Distributions propres invariantes sur la paire sym\' etrique (gl(4,R),gl(2,R)*gl(2,R))

TL;DR

This paper investigates invariant distributions and orbital integrals for a specific symmetric pair involving gl(4,R) and gl(2,R)*gl(2,R), providing asymptotic analysis and explicit bases for eigendistributions near semisimple elements.

Contribution

It offers a detailed study of orbital integrals and eigendistributions for the symmetric pair (gl(4,R), gl(2,R)*gl(2,R)), including asymptotic behavior and explicit basis construction.

Findings

Asymptotic behavior of orbital integrals near nonzero semisimple elements.

Explicit basis of eigendistributions on nilpotent set q-N.

Locally integrable functions representing eigendistributions.

Abstract

We study orbital integrals and invariant eigendistributions for the symmetric pair (g,h)=(gl(4,R),gl(2,R)*gl(2,R)). Let q=g/h and let N be the set of nilpotents of q. We first obtain an asymptotic behavior of orbital integrals around nonzero semisimple elements of q. We study eigendistributions around such elements and give an explicit basis of eigendistributions on q-N given by a locally integrable function on q-N.