MIT Lie Groups Seminar
2022  2023
Meetings: 4:00pm on Wednesdays
This seminar will take place either inperson or online. For inperson seminars, it will be held at 2142. You are welcome to join inperson seminars by Zoom. For remote participation, the Zoom link is the same as last year's. You can email JuLee Kim for the Zoom meeting link. To access videos of talks, please email for the password.
Spring 2023

Feb 8
Kari Vilonen
(Melbourne)Character sheaves for graded Lie algebras and invariant systems
Abstract: A variant of the nearby cycle construction, due initially to Grinberg, is the main geometric ingredient in our construction of character sheaves for graded Lie algebras. In this talk I will explain this construction and how it allows us to associate character sheaves to representations of Hecke algebras at roots of unity. In particular, I will explain how the Hecke relations arise from appropriate bfunctions associated to sections of certain Dmodules. The bfunctions that arise in our applications can be deduced from previous results of Fumihiro Saito on prehomogenous vector spaces. I will also explain the relationship to invariant systems of differential equations. This is joint work with Ting Xue and Misha Grinberg.

Feb 15
Zhiwei Yun
(MIT)An irregular DeligneSimpson problem and Cherednik algebras
Abstract: The DeligneSimpson problem asks for a criterion of the existence of connections on an algebraic curve with prescribed singularities at punctures. We give a solution to a generalization of this problem to Gconnections on P^1 with a regular singularity and an irregular singularity (with a condition called isoclinic). Here G can be any complex reductive group. Perhaps surprisingly, the solution is expressed in terms of rational Cherednik algebras. This is joint work with Konstantin Jakob.

Feb 22
George Lusztig
(MIT)Unipotent character sheaves and strata of reductive groups
Abstract: Let G be a connected reductive group over an algebraically closed field. There is a natural partition of G into finitely many subsets called strata. Each stratum is a union of conjugacy classes of fixed dimension. The strata of G can be indexed by a set Str(G) which is independent of the ground field. Let CS(G) be the set of unipotent character sheaves (up to isomorphism). These are certain simple perverse sheaves on G. It turns out that there is a natural surjective map from CS(G) to Str(G).

Mar 1
Elad Zelingher
(U. Michigan)On gamma factors and Bessel functions of finite general linear groups
Abstract: In the representation theory of padic groups, one way to study irreducible representations is by attaching local factors to them. These local factors are complex valued functions of a complex variable, and they often encode properties of the representations in question. In this talk, I will explain an analog of these local factors for irreducible generic representations of GL(n, F_q). I will talk about two families of gamma factors for the tensor product representation (based on constructions by JacquetPiatetskiShapiroShalika and Shahidi) and about the Bessel function of a generic representation (defined by GelfandGraev). I will give a conceptual quantitative interpretation for the absolute value of these gamma factors. I will also explain a relation between the Bessel function and Kloosterman sums and sheaves. This talk is based on my two recent preprints, where the most recent one is a joint work with David Soudry.

March 8
Xinwen Zhu
(Stanford)The unipotent categorical local Langlands correspondence
Abstract: I will discuss a conjectural categorical form of the (arithmetic) local Langlands correspondence for padic groups and establish the unipotent part of such correspondence (for characteristic zero coefficient field). As an application, we show that the coherent Springer sheaf on the moduli of unipotent Langlands parameter is a honest coherent sheaf (i.e. a complex concentrated in degree zero). Joint work with Tamir Hemo.

March 15
Tiep H. Pham
Character bounds for finite groups of Lie type
Abstract: Given the current knowledge of complex representations of finite groups of Lie type, obtaining good upper bounds for their character values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We will report on recent results that produce such character bounds, and discuss some applications of them, in and outside of group theory.

March 22
Artem Kalmykov
Whittaker vectors and vertexIRF transformation
Abstract: In this talk I will explain how a certain construction from the theory of integrable systems (a vertexIRF transformation) can be interpreted in terms of familiar objects from (geometric) representation theory and how this interpretation gives new results of independent interest. For instance, I will show how to construct Whittaker vectors for representations of \mathfrak{gl}_n using the GelfandTsetlin subalgebra and how it is related to nonstandard quantizations of the group GL_n.

March 29
Spring Break (no seminar)

April 5
Weinan Zhang
(U. Virginia)Relative braid group symmetries on iquantum groups
Abstract: Introduced by Lusztig in the 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The iquantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, which are now viewed as natural generalizations of quantum groups. In this talk, I will present our construction of relative braid group symmetries (associated to the relative Weyl group of a symmetric pair) on iquantum groups. These new symmetries admit many properties similar to Lusztig's symmetries. Our construction has further led to compatible relative braid group actions on modules. This is joint work with Weiqiang Wang.

April 12
Sam Raskin
(Texas)Recent progress in the geometric Langlands program
Abstract: The geometric Langlands conjecture of BeilinsonDrinfeld describes the category of Dmodules on the moduli stack of Gbundles on a smooth projective curve in terms of spectral data involving the dual group. It can be thought of as an analogue of Langlands's conjectures in the unramified function field case, but where all of the objects have purely algebrogeometric nature. In this talk, I will describe ongoing work with Dennis Gaitsgory settling this problem. I will aim to keep the talk as accessible as possible while emphasizing the new aspects of our work.

April 19
Andrew Salmon
(MIT)Nearby cycles over general bases and the tamely ramified Langlands correspondence for function fields
Abstract: Vincent Lafforgue gave a construction of a semisimple Langlands parameter from an automorphic form for any reductive group over a function field by using excursion operators. Our aim is to give a general approach for proving certain localglobal compatibilities satisfied by these Langlands parameters. The main consequence for the Langlands correspondence is to show that Lafforgue's construction is compatible with Lusztig's theory of character sheaves at a given point of a smooth curve over a finite field. Namely, using the theory of character sheaves, one attaches a torus character and a twosided cell to an irreducible representation of a reductive group over a finite field. If our automorphic form lives in an isotypic component determined by this irreducible representation, we show that the torus character and twosided cell determine the semisimple and unipotent parts of the image of the tame generator under the Langlands correspondence, respectively.
The main theorem in the general approach to localglobal compatibility is that nearby cycles commute with pushforward of certain perverse sheaves from the stack of global shtukas to a power of a curve. To prove this result, the main technical ingredient is the notion of what we call $\Psi$factorizability, where nearby cycles over a general base are independent of the composition of specializations chosen, and the $\Psi$factorizability statements we make give some answers to a question raised by GenestierLafforgue. Using this theorem, to compute the action of framed excursion operators, one may instead compute the monodromy of certain nearby cycles sheaves on certain restricted shtukas. In our case of interest, perverse sheaves over restricted shtukas are related to the monodromic affine Hecke category by a horocycle correspondence. That is, one may reduce certain questions in the global function field Langlands program to questions in local geometric Langlands. 
April 26
Julia Pevtsova
(U. Washington)Fiberwise approach to support theories
Abstract: Given a tensor triangulated category T one can try to study it by establishing an “exhaustive” family of fiber functors $F_\lambda: T \to T_\lambda$ where the categories $T_\lambda$ are already “understood”. I’ll focus on two specific examples where this strategy leads to some (partial) information about the Balmer spectrum for the category T. One example deals with the stable category of lattices for a finite group scheme defined over a ring unifying the classical results of Hopkins, Neeman for a derived category of a commutative ring with more recent theorems in modular representation theory. The other considers the spectrum of a stable category of a small quantum group. Joint work with T. Barthel, D. Benson, S. Iyengar, H. Krause; and C. Negron.

May 3
Nate Harman
(U. Michigan)Discrete preTannakian categories
Abstract: PreTannakian categories are a purely categorical axiomatization of the additional structure the category of finite dimensional representations of a group has. If a PreTannakian category C admits a structure preserving "fiber functor" to vector spaces then Tannakian reconstruction says that necessarily C = Rep(G) for some group (scheme) G. However there are a number of interesting examples of preTannakian categories which do not admit a fiber functor, and classifying them is an important open problem. We introduce a notion of a discrete preTannakian category, with the property that Rep(G) is discrete if and only if G is a profinite group, and show that all discrete preTannakian categories come from oligomorphic groups via a previous construction of ours.

May 10
Oron Propp
(MIT)A coherent categorification of the asymptotic affine Hecke algebra
Abstract: We describe a new realization of Lusztig's asymptotic affine Hecke algebra $J$ in terms of coherent sheaves on a moduli stack of Deligne–Langlands parameters. More precisely, we show that $J$ arises from a certain restriction of the "coherent Springer sheaf," which is associated to the (usual) affine Hecke algebra via the categorical local Langlands correspondence. We will then explain how to "upgrade" this sheaftheoretic realization of $J$ to a categorification using certain coherent sheaves on Springer fibers, following a conjecture of Qiu–Xi. Finally, we will use the asymptotic picture to motivate the result, independently announced by Hemo–Zhu, that the coherent Springer sheaf lies in cohomological degree $0$ (i.e., is a sheaf rather than a complex).

May 17
Roman Bezrukavnikov
(MIT)Commuting pairs and unipotent representations
Abstract: I will report on a joint project with Dan Ciubotaru, David Kazhdan and Yakov Varshavsky, in which we describe unipotent invariant distributions on a padic group in terms of the Langlands dual group. Some of our constructions are related to the results by Li, Nadler, Yun presented by Zhiwei Yun in his recent seminar talk.
Fall 2022

Sept 7
2142
Local geometric langlands and affine BeilinsonBernstein localization
Abstract: We will exposit some tenets of the local geometric Langlands philosophy with concrete representationtheoretic consequences. In particular, we will focus on the problem of affine BeilinsonBernstein localization at the critical level. Then, we will explain how to actually prove some of these statements. Some of this work is joint with Sam Raskin.

Sept 14
(10am)ChengChiang Tsai
(Taiwan)Zoom
Wavefront set and graded Springer theory
Abstract: For characters of padic reductive groups there is the notion of wavefront set, which is a set of nilpotent orbits that describes the asymptotic behavior of a character near the identity. (Maybe one can think of the closure of wavefront set as singular support.) There is a longstanding conjecture that any wavefront set is contained in a single geometric orbit, as worked out by many authors for several types of depth0 representations. In this talk, we explain how the above conjecture cannot hold in general, because an analogous assertion does not hold for graded Lie algebras. We will discuss this last statement in the context of graded Springer theory.

Sept 21
2142
Wavefront sets and unipotent representations of padic groups
Abstract: An important invariant for admissible representations of reductive padic groups is the wavefront set, the collection of the maximal nilpotent orbits in the support of the orbital integrals that occur in the HarishChandraHowe local character expansion. We compute the geometric and Okada's canonical unramified wavefront sets for representations in Lusztig's category of unipotent reduction for a split group in terms of the KazhdanLusztig parameters. We use this calculation to give a new characterisation of the antitempered unipotent Arthur packets. Another interesting consequence is that the geometric wavefront set of a unipotent supercuspidal representation uniquely determines the nilpotent part of the Langlands parameter; this is an extension to padic groups of Lusztig's result for unipotent representations of finite groups of Lie type. The talk is based on joint work with Lucas MasonBrown and Emile Okada.

Sept 23
Miller, Adams
Achar, Lusztig2190

Sept 24
MasonBrown
Nevins, Trapa2190

Sept 28
(3pm)Raphael BeuzartPlessis (Marseille)
Zoom
On the formal degree conjecture for classical groups
Abstract: A conjecture of Hiraga, Ichino and Ikeda expresses the formal degree of a discrete series of a (algebraic) reductive group over a local field in terms of the adjoint gamma factor of its Langlands parameter. It can be checked for real reductive groups using work of Harish Chandra and the explicit form of the Langlands correspondence in this case. For classical groups over a padic field, the conjecture was established for odd orthogonal groups as well as unitary groups by two completely different methods. In this talk, I will explain a proof for symplectic and even orthogonal padic groups based on properties of twisted endoscopy and ideas originating from Shahidi relating residue of intertwining operators to twisted orbital integrals. This method can actually be readily adapted to treat odd orthogonal and unitary groups as well.

Oct. 5
2142
Affine Springer fibers and sheaves on Hilbert scheme of points on the plane.
Abstract: My talk is based on the joint work with E. Gorsky and O. Kivinen. I will explain a construction that associates a coherent sheaf on the Hilbert scheme of points on the plane to plane curve singularity. The global sections of the sheaf are equal to cohomology of the corresponding Affine (type A) Springer fiber. The construction categorifies HOMFLYPT homology/cohomogy of compactified Jacobian conjecture if combined with Soergel bimodule/ Sheaves of Hilbert scheme theorem of OblomkovRozansky. I will also discuss generalizations outside of type A.

Oct. 12
Dennis Gaitsgory
(Bonn)2142
From geometric Langlands to classical via the trace of Frobenius
Abstract: I'll start by summarizing the main results of the series [AGKRRV], where it is shown that the trace of Frobenius on the category of automorphic sheaves with nilpotent singular support identifies with the space of unramified automorphic functions. We'll then discuss conjectural counterparts of this statement in the local and global ramified settings.

Oct. 19
2142
Multiplicative Hitchin fibration and Fundamental Lemma
Abstract: Given a reductive group G and some auxiliary data, one has the Hitchin fibration associated with the adjoint action of G on Lie(G), which is successfully used by B. C. Ngô to prove the endoscopic fundamental lemma for Lie algebras. Following the same idea, there is a group analogue called the multiplicative Hitchin fibration by replacing the Lie algebra with reductive monoids, and one can hope to directly prove the fundamental lemma at group level. This project is close to completion and we report the results so far. There are many new features that are not present in the additive case, among which is a pleasant surprise that there might be some strata in the support theorem that are not explained by endoscopy.

Oct. 26
2142
On some Hecke algebra modules arising from theta correspondence and it’s deformation
Abstract: This talk is based on the joint work with Jiajun Ma and Congling Qiu on theta correspondence of type I dual pairs over a finite field F_q. We study the Hecke algebra modules arising from theta correspondence between certain HarishChandra series for these dual pairs. We first show that the normalization of the corresponding Hecke algebra is related to the first occurrence index, which leads to a proof of the conservation relation. We then study the deformation of this Hecke algebra module at q=1 and generalize the results of AubertMichelRouquier and Pan on theta correspondence between unipotent representations along this way.

Nov. 2
Alexander Bertoloni Meli
(U. Michigan)2142
Comparison of local Langlands correspondences for odd unitary groups
Abstract: I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a padic field, there is a Langlands correspondence using trace formula techniques due to Mok and KalethaMinguezShinWhite. Another correspondence was constructed recently by FarguesScholze using padic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.

Nov. 9
Joel Kamnitzer
(Toronto)2142
Moduli space of flower curves
Abstract: The DeligneMumford moduli space of genus 0 curves plays many roles in representation theory. For example, the fundamental group of its real locus is the cactus group which acts on tensor products of crystals. I will discuss a variant on this space which parametrizes "flower curves". The fundamental group of the real locus of this space is the virtual cactus group. This moduli space of flower curves is also the parameter space for inhomogeneous Gaudin algebras.

Nov. 16
Carl Mautner
(UC Riverside)2142
Perverse sheaves on symmetric products of the plane
Abstract: In joint work with Tom Braden we give a purely algebraic description of the category of perverse sheaves (with coefficients in any field) on $S^n(C^2)$, the nfold symmetric product of the plane. In particular, using the geometry of the Hilbert scheme of points, we relate this category to the symmetric group and its representation ring. Our work is motivated by analogous structure appearing in the Springer resolution and HilbertChow morphism..

Nov. 23
Asilata Bapat
Zoom
Categorical qdeformed rational numbers and compactifications of stability space
Abstract: We discuss new categorical interpretations of two distinct qdeformations of the rational numbers. The first one was introduced in a different context by MorierGenoud and Ovsienko, and enjoys fascinating combinatorial, topological, and algebraic properties. The second one is a natural partner to the first, and is new. We obtain these deformations via boundary points of a compactification of the space of Bridgeland stability conditions on the 2CalabiYau category of the A2 quiver. The talk is based on joint work with Louis Becker, Anand Deopurkar, and Anthony Licata.

Nov. 30
Elijah Bodish
(MIT)2142
Webs for symplectic and orthogonal groups
Abstract: In joint work with Ben Elias, David Rose, and Logan Tatham, we give a generators and relations presentation of the monoidal category generated by fundamental representations of the symplectic group (and its qanalogue). I will discuss our work, which generalizes both the TemperelyLieb algebra description of representations of SL_2 and Kuperberg’s ``spider" description of representations of Sp_4. I will then comment on joint work in progress with Haihan Wu which gives an analogous presentation of the monoidal category generated by exterior powers of the vector representation of the orthogonal group (and its qanalogue).

Dec. 7
Calder MortonFerguson
(MIT)2142
KazhdanLaumon Category O, Schwartz space, and the semiinfinite flag variety
Abstract: We define an analogue of Category O in the context of Kazhdan and Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U. We explicitly classify its simple objects, and then use our understanding of the structure of this category to discuss its connections to some other interesting objects in representation theory, namely BravermanKazhdan's Schwartz space on G/U and perverse sheaves on the semiinfinite flag variety.
Archive
Contact:
Roman Bezrukavnikov
Pavel Etingof
JuLee
Kim
Zhiwei Yun