MIT Lie Groups Seminar
2021 - 2022
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 Andre Dixon or Pavel Etingof for the Zoom meeting Link. To access videos of talks, please email Andre Dixon for the password.
Constructing unipotent representations
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.)
From braids to transverse slices in reductive groups
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.
Total positivity in symmetric spaces
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.
Harish-Chandra modules over quantizations of nilpotent orbits
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).
Oct 6 10AM
Frobenius-twisted conjugacy classes of loop groups and Demazure product of Iwahaori-Weyl groups
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.
Derived Chevalley isomorphisms
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.
Frobenii on Morava E-theoretical quantum groups
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.
Topological realization of rings of quasi-invariants of finite reflection groups
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. `
Universal symplectic quotients via Lie theory
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.
Quantum symmetric pairs via star products
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.
Examples of Hecke eigen-functions for moduli spaces of bundles over local non-archimedean field and an analog of Eisenstein series
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).
Characterization and construction of the local Langlands correspondence for supercuspidal parameters
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.
A nonabelian Fourier transform for tempered unipotent representations of p-adic groups
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.
Contact: Andre Dixon