MIT Lie Groups Seminar

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

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.

Video

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.

Video

Abstract: I will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. I will also explain how to match the modular group action on the characters of representations with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. If time permits, I will discuss some extension of these results to non-admissible levels and some conjectures about their associated varieties.

This is based on joint work with Dan Xie, Wenbin Yan and Qixian Zhao.

Slides

Video

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. We will also address proofs of these conjectures in the cases where the identity component is a torus, or an adjoint group over the real numbers, where interesting new phenomena appear. Time permitting, we will mention the beginnings of a global program, and an application to the reduction of the refined local Langlands correspondence, both for connected and disconnected groups, to the case of discrete parameters.

Video

Abstract: We will first explain how to construct a Galois analogue of the Bernstein decomposition of the category of smooth representations of p-adic reductive groups. The role of irreducible objects will be played by conjugacy classes of enhanced Langlands parameters. These latter are partitioned into series thanks to the generalized Springer correspondence, and each series is in bijection with the simple modules of a (possibly twisted) extended affine Hecke algebra. Next, guided by the work of Lusztig for unipotent representations of p-adic groups, we will apply this construction to build an explicit local Langlands correspondence in numerous situations.

Slides

Video

Abstract: I will describe the integral transforms on the Steinberg-Hitchin base that give rise to the projectors in the Bernstein decomposition of the stable cocenter of SL(2, F) where F is a nonarchimedean local field.

Video

Abstract: The geometric Satake equivalence is a celebrated construction (with contributions by Lusztig, Ginzburg, Drinfeld and Mirkovic-Vilonen) that realizes the category of representations of a connected reductive group as a category of perverse sheaves on the affine Grassmannian of the Langlands dual group. In the setting of l-adic coefficients, Zhu and Richarz have studied a variant of this construction in a "ramified" situation, where the group of which one takes the affine Grassmannian can be a non constant group scheme over formal loops. In this talk I'll explain a version of this equivalence for general coefficients; the Tannakian group on the dual side is then a certain group of fixed points for automorphisms of a reductive group, which is not necessarily smooth. This is joint work with P. Achar, J. Lourenço and T. Richarz.

Video

Abstract: An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will give an overview of what we know about the structure of the Bernstein blocks. In particular, I will discuss a joint project in progress with Adler, Mishra and Ohara in which we show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

Video

Abstract: Lusztig's theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give a brief overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the "generic" part of this desired theory. In the simplest nontrivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne--Lusztig induction. This talk is intended to describe what we know about these generic character sheaves, especially within the context of the Langlands program.

Video

Abstract:Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular semisimple element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $G(F)$ whose central character is unitary and the formal degree of $\pi$ tends to infinity. I will sketch the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open compact modulo center subgroup. If time allows I could also discuss progress on optimizing the bound.

Video

Abstract: For compact Kaehler manifolds, Corlette and Simpson have established a nonabelian Hodge correspondence between representations of their fundamental groups and Higgs bundles on them subject to certain semistability conditions. Since the work of Faltings, analogy has been sought for p-adic rigid varieties. Inspired by the recent work of Heuer, we found that for a smooth rigid space over a perfectoid field, there is a natural etale Gm-gerbe on its cotangent bundle, whose coherent sheaf theory realizes a p-adic Simpson correspondence. This shows great similarity to results in prime characteristics discovered by Ogus-Vologodsky. In this talk, I will explain this perspective.

This is joint work in progress with Bhargav Bhatt.

Video

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

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

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

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

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

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

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

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

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

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

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

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

Abstract: