MIT Lie Groups Seminar
2024  2025
Meetings: 4:00pm on Wednesdays
This seminar will take place either inperson or online. For inperson seminars, it will be held at 2142. You are welcome to join inperson seminars by Zoom. For remote participation, the Zoom link is the same as last year's. You can email JuLee Kim for the Zoom meeting Link and for the passcode to access videos of talks.
Fall 2024

Sep 11
David Vogan
(MIT)2142
Restricting real group representations to K
Abstract: It's an old idea of HarishChandra 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 CartanWeyl 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 HarishChandra 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 nonnegative 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 Ktypes 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 padic groups.
Slides
Video 
Sep 18
Ting Xue
(University of Melbourne, Australia)2142
Cuspidal character sheaves on graded Lie algebras
Abstract: We explain a uniform construction of cuspidal character sheaves on Z/mZgraded Lie algebras. We show that they all arise from the FourierSato 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 LusztigYun’s work on graded Lie algebras where all simple perverse sheaves on the nilpotent cone (FourierSato transforms of character sheaves) are produced via spiral induction. This is based on joint work with Wille Liu, ChengChiang Tsai and Kari Vilonen.
Video 
Sep 25
Xinchun Ma
(University of Chicago)2142
Rational Cherednik algebras and torus knots
Abstract: In this talk, we will explore how the KhovanovRozansky homology of the (m,n)torus knot can be extracted from the finitedimensional 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 Dmodules.
Video 
Oct 2
Yakov Varshavskyy
(Hebrew University, visiting MIT)2142
From character sheaves to characters of DeligneLusztig representations via categorical traces
Abstract: A very important result of Lusztig asserts that characters of DeligneLusztig representations are obtained by the sheaftofunction 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 
Oct 9
Max Gurevich
(Technion)ZOOM
Ramification of weak Arthur packets for padic groups
Abstract: A microlocal approach to Arthur theory for padic groups was revisited by CiubotaruMasonBrownOkada, suggesting that some endoscopic Arthur packets can be closely approximated by an analogue: The collection of irreducible representations whose GelfandKazhdan 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 nonzero 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 
Oct 16
Dennis Gaitsgory
(MPI Bonn)2142
Algebrogeometric version of the FarguesScholze Bun_G
Abstract: I’ll describe an ongoing project, joint with Arnaud Eteve, Alain Genestier and Vincent Lafforgue. We start with an algebrogeometric incarnation of the FarguesScholze D(Bun_G), given by the category of sheaves on the loop group equivariant with respect to Frobeniustwisted 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 multidimensional base.

Oct 23
Tom Gannon
(UCLA)2142

Oct 30
Peter Dillery
(U Maryland)2142

Nov 6
Kevin McGerty
(Oxford)2142

Nov 9
2190

Nov 10
2190

Nov 11
2190

Nov 13
Tom Haines
(U Maryland)2142

Nov 20
Eric Opdam
(Univ. Amsterdam, visiting MIT)2142

Nov 27
Thanksgiving week
No Seminar 
Dec 4
Jeremey Taylor
(Berkeley)2142

Dec 11
Nigel Higson
(Penn State)2142
Archive
Contact:
Roman Bezrukavnikov
JuLee
Kim
Zhiwei Yun