Tempered Representations and Nilpotent Orbits

TL;DR

This paper investigates conditions under which nilpotent orbits relate to tempered representations in real reductive groups, expressing wave front cycle coefficients via volumes of specific submanifolds, thus linking geometric and representation-theoretic properties.

Contribution

It provides a necessary condition for the existence of tempered representations associated with given nilpotent orbits and expresses wave front cycle coefficients in geometric terms.

Findings

Established a necessary condition for nilpotent orbits to occur in tempered representations.

Expressed wave front cycle coefficients using volumes of precompact submanifolds.

Connected geometric measures with representation-theoretic invariants.

Abstract

Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.