Spherical analysis on homogeneous vector bundles

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 rôle 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 $G=K⋉H$ 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}