MIT Lie Groups Seminar

Abstract: It's an old idea of Harish-Chandra that representations of a real reductive G can be studied by understanding their restrictions to a maximal compact K. Traditionally this was done using the Cartan-Weyl highest weight theory for K representations.

I will recall a (closely related but very different) description of K representations, in which the highest weight is replaced by the Harish-Chandra parameter of a discrete series on a Levi subgroup of G. Using this new parameter, one can define a very natural "height" of a representation of K, taking non-negative integer values. As evidence that height is natural, I offer

Conjecture. Suppose G is split, and pi_1 and pi_2 are two (minimal) principal series representations of G. If pi_1 and pi_2 have the same restriction to the center of G, then the sets of heights of K-types of pi_1 and of pi_2 are the same.

For example, split G_2 has principal series having two different restrictions to K. In each, the first heights appearing are

0, 3, 6, 9, 10,12, 15...

F4 has principal series representations having three different restrictions to K. But in all three, the first heights appearing are

0, 8, 11, 15,16, 21, 24, 26, 27, 29, 30, 32, 33, 36....

The numbers for split E_8 (which also has three types of principal series) are

0, 29, 46, 57, 58, 68, 75, 84, 87, 91, 92...

It seems more or less impossible to find a similar statement using highest weights. I will give some evidence for this conjecture.

Of course there should be an elementary construction of these sequences of heights from the root system of G and the central character; I have no idea how to make such a construction.

If there is time, I will talk about the possibility of finding parallel statements for p-adic groups.

Slides
Video

Abstract: We explain a uniform construction of cuspidal character sheaves on Z/mZ-graded Lie algebras. We show that they all arise from the Fourier-Sato transform of nearby cycle sheaves associated to very special supercuspidal data. These data essentially come from Lusztig’s cuspidal character sheaves on (ungraded) reductive Lie algebras. We make use of Lusztig-Yun’s work on graded Lie algebras where all simple perverse sheaves on the nilpotent cone (Fourier-Sato transforms of character sheaves) are produced via spiral induction. This is based on joint work with Wille Liu, Cheng-Chiang Tsai and Kari Vilonen.

Video

Abstract: In this talk, we will explore how the Khovanov-Rozansky homology of the (m,n)-torus knot can be extracted from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. This result confirms a conjecture of Gorsky, Oblomkov, Rasmussen, and Shende. Our approach involves constructing a family of coherent sheaves on the Hilbert scheme of points on the plane, coming from cuspidal mirabolic D-modules.

Video

Abstract: A very important result of Lusztig asserts that characters of Deligne--Lusztig representations are obtained by the sheaf-to-function correspondence from character sheaves. The goal of my talk is to outline a proof of a generalization of this result using the categorical trace machinery. This is a joint work in progress with Dennis Gaitsgory and Nick Rozenblyum.

Video

Abstract: A microlocal approach to Arthur theory for p-adic groups was revisited by Ciubotaru-Mason-Brown-Okada, suggesting that some endoscopic Arthur packets can be closely approximated by an analogue: The collection of irreducible representations whose Gelfand-Kazhdan dimension is minimal among those that admit a prescribed infinitesimal character.

In joint work with Emile Okada, we refined this heuristic into a precise characterization for split symplectic and odd orthogonal groups. A key added ingredient are the 'weakly spherical' representations - those admitting non-zero vectors invariant under a maximal compact, not necessarily hyperspecial, subgroup.

Roughly, we discovered that the microlocal packets are unions of those endoscopic packets that contain a weakly spherical representation, with the union's structure governed by nilpotent cone geometry. In particular, by relating Arthur parameters to Springer theory, it appears that weak sphericity corresponds to parameters factoring through Lusztig's canonical quotient associated with the relevant nilpotent orbit.

slides

Abstract: I’ll describe an ongoing project, joint with Arnaud Eteve, Alain Genestier and Vincent Lafforgue. We start with an algebro-geometric incarnation of the Fargues-Scholze D(Bun_G), given by the category of sheaves on the loop group equivariant with respect to Frobenius-twisted conjugacy, and our goal is to define on it a ‘spectral’ action of QCoh of the stack of arithmetic local systems. This is achieved by a fusion procedure. In the process one has to resolve difficulties associated with taking nearby cycles along a multi-dimensional base.