Unipotent representations of real reductive groups

Abstract: Let $G$ be a real reductive group and let ${\widehat G}$ be the set of irreducible unitary representations of $G$. The determination of $\widehat G$ (for arbitrary $G$) is one of the fundamental unsolved problems in representation theory. In the early 1980s, Arthur introduced a finite set Unip($G$) of (conjecturally unitary) irreducible representations of $G$ called {\it unipotent representations}. In a certain sense, these representations form the building blocks of $\widehat G$. Hence, the determination of $\widehat G$ requires as a crucial ingredient the determination of Unip($G$). In this thesis, we prove three results on unipotent representations. First, we study unipotent representations by restriction to $K\subset G$, a maximal compact subgroup. We deduce a formula for this restriction in a wide range of cases, proving (in these cases) a long-standing conjecture of Vogan. Next, we study the unipotent representations attached to induced nilpotent orbits. We find that Unip($G$) is ‘generated’ by an even smaller set $\hbox{Unip}'(G)$ consisting of representations attached to rigid nilpotent orbits. Finally, we study the unipotent representations attached to the principal nilpotent orbit. We provide a complete classification of such representations, including a formula for their $K$-types.

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.