Multiplicity bounds and the subrepresentation theorem for real spherical spaces

TL;DR

This paper establishes finite bounds on the multiplicity of H-fixed distribution vectors in Harish-Chandra modules for certain real spherical spaces and proves a related subrepresentation theorem, advancing understanding of representation restrictions.

Contribution

It provides the first sharp finite bounds for multiplicities and introduces a subrepresentation theorem for real spherical spaces, extending previous results in representation theory.

Findings

Finite bounds for H-fixed distribution vectors are established.

A subrepresentation theorem for Harish-Chandra modules is proved.

Results apply to real spherical spaces with open orbits on flag manifolds.

Abstract

Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of H-fixed distribution vectors for V and a related subrepresentation theorem is derived. Extended final version. To appear in Trans. Amer. Math. Soc.