Corwin-Greenleaf multiplicity function for compact extensions of the Heisenberg group

TL;DR

This paper investigates the Corwin-Greenleaf multiplicity function for certain group extensions of the Heisenberg group, providing conditions for when this multiplicity is at most one and exploring its relation to representation multiplicities.

Contribution

It offers new sufficient conditions for the multiplicity function to be at most one and analyzes its connection to representation theory in the context of compact extensions of the Heisenberg group.

Findings

Provided two sufficient conditions for the multiplicity function to be ≤ 1.

Explored the relationship between the multiplicity function and representation multiplicities.

Identified cases where the multiplicity function differs from the representation multiplicity.

Abstract

Let $\mathbb{H}_n$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a closed subgroup of $U(n)$ acting on $\mathbb{H}_n$ by automorphisms such that $(K,\mathbb{H}_n)$ is a Gelfand pair. Let $G=K\ltimes\mathbb{H}_n$ be the semidirect product of $K$ and $\mathbb{H}_n$. Let $\mathfrak{g}\supset\mathfrak{k}$ be the respective Lie algebras of $G$ and $K$, and $\operatorname{pr}: \mathfrak{g}^{*}\longrightarrow\mathfrak{k}^{*}$ the natural projection. For coadjoint orbits $\mathcal{O}^{G}\subset\mathfrak{g}^{*}$ and $\mathcal{O}^{K}\subset\mathfrak{k}^{*}$, we denote by $n\big(\mathcal{O}^{G},\mathcal{O}^{K}\big)$ the number of $K$-orbits in $\mathcal{O}^{G}\cap \operatorname{pr}^{-1}(\mathcal{O}^{K})$, which is called the Corwin-Greenleaf multiplicity function. In this paper, we give two sufficient conditions on $\mathcal{O}^G$ in order that $$n\big(\mathcal{O}^G,\mathcal{O}^K\big)\leq 1\:\:\text{for any $K$-coadjoint orbit}\:\:\mathcal{O}^{K}\subset\mathfrak{k}^{*}.$$ For $K=U(n)$, assuming furthermore that $\mathcal{O}^{G}$ and $\mathcal{O}^{K}$ are admissible and denoting respectively by $\pi$ and $\tau$ their corresponding irreducible unitary representations, we also discuss the relationship between $n\big(\mathcal{O}^G,\mathcal{O}^K\big)$ and the multiplicity $m(\pi,\tau)$ of $\tau$ in the restriction of $\pi$ to $K$. Especially, we study in Theorem 4 the case where $n(\mathcal{O}^{G},\mathcal{O}^{K})\neq m(\pi,\tau)$. This inequality is interesting because we expect the equality as the naming of the Corwin-Greenleaf multiplicity function suggests.