Regularity of some invariant distributions on nice symmetric pairs

TL;DR

This paper investigates the regularity of invariant distributions on certain symmetric pairs, establishing conditions under which these distributions are locally integrable functions, with a focus on nice symmetric pairs and specific Lie algebra cases.

Contribution

It extends previous work by analyzing the regularity of invariant eigendistributions on nice symmetric pairs, including the case (gl(4,R), gl(2,R)*gl(2,R)), showing their local integrability.

Findings

Invariant eigendistributions on (gl(4,R), gl(2,R)*gl(2,R)) are locally integrable functions.

Identifies conditions under which invariant distributions are regular on nice symmetric pairs.

Provides a method to study the regularity of invariant distributions annihilated by Casimir polynomial.

Abstract

J.~Sekiguchi determined the semisimple symmetric pairs (g,h), called nice symmetric pairs, on which there is no non-zero invariant eigendistribution with singular support. On such pairs, we study regularity of invariant distributions annihilated by a polynomial of the Casimir operator. We deduce that invariant eigendistributions on (gl(4,R),gl(2,R)*gl(2,R)) are locally integrable functions.