MIT Lie Groups Seminar

Abstract:I will explain how mixed Hodge modules can be utilized to understand representation theory of real groups. In particular, we obtain a refinement of the Lusztig-Vogan polynomials in this setting. Adams, van Leeuwen, Trapa, and Vogan (ALTV) have given an algorithm to determine the unitary dual of a real reductive group. As a corollary of our results we obtain a proof of a key result of (ALTV) on signature polynomials.

This is joint work with Dougal Davis.

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Abstract: The wavefront set is a powerful harmonic analytic invariant attached to representations of p-adic groups that is expected to play an important role in the construction of Arthur packets. In this talk I will present new results relating it to the local Langlands correspondence for representations in the principal block. In the process I will introduce a natural refinement of the (geometric) wavefront set with many nicer properties and use it to construct some unipotent Arthur packets of arbitrary split groups. The results are based on joint work with Dan Ciubotaru and Lucas Mason-Brown.

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Abstract:We construct an equivalence between Iwahori-integrable representations of affine Lie algebras and representations of the "mixed" quantum group, thus confirming a conjecture by Gaitsgory. Our proof utilizes factorization methods: we show that both sides are equivalent to algebraic/topological factorization modules over a certain factorization algebra, which can then be compared via Riemann-Hilbert. On the quantum group side this is achieved via general machinery of homotopical algebra, whereas the affine side requires inputs from the theory of (renormalized) ind-coherent sheaves as well as compatibility with global Langlands over P1. This is joint work with Lin Chen.

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Abstract: We will formulate a list of properties that uniquely characterize the local Langlands correspondence for discrete Langlands parameters with trivial monodromy. Suitably interpreted, this characterization holds for any local field, but requires an assumption on p in the non-archimedean case. We will then discuss an explicit construction of this correspondence, as a realization of functorial transfer from double covers of elliptic maximal tori.

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Abstract: Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group W (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from W. Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with W (roughly speaking, families correspond to ℂ×-fixed points, Harish-Chandra series correspond to symplectic leaves, blocks correspond to symplectic leaves in the fixed point subvariety under the action of a root of unity). The aim of this talk is to explain these observations, state precise conjectures and provide general facts and examples supporting these conjectures.

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Abstract: We propose a connection between the horizontal trace of the affine Hecke category and the elliptic Hall algebra, mirroring known constructions for the finite Hecke category. Explicitly, we construct a family of generators of the affine Hecke category, compute certain categorified commutators between them, and show that their K-theoretic shadows match certain commutators in the elliptic Hall algebra. Joint work with Eugene Gorsky.

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Abstract: The main theorem of this talk will be that one can understand a "dense open" subset of DG categories with an action of a split reductive group G over a field of characteristic zero entirely in terms of its root datum. We will start by introducing the notion of a categorical representation of G and discuss some motivation. Then, we will discuss some of the main technical tools involved in making the statement of the main theorem precise, including discussion of the "coarse quotient" of the dual maximal Cartan by the affine Weyl group. We will also discuss how sheaves on this coarse quotient can be identified with bi-Whittaker sheaves on G, obtaining symmetric monoidal upgrade of a result of Ginzburg and Lonergan, and then give an outline of the proof of the main theorem. Time permitting, we will discuss some applications of these categorical representation theoretic ideas which prove a modified version of a conjecture of Ben-Zvi and Gunningham on the essential image of parabolic restriction.

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Abstract: Perverse sheaves have important applications in representation theory and number theory. In this talk we will consider the case of mod p étale sheaves on affine flag varieties over a field of characteristic p. Despite the pathological behavior of such sheaves, they encode the structure of mod p Hecke algebras. We will primarily focus on a version of the geometric Satake equivalence for the affine Grassmannian. Time permitting, we may also discuss central sheaves on the Iwahori affine flag variety. Part of this is joint work with Cédric Pépin.

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Abstract: For a real form of an algebraic group acting on the flag variety we define a t-structure on the category of equivariant-monodromic sheaves and develop the theory of tilting sheaves. In case of a quasi-split real form we construct an analog of a Soergel functor, which full-faithfully embeds the subcategory of tilting objects to the category of coherent sheaves on a block variety. We apply the results to give a new, purely geometric, proof of the Soergel's conjecture for quasi-split groups.

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Abstract: Hessenberg varieties are fibers of certain proper maps to a simple Lie algebra. These maps are generalizations of the Springer and Grothendieck-Springer resolutions. In this talk, we describe some new properties of nilpotent Hessenberg varieties. In particular, we show that their cohomology satisfies a modular law as we vary the maps. This law generalizes one of De Concini, Lusztig, and Procesi and coincides with a combinatorial law of Guay-Paquet and Abreu-Nigro in type A. We also study the push-forward of the constant sheaf of these maps and show that only intersection cohomology sheaves with local systems coming from the Springer correspondence appear in the decomposition, resolving a conjecture of Brosnan. This is joint work with Martha Precup.

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Abstract: We define for a local or global function field F a gerbe E over F banded by a profinite group scheme whose set of G-torsors parametrizes all inner twists of an arbitrary connected reductive group G, generalizing the Kottwitz gerbe whose torsors parametrize extended pure inner forms of G. We discuss local and global duality results for these sets of torsors, and use them to state conjectures regarding the local and global Langlands correspondence and endoscopy. Locally, we give a conjectural parametrization of L-packets and construct a w-normalized absolute transfer factor for an endoscopic datum. Globally, we relate these new local transfer factors to the adelic transfer factor and construct a pairing involving L-packets which is used in the conjectural multiplicity formula for discrete automorphic representations. An emphasis will be placed on understanding torsors on the category- theoretic object E in a concrete way (via torsors twisted by a Cech 2-cocycle).

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Abstract: Let G be a connected reductive group over an algebraically closed field, and let C be a nilpotent orbit for G. If L is an irreducible G-equivariant vector bundle on C, then one can define a "coherent intersection cohomology complex" IC(C,L). These objects play an important role in various results related to the local geometric Langlands program.

When G has positive characteristic, instead of an irreducible bundle L, one might consider a tilting bundle T on C. I will explain a new construction that associates to the pair (C,T) a complex of coherent sheaves S(C,T) with remarkable Ext-vanishing properties. This construction leads to a proof of a conjecture of Humphreys on (relative) support varieties for tilting modules, and hints at a kind of "recursive" structure in the tensor category of tilting G-modules. This work is joint with W. Hardesty (and also partly with S. Riche).

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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.)

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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.

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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.

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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).

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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.

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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.

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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.

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Abstract: Quasi-invariants are natural geometric generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics in the early 1990s, and since then have found applications in a number of other areas (most notably, representation theory, algebraic geometry and combinatorics).

In this talk, I will explain how the algebras of quasi-invariants can be realized topologically: as (equivariant) cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result can be viewed as a generalization of a well-known theorem of A. Borel that realizes the algebra of invariant polynomials of a Weyl group W as the cohomology ring of the classifying space BG of the corresponding Lie group G. Replacing equivariant cohomology with equivariant K-theory gives a multiplicative (exponential) analogues of quasi-invariants of Weyl groups. But perhaps more interesting is the fact that one can also realize topologically the quasi-invariants of some non-Coxeter groups: our `spaces of quasi-invariants' can be constructed in a purely homotopy-theoretic way, and this construction extends naturally to (p-adic) pseudoreflection groups. In this last case, the compact Lie groups are replaced by p-compact groups (a.k.a. homotopy Lie groups). The talk is based on joint work with A. C. Ramadoss. `

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Abstract: In its most basic form, symplectic geometry is a mathematically rigorous framework for classical mechanics. Noether's perspective on conserved quantities thereby gives rise to quotient constructions in symplectic geometry. The most classical such construction is Marsden-Weinstein-Meyer reduction, while more modern variants include Ginzburg-Kazhdan reduction, Kostant-Whittaker reduction, Mikami-Weinstein reduction, symplectic cutting, and symplectic implosion.

I will provide a simultaneous generalization of the quotient constructions mentioned above. This generalization will be shown to have versions in the smooth, holomorphic, complex algebraic, and derived symplectic contexts. As a corollary, I will derive a concrete and Lie-theoretic construction of "universal" symplectic quotients.

This represents joint work with Maxence Mayrand.

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Abstract: The systematic study of quantum symmetric pairs (QSPs) was initiated by Gail Letzter in 1999. The area has been greatly developed in recent years. We will present a new approach to the theory of quantum symmetric pairs for symmetrizable Kac-Moody algebras based on star products on noncommutative graded algebras. It will be used to give solutions to two main problems in the area: (1) determine the defining relations of QSPs and (2) find a Drinfeld type formula for universal $K$-matrices as sums of tensor products over dual bases. This is a joint work with Stefan Kolb.

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Abstract: Let X be a smooth projective curve over a finite field k, and let G be a reductive group. The unramified part of the theory of automorphic forms for the group G and the field k(X) studies functions on the k-points on the moduli space of G-bundles on X and the eigen-functions of the Hecke operators (to be reviewed in the talk!) acting there. The spectrum of the Hecke operators has continuous and discrete parts and it is described by the global Langlands conjectures (which in the case of functional fields are essentially proved by V.Lafforgue).

After recalling the above notions and constructions I will discuss what happens when k is replaced by a local field. The corresponding Hecke operators were essentially defined by myself and Kazhdan about 10 years ago, but the systematic study of eigen-functions has begun only recently. It was initiated several years ago by Langlands when k is archimedean and then Etingof, Frenkel and Kazhdan formulated a very precise conjecture describing the spectrum in terms of the dual group. Contrary to the classical case only discrete spectrum is expected to exist. I will discuss what is is known in the case when k is a local non-archimedean field K. In particular, I will talk about some version of the Eisenstein series operator which allows to construct a Hecke eigen-function over K starting from a cuspidal Hecke eigen-function over finite field (joint work in progress with D.Kazhdan and A.Polishchuk).

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Abstract: We will formulate a list of properties that uniquely characterize the local Langlands correspondence for discrete Langlands parameters with trivial monodromy. Suitably interpreted, this characterization holds for any local field, but requires an assumption on p in the non-archimedean case. We will then discuss an explicit construction of this correspondence, as a realization of functorial transfer from double covers of elliptic maximal tori.

Abstract: In the representation theory of finite reductive groups, an essential role is played by Lusztig's nonabelian Fourier transform, an involution on the space of unipotent characters the group. This involution is the change of bases matrix between the basis of irreducible characters and the basis of `almost characters', certain class functions attached to character sheaves. For reductive p-adic groups, the unipotent local Langlands correspondence gives a natural parametrization of irreducible smooth representations with unipotent cuspidal support. However, many questions about the characters of these representations are still open. Motivated by the study of the characters on compact elements, we introduce in joint work with A.-M. Aubert and B. Romano (arXiv:2106.13969) an involution on the spaces of elliptic and compact tempered unipotent representations of pure inner twists of a split simple p-adic group. This generalizes a construction by Moeglin and Waldspurger (2003, 2016) for elliptic tempered representations of split orthogonal groups, and potentially gives another interpretation of a Fourier transform for p-adic groups introduced by Lusztig (2014). We conjecture (and give supporting evidence) that the restriction to reductive quotients of maximal compact open subgroups intertwines this involution with a disconnected version of Lusztig's nonabelian Fourier transform for finite reductive groups.