Temperedness of reductive homogeneous spaces

TL;DR

This paper determines the optimal integrability exponent for the representation of a semisimple algebraic Lie group on homogeneous spaces and provides a criterion to identify when these representations are tempered.

Contribution

It introduces a geometric method to find the best even integer p for the L^p representation of G in L^2(G/H) and establishes a criterion for temperedness.

Findings

Identifies the optimal p for L^p representation of G on G/H

Provides a geometric criterion for temperedness of the representation

Enhances understanding of harmonic analysis on homogeneous spaces

Abstract

Let G be a semisimple algebraic Lie group and H a reductive subgroup. We find geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^{p}. As an application, we give a criterion which detects whether this representation is tempered.