Lie Groups Days
(in honor of David Vogan)

https://math.mit.edu/events/lie-groups-day/

September 23-24, 2022 (Friday & Saturday)
MIT

Registration 

Schdedule of Speakers :

Friday, September 23

9:30am - 10:30am -- Stephen Miller (Rutgers University)

Miller -- Title:  David Vogan and the unitary dual in 2022: the beachhead of Arthur's unipotent representations.
Abstract:  David Vogan has devoted much of his career to understanding the unitary representations of real reductive Lie groups.  A particular focus over the last several decades has been the "unipotent" representations introduced by Jim Arthur in the 1980s.  I'll describe Vogan's recent work on this, as well as other ideas originating from string theory, which together prove the unitarity of unipotent representations for exceptional groups (including E8).

11:00am - 12:00pm -- Jeffrey Adams (Univeristy of Maryland)

1:45pm - 2:45pm -- Pramod Achar (Louisiana State University)

3:00pm - 4:00pm -- George Lusztig (MIT)

Lusztig -- Title: Parametrization of canonical bases.
Abstract: Let U be the Drinfeld-Jimbo quantum group attached to a root datum, let
U^+ be its plus part, and let V_\lambda be its simple module with highest
weight \lambda. The canonical bases of U^+ and V_\lambda were
defined in 1990 and were parametrized in terms which were later
interpreted in terms of objects over the semifield Z. We will describe
a parametrization in a similar spirit for the canonical basis of \dot U, a modified form of U.

Saturday, September 24

10:00am - 11:00am -- Lucas Mason-Brown (Oxford University)

Mason-Brown -- Title: Some Comments on the Structure of the Unitary Dual
Abstract: In his `Orange Book', David Vogan formulates some general expectations about the structure of the unitary dual of a real reductive group. These expectations can be summarized as follows: every irreducible unitary representation can be constructed from some elementary building blocks (called `unipotent representations') through some unitarity-preserving operations (unitary induction, cohomological induction, and complementary series). Turning this philosophy into a precise mathematical conjecture turns out to be a subtle and difficult problem. In this talk, I will attempt to do so in the case of spherical representations of a complex group. This talk is partially based on joint work with Ivan Losev.

11;30am - 12:30pm -- Monica Nevins (Univeristy of Ottawa)

Nevins -- Title: The p-adic local character expansion as a branching rule
Abstract:  The character of an admissible representation $\pi$ of a $p$-adic group $G$ can be expressed, in a neighbourhood of the identity, as a linear combination of functions arising from the finitely many nilpotent orbits. In this talk, we propose an interpretation of the local character expansion as branching rules of the restriction of $\pi$ to a maximal compact open subgroup, with a view towards understanding a conjecture of Adams--Vogan.  We elaborate with the example of $\mathrm{SL}(2).$

2:15pm - 3:15pm -- Peter Trapa (University of Utah)

Organizers:

Roman Bezrukavnikov (MIT)

Pavel Etingof (MIT)

Ju-Lee Kim (MIT)

Zhiwei Yun (MIT)