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MIT Lie Groups Seminar

2023 - 2024

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 Ju-Lee Kim for the Zoom meeting Link and for the passcode to access videos of talks.

Spring 2024

  • Feb 7

    Dougal Davis
    (Melbourne)

    2-142

    Unitary representations of real groups and localization theory for Hodge modules

    Abstract: Wilfried Schmid and Kari Vilonen have conjectured that the unitarity of an irreducible representation of a real reductive group can be read off from a canonical filtration, the Hodge filtration. This filtration arises naturally from Beilinson-Bernstein localization and the deep theory of mixed Hodge modules on the complex flag variety. In this talk, I will explain a proof of this conjecture, obtained in recent joint work with Vilonen (arXiv:2309.13215). Our proof completes a sketch of Adams, Trapa and Vogan (based on the Atlas algorithm) by establishing two missing ingredients: a version of localization for Hodge modules and a wall-crossing theory for their Hodge filtrations. Time permitting, I may also indicate some applications to the orbit method, to appear in joint work with Lucas Mason-Brown.

    Video
  • Feb 14

    Elon Lindenstrauss

  • Feb 21

    Jianqiao Xia
    (Harvard)

    2-142

    Equivalence of Hecke Categories with Deeper Level Structures

    Abstract: Inspired by the theory of positive depth representations of p-adic reductive groups, we study Hecke categories associated to certain open compact subgroups smaller than the Iwahori subgroup. In this talk, I will prove that in some cases these Hecke categories are monoidally equivalent to affine Hecke categories of smaller groups, therefore having applications to the local geometric Langlands correspondence. Using sheaf-function correspondence, our categorical equivalence recovers a family of Hecke algebra isomorphisms already proven by Ju-Lee Kim.

  • Feb 28

    David Ben-Zvi
    (U.Texas, Austin)

    2-142

    Coherent and Constructible Local Langlands

    Abstract: Refined forms of the Local Langlands Correspondence (LLC) relate representations of reductive groups over local fields with sheaves on spaces of Langlands parameters -- either perverse sheaves, as in Vogan's conjectures and Lusztig's unipotent LLC, or coherent sheaves, as in Fargues' conjecture. In this talk I will explain joint work with Harrison Chen, David Helm and David Nadler relating the two settings using circle actions in derived algebraic geometry. Specifically, the constructible LLC emerges as a deformation (with respect to an equivariant parameter) of coherent LLC on a full subcategory.

  • Mar 6

    Peng Shan

    2-142

  • Mar 13

    Tasho Kaletha

    2-142

  • Mar 20

    Anne-Marie Aubert

    2-142

  • Mar 27

    Spring break

  • Apr 3

    Bảo Châu Ngô

    2-142

  • Apr 10

    Simon Riche

    2-142

  • Apr 17

    Charles Fu

    2-142

  • Apr 24

    Charlotte Chan

    2-142

  • May 1

    Anna Szumowicz

    2-142

  • May 8

    Mingjia Zhang
    (Princeton University)

    2-142

Fall 2023

  • September 13

    David Vogan
    (MIT)

    2-142

    Generalizing endoscopic transfer

    Abstract: The notion of endoscopic group was created by Langlands, Shelstad, and others beginning in the 1970s, in order to study problems in harmonic analysis on a reductive group $G$: for example, the nature of {\em characters} of irreducible representations of $G$. An endoscopic group $H$ for $G$ is a smaller reductive group, equipped with a natural ``endoscopic transfer'' map from characters of $H$ to characters of $G$.

    Perhaps the simplest example of such an endoscopic group is the Levi subgroup $M$ of a rational parabolic subgroup $P = MU$. Endoscopic transfer in this case is just the Mackey-Gelfand notion of {\em parabolic induction} from $M$ to $G$.

    This example of rational parabolic induction is almost never mentioned in the literature on endoscopy, because endoscopy offers little that is new in that case. But I believe that it sheds some light on the nature of the Langlands-Shelstad theory.

    I will talk about a (real groups) generalization of endoscopic groups and endoscopic transfer, for which the simplest example is the Levi subgroup $L$ of a theta-stable parabolic; and transfer is {\em cohomological induction} from $L$ to $G$. The formalism appears to make sense for any local field, and should lead to a generalized definition of endoscopic transfer once an appropriate local Langlands conjecture is proved.

    This is joint work with Lucas Mason-Brown and Jeffrey Adams.

    Slides

    Video

  • September 20

    Huanchen Bao
    (Singapore)

    2-142

    Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs

    Abstract: Let G be a connected reductive group over an algebraically closed field. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. In this talk, we shall first recall the construction of Chevalley group schemes by Lusztig using quantum groups. Then we shall discuss the construction of symmetric subgroup schemes parameterising symmetric subgroups K of G using quantum symmetric pairs. The existence of such group schemes allows us to apply characteristic p methods to study the geometry of K-orbits on the flag variety of G. This leads to a construction of Frobenius splittings via quantum symmetric pairs, generalising the algebraic Frobenius splittings by Kumar-Littelmann. This is based on joint work with Jinfeng Song (NUS).

    Slides

    Video

  • September 27

    George Lusztig
    (MIT)

    2-142

    Pre-cuspidal families and indexing of Weyl group representations

    Abstract: We define pre-cuspidal families in proper parabolic subgroups of a Weyl group and show how to use them to index the irreducible representations of that Weyl group in terms of certain pairs of finite groups.

    Slides

  • October 4

    Kenta Suzuki
    (MIT)

    2-142

    The explicit Local Langlands Correspondence for G2 and GSp4, character formulas and stability

    Abstract: I will talk about my recent joint work with Yujie Xu, where we prove the Local Langlands Conjecture explicitly for G2 (combined with previous work of Aubert-Xu) and GSp(4). I will discuss Aubert-Moussaoui-Solleveld's construction of the cuspidal support map using the generalized Springer correspondence, which serves as a key property of our correspondence. I will also discuss character formulas in terms of generalized Green functions to pin down the "mixed packets," i.e., L-packets with multiple cuspidal supports. In the case of GSp(4), we classify the stable distributions on GSp(4), extending the methods of DeBacker-Kazhdan, using the homogeneity result of DeBacker and Waldspurger.

    Video

  • October 11

    Pablo Boixeda Alvarez
    (Yale)

    2-142

    The center of the small quantum group and affine springer fibers

    Abstract: The quantum group U_q is Hopf-algebra deforming the enveloping al- gebra introduced by Lusztig. The representation theory of this algebra is particularly interesting at l-th roots of one, where it includes a finite dimensional subalgebra known as the small quantum group. In joint work with Bezrukavnikov, Shan and Vasserot we construct an injective map to the center of this algebra from the cohomology of a certain affine Springer fiber Fl_{ts} for s a regular semisimple element. In recent progress we check that this map is surjective in type A and get a bound on dimension in general types related to the diagonal coinvariant algebra. We also give an algebro geometric description of the spectrum of the cohomology of the Springer fiber. The work relies on understanding the representation category through a filtration coming from intersection with G[[t]]-orbits in Fl_{ts}. In this talk I will present the result and related properties of this filtration of the category

    Video

  • October 18

    David Yang
    (MIT)

    2-142

    A stratification of the moduli space of local systems on the punctured disc and applications

    Abstract:We will define a stratification on the moduli space of local systems on the punctured disc and prove some properties of it. This stratification has a counterpart for categories with an action of the loop group, and we will describe a conjectural application of this formalism to the representation theory of W-algebras.

    Video

  • October 25

    Chengze Duan
    (U. Maryland)

    2-142

    Good position braids and transversal slices

    Abstract: Let G be a reductive group over an algebraically closed field and W be its Weyl group. Using Coxeter elements, Steinberg constructed cross-sections of the adjoint quotient of G which also yield transversal slices of regular unipotent classes. In 2012, He and Lusztig constructed transversal slices using minimal length elements in elliptic conjugacy classes in W, which yield transversal slices of basic unipotent classes. In this talk, we generalize minimal length elements to good position braids in the associated braid monoid of W and use these elements to construct transversal slices of all unipotent classes in G. We shall see these new elements also appear in many other aspects of representation theory, such as affine Springer fibers and the partial order on unipotent classes, etc.

    Video

  • November 1

    Minh-Tâm Trinh
    (MIT)

    2-142

    Φ-Harish-Chandra Series, Level-Rank Duality, and Affine Springer Fibers

    Abstract: Broué–Malle–Michel noticed that for any m, the unipotent irreducible characters of a finite reductive group G can be partitioned into “m-twisted” Harish-Chandra (HC) series with “m-twisted” Howlett–Lehrer parametrizations by irreducible characters of relative Weyl groups, recovering the usual notions when m = 1. Ting Xue and I conjecture that for any G and ℓ and m, the intersection of an ℓ-twisted HC series and an m-twisted HC series is simultaneously parametrized by a union of m-blocks for a certain Hecke algebra on the ℓ side and a union of ℓ-blocks for a certain Hecke algebra on the m side, in a way that matches up blocks. We show that when G = GL(n), this is Uglov’s level-rank duality in disguise. More surprisingly, we conjecture that these bijections are (essentially) realized by bimodules that Oblomkov–Yun and Boixeda Alvarez–Losev construct from the cohomology of affine Springer fibers.

    Video

  • November 8

    Junliang Shen
    (Yale/MIT)

    2-142

    Fourier transforms and perverse filtrations for abelian fibrations

    Abstract: The perverse filtration captures interesting homological information of algebraic maps. In recent years, perverse filtrations are found to share surprising connections to studies of non-abelian Hodge theory (the P=W conjecture), enumerative geometry (refined BPS invariants), and planar singularities (DAHA, knot invariants). In this talk, I will explain a theory of Fourier transform for abelian fibrations, which provides a uniform explanation of certain mysterious features predicted by the connections mentioned above. This Fourier theory can be viewed as an extension of the Beauville decomposition from abelian schemes to certain abelian fibrations with singular fibers. In particular, I will discuss how/why such an extension is possible when singular fibers break (most) symmetries of the geometry. Based on joint work with Davesh Maulik and Qizheng Yin.

    Video

  • November 15

    Akshay Venkatesh
    (IAS, Princeton)

    Some examples in relative Langlands duality

    Abstract: This talk is based on joint work with Ben-Zvi and Sakellaridis. In our preprint "relative Langlands duality" we propose that duality switches not only reductive groups but a distinguished class of Hamiltonian spaces under those groups.

    I will discuss some of the simplest examples, emphasizing some nice aspects of their geometry. Then I will discuss a still-tentative picture of how this relative duality manifests itself in local Langlands. If time permits, I will discuss how this story relates to Regge symmetry of 3j and 6j symbols.

    Video

  • November 29

    Tsao Hsien Chen
    (U. Minnesota)

    2-142

    Mars-Springer slices for loop spaces of symmetric varieties

    Abstract: Let X be a symmetric variety. J. Mars and T. Springer constructed conical transversal slices to the closure of Borel orbits on X and used them to show that the IC-complexes for the orbit closures are pointwise pure. This is an important geometric ingredient in their work on Hecke algebra representations associated to symmetric varieties providing a more geometric approach to the results of Lusztig-Vogan. In the talk, I will discuss a generalization of Mars-Springer's construction of transversal slices to the setting of the loop space LX of X where we consider closures of spherical orbits on LX. I will explain applications to relative Langlands duality. If time permits, I will discuss the case of closures of Iwahori orbits on LX. This is a joint work with Lingfei Yi.

    Video

  • December 6

    Spencer Leslie
    (Boston College)

    2-142

    Endoscopic symmetric varieties and rationality

    Abstract: Let G be a quasi-split reductive group over a field k and let X be a spherical variety. Motivated by applications toward relative trace formulae, we review and refine rationality results of Borovoi-Gagliardi, giving a solution (away from characteristic two) when X is symmetric. In good cases, the answer is intimately connected with the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I then discuss the source of these questions in the theory of endoscopy in the context of the relative Langlands program. Finally, I will outline the construction of endoscopic varieties, which are symmetric varieties of an associated endoscopic group. The construction works for most hyperspherical varieties induced from symmetric varieties.

    Video

  • December 13

    Pavel Etingof
    (MIT)

    2-142

    Analytic Langlands correspondence

    Abstract:




Archive

Contact: Roman Bezrukavnikov
Ju-Lee Kim
Zhiwei Yun

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