Dimension of the space of intertwining operators from degenerate principal series representations

TL;DR

This paper investigates the multiplicity of intertwining operators in the regular representation of a real reductive Lie group, revealing cases where infinite multiplicity occurs despite finitely many orbits, contrasting previous minimal parabolic results.

Contribution

It identifies new examples of infinite multiplicity of degenerate principal series representations in regular representations, expanding understanding beyond minimal parabolic subgroup cases.

Findings

Infinite multiplicity can occur for degenerate principal series representations from non-minimal parabolics.

Finite orbit count does not guarantee finite multiplicity for these representations.

Contrasts with prior results where minimal parabolics had finite multiplicities.

Abstract

Let $X$ be a homogeneous space of a real reductive Lie group $G$. It was proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(X)$ contains each irreducible representation of $G$ at most finitely many times if a minimal parabolic subgroup $P$ of $G$ has an open orbit in $X$, or equivalently, if the number of $P$-orbits on $X$ is finite. In contrast to the minimal parabolic case, for a general parabolic subgroup $Q$ of $G$, we find a new example that the regular representation $C^{\infty}(X)$ contains degenerate principal series representations induced from $Q$ with infinite multiplicity even when the number of $Q$-orbits on $X$ is finite.