Spherical analysis on homogeneous vector bundles

TL;DR

This paper explores spherical analysis on homogeneous vector bundles over Lie groups, focusing on Gelfand pairs, invariant differential operators, and the structure of commutative algebras in specific group settings.

Contribution

It establishes conditions for commutativity of bi-$ au$-equivariant functions, analyzes the Gelfand spectrum, and characterizes representations leading to commutative algebras for certain semidirect product groups.

Findings

Gelfand pair property for (G,K) under specified assumptions

Homeomorphic embeddings of Gelfand spectrum in complex space

Characterization of representations for commutative algebra in specific groups

Abstract

Given a Lie group $G$, a compact subgroup $K$ and a representation $\tau\in\hat K$, we assume that the algebra of $\text{End}(V_\tau)$-valued, bi-$\tau$-equivariant, integrable functions on $G$ is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the r\^ole of the algebra of $G$-invariant differential operators on the homogeneous bundle $E_\tau$ over $G/K$. In particular, we observe that, under the above assumptions, $(G,K)$ is a Gelfand pair and show that the Gelfand spectrum for the triple $(G,K,\tau)$ admits homeomorphic embeddings in $\mathbb C^n$. In the second part, we develop in greater detail the spherical analysis for $G=K\ltimes H$ with $H$ nilpotent. In particular, for $H=\mathbb R^n$ and $K\subset SO(n)$ and for the Heisenberg group $H_n$ and $K\subset U(n)$, we characterize the representations $\tau \in \hat K$ giving a commutative algebra. \end{abstract}