Lie Groups and Representation Theory

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Date, time & place Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.)

2013/10/22

17:00-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Benjamin Harris (Louisiana State University (USA))
Representation Theory and Microlocal Analysis (ENGLISH)
[ Abstract ]
Suppose $H\\subset K$ are compact, connected Lie groups, and suppose $\\tau$ is an irreducible, unitary representation of $H$. In 1979, Kashiwara and Vergne proved a simple asymptotic formula for the decomposition of $Ind_H^K\\tau$ by microlocally studying the regularity of vectors in this representation, thought of as vector valued functions on $K$. In 1998, Kobayashi proved a powerful criterion for the discrete decomposability of an irreducible, unitary representation $\\pi$ of a reductive Lie group $G$ when restricted to a reductive subgroup $H$. One of his key ideas was to restrict $\\pi$ to a representation of a maximal compact subgroup $K\\subset G$, view $\\pi$ as a subrepresentation of $L^2(K)$, and then use ideas similar to those developed by Kashiwara and Vergne.

In a recent preprint the speaker wrote with Hongyu He and Gestur Olafsson, the authors consider the possibility of studying induction and restriction to a reductive Lie group $G$ by microlocally studying the regularity of the matrix coefficients of (possibly reducible) unitary representations of $G$, viewed as continuous functions on the (possibly noncompact) Lie group $G$. In this talk, we will outline the main results of this paper and give additional conjectures.