Asymptotic K-Support and Restrictions of Representations

TL;DR

This paper revisits criteria for the discrete decomposability of unitary representations of reductive Lie groups, replacing hyperfunction techniques with microlocal analysis in the smooth category to advance understanding of representation restrictions.

Contribution

It introduces a microlocal analysis approach in the smooth category as an alternative to hyperfunction methods for studying representation restrictions.

Findings

Replaces hyperfunction techniques with microlocal analysis in the $C^ abla$ category.

Provides new insights into the asymptotic support of representations.

Enhances the theoretical framework for analyzing restrictions of unitary representations.

Abstract

In the late nineties T. Kobayashi wrote a series of papers in which he established a criterion for the discrete decomposablity of restrictions of unitary representations of reductive Lie groups to reductive subgroups. A key tool in the proof of sufficiency of his criterion was the use of the theory of hyperfunctions to study the microlocal behavior of characters of restrictions to compact subgroups. In this paper we show how to replace this tool by microlocal analysis in the $C^\infty$ category.