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.)

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.