalized affine Springer fibers. The key observation is that the affine Deligne-Lusztig varieties, in some sense, may be regarded as the ``shadow'' of generalized affine Springer fibers. I will also explain some ingredients used to deduce some geometric properties (nonemptiness, dimension, irreducible components) of generalized affine Springer fibers from the properties of the affine Deligne-Lusztig varieties. This talk is based on a work in progress.

On the derived category of the Iwahori-Hecke algebra, I

When: Mon, November 16, 2020 - 2:00pm
Where: Online
Speaker: Eugen Hellmann (U. Münster) -
Abstract: We will discuss a conjecture that relates the derived category of smooth representations of a (split) p-adic reductive group to the derived category of quasi-coherent sheaves on a stack of L-parameters. Conjectures like that also were recently proposed by several people (X. Zhu, Ben-Zvi-Chen-Helm-Nadler, Fargues-Scholze).
In the case of the principal block of GL_n we will make the conjecture precise and relate it to the family of GL_n representations on the stack of L-parameters that interpolates the local Langlands correspondence, as proposed by Emerton and Helm. We will recall how to construct this Emerton-Helm family and explain why the (derived) tensor product with this family should realize the conjectured relation between categories of representations and categories of sheaves.

On the derived category of the Iwahori-Hecke algebra, II

When: Wed, November 18, 2020 - 2:00pm
Where: Online
Speaker: Eugen Hellmann (U. Münster) -
Abstract: We will discuss a conjecture that relates the derived category of smooth representations of a (split) p-adic reductive group to the derived category of quasi-coherent sheaves on a stack of L-parameters. Conjectures like that also were recently proposed by several people (X. Zhu, Ben-Zvi-Chen-Helm-Nadler, Fargues-Scholze).
In the case of the principal block of GL_n we will make the conjecture precise and relate it to the family of GL_n representations on the stack of L-parameters that interpolates the local Langlands correspondence, as proposed by Emerton and Helm. We will recall how to construct this Emerton-Helm family and explain why the (derived) tensor product with this family should realize the conjectured relation between categories of representations and categories of sheaves.

Representations of p-adic groups

When: Mon, November 23, 2020 - 2:00pm
Where: Online
Speaker: Jessica Fintzen (Cambridge U./Duke U.) -
Abstract: The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress, and outline some applications.


From representations of p-adic groups to congruences of automorphic forms

When: Mon, November 30, 2020 - 2:00pm
Where: Online
Speaker: Jessica Fintzen (Cambridge U. / Duke U.) -
Abstract: The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.
This is joint work with Sug Woo Shin.

A local Langlands conjecture for disconnected groups

When: Mon, December 7, 2020 - 2:00pm
Where: Online
Speaker: Tasho Kaletha (U. Michigan) -
Abstract: Langlands' conjectures are usually phrased in the setting of connected reductive groups. In this talk we will explore a generalization of the statement of the local conjectures to the setting of disconnected groups, subject to a certain mild restriction. Proving these conjectures in the simplest case -- that of a disconnected group whose identity component is a torus -- already shows an interesting new phenomenon. Time permitting, we will mention an application to the reduction of the refined local Langlands correspondence, both for connected and disconnected groups, to the case of discrete parameters.

What is a unipotent representation?

When: Mon, December 14, 2020 - 2:00pm
Where: Online
Speaker: Lucas Mason-Brown (Oxford University) -
Abstract: Let k be a finite field and let G(k) be the k-points of a connected reductive algebraic group. In 1984, Lusztig classified the irreducible finite-dimensional representations of G. He showed:

1) All irreducible representations of G(k) can be constructed out of a finite set of building blocks , called `unipotent representations.'

2) Unipotent representations can be classified by geometric data related to nilpotent co-adjoint orbits for a certain complex group associated to G(k).

Now, replace k with C, the field of complex numbers, and replace G(k) with G(C). There is a striking analogy between the finite-dimensional representation theory of G(k) and the unitary representation theory of G(C). This analogy suggests that all unitary representations of G(C) can be constructed out of a finite set of building blocks, called `unipotent representations', classified by nilpotent co-adjoint orbits. In this talk I will propose a definition of these building blocks. The definition I propose is geometric and case-free. After sketching the definition, I will give a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for Arthur's representations.

Mod p points on Shimura varieties of parahoric level, I

When: Mon, February 8, 2021 - 2:00pm
Where: Online
Speaker: Pol Van Hoften (King's College London) -
Abstract: The conjecture of Langlands-Rapoport gives a conjectural description of the mod p points of Shimura varieties, with applications towards computing the (semi-simple) zeta function of these Shimura varieties. The conjecture was proven by Kisin for abelian type Shimura varieties at primes of (hyperspecial) good reduction, after having constructed smooth integral models. For primes of (parahoric) bad reduction, Kisin and Pappas have constructed ”good” integral and the conjecture was generalised to this setting by Rapoport. In this talk I will discuss recent results towards the conjecture for these integral models, under minor hypothesis, building on earlier work of Zhou. Along the way we will see irreducibility results for various stratifications on special fibers of Shimura varieties, including irreducibility of central leaves and Ekedahl-Oort strata.

Mod p points on Shimura varieties of parahoric level, II

When: Wed, February 10, 2021 - 2:00pm
Where: Online
Speaker: Pol Van Hoften (Kings College London) -
Abstract: The conjecture of Langlands-Rapoport gives a conjectural description of the mod p points of Shimura varieties, with applications towards computing the (semi-simple) zeta function of these Shimura varieties. The conjecture was proven by Kisin for abelian type Shimura varieties at primes of (hyperspecial) good reduction, after having constructed smooth integral models. For primes of (parahoric) bad reduction, Kisin and Pappas have constructed ”good” integral and the conjecture was generalised to this setting by Rapoport. In this talk I will discuss recent results towards the conjecture for these integral models, under minor hypothesis, building on earlier work of Zhou. Along the way we will see irreducibility results for various stratifications on special fibers of Shimura varieties, including irreducibility of central leaves and Ekedahl-Oort strata.

Lifting involutions in a Weyl group to the normalized of the torus

When: Wed, March 10, 2021 - 2:00pm
Where: https://umd.zoom.us/j/96890967721
Speaker: Moshe Adrian (Queens College) -

Abstract: Let N be the normalizer of a maximal torus T in a split reductive group over F_q, and let w be an involution in the Weyl group N/T. Recently, Lusztig explicitly constructed a lift n of w in N, such that the image of n under the Frobenius map is equal to the inverse of n.

Lusztig’s construction is quite complicated, and the proof of his result is involved. We give a completely elementary construction, with even stronger properties.

Howe to transfer Harish-Chandra character via Weil’s representation

When: Mon, April 5, 2021 - 9:00am
Where: https://www-math.umd.edu/research/seminars/lie-groups-and-rep-theory-seminar.html
Speaker: Wee Teck Gan (University of Singapore) -
Abstract: Over a nonarchimedean field, we discuss the question ofrelating the Harish-Chandra characters of two representations which are
theta lifts of each other, in the setting of the dual pair SO(2n+1) x
Mp(2n).

On the computation of Green functions

When: Wed, April 7, 2021 - 2:00pm
Where: Online
Speaker: Meinolf Geck (Stuttgart) -
Abstract: We report on recent progress on the computation of the
Green functions of a reductive group over a finite field,
as introduced by Deligne and Lusztig in the 1970s. By work
of Lusztig and Shoji, it is known that these Green functions
coincide with another type of Green functions defined in
terms of character sheaves. And there is a purely combinatorial
algorithm for computing the values of these functions, up to
certain signs. These signs have been explicitly determined in
almost all cases. We show how the missing cases, which occur
in groups of exceptional type in bad characteristics, can be
solved by a purely group-theoretical computation.

New tools for the computation of discrete series multiplicities for classical groups over Z

When: Wed, April 28, 2021 - 2:00pm
Where: Online
Speaker: Olivier Taibi (ENS de Lyon) - https://otaibi.perso.math.cnrs.fr/
Abstract: In 2014 I gave an algorithm to compute certain orbital integrals for the unit element of the unramified Hecke algebra of a p-adic classical group. This allowed me to compute the geometric side of Arthur's "L^2 Lefschetz" trace formula giving averaged discrete series multiplicities in the level one discrete automorphic spectrum of classical groups over Z of rank at most 6. Equivalently, this gives (non-trivially) explicit formulas for the number of self-dual level one cuspidal algebraic regular automorphic representations of general linear groups over Q in dimension at most 13. This followed work of Ga\"etan Chenevier and David Renard using definite classical groups over Z. More recently Ga\"etan Chenevier and I introduced a new method, using the Weil explicit formula, to explicitly compute the geometric side of the above trace formula without computing any orbital integral, up to rank 8 (i.e. up to
dimension 17). In this talk I will recall these methods and explain how to bring them together and other tools to handle cases of even higher rank (up to dimension 24). This is part of a long-term project with Ga\"etan Chenevier.