Lie Groups Days
(in honor of David Vogan)

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).

Abstract

According to the Local Langlands conjecture, representations of a real reductive group G(R) are parametrized by pairs consisting of an L-homomorphism and a character of the component group of its centralizer. This construction is natural with respect to various constructions, such as infinitesimal and central character, parabolic induction, endoscopy, etc.

One of the main invariants of a representation is its set of lowest K-types. In this talk I will describe how to understand this set in terms of data on the dual group. This is most conveniently expressed in the language of the Atlas of Lie Groups and Representations. This is joint work with Alexandre Afgoustidis.

Abstract

n previous work, W. Hardesty and I used the machinery of co-t-structures to prove the relative Humphreys conjecture for reductive groups. (In fact, this was the topic of my last talk in the MIT Lie Groups Seminar.) This machinery produces some distinguished objects in the derived category of coherent sheaves on the nilpotent cone of a reductive group or on the Springer resolution. In this talk, I will review this machinery, and I will report on work in progress (mostly partial results and conjectures at this point) to generalize this theory to the setting of symmetric pairs. This work in progress is joint with W. Hardesty and L. Liu.

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.

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.

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 $\mathrm{SL}(2).$

Abstract

A fundamental problem in the representation theory of reductive groups is to write the characters of irreducible representations in terms of (better-understood) characters of standard modules. For complex or real groups, for example, this amounts to computing Kazhdan-Lusztig or Lusztig-Vogan polynomials. For certain kinds of representations of split $p$-adic groups, including the Iwahori-spherical ones, Lusztig gave an explicit algorithm to compute the corresponding $p$-adic polynomials. The real and p-adic algorithms have quite different structure, but the polynomials that emerge are very similar. For example, Ciubotaru-Trapa showed that every $p$-adic polynomial for $\mathrm{GL}(n,\mathbb{Q}_p)$ is a Lusztig-Vogan polynomial for $\mathrm{GL}(n,\mathbb{R})$ in a natural way. In this talk, we generalize the Ciubotaru-Trapa result to other classical groups and show that the $p$-adic polynomials are a subset of the corresponding Lusztig-Vogan polynomials in an explicitly computable way. Since the latter are accessible in the ${\tt atlas}$ software, this gives a practical way to compute examples of the $p$-adic polynomials for classical groups. This is based on joint work with Leticia Barchini.