Orthogonality relations on certain homogeneous spaces

Abstract:

Let $G$ be a locally compact group and let $K$ be its closed subgroup. Write $\widehat {G}_{K}$ for the set of irreducible representations with non-zero $K$-invariant vectors. We call a pair $(G,K)$ admissible if for each irreducible representation $(\pi , V_{\pi })$ in $\widehat {G}_{K}$, its $K$-invariant subspace $V_{\pi }^{K}$ is of finite dimension. For each $\pi$ in $\widehat {G}_{K}$, let $\pi _{v_{i}, \overline {\xi }_{j}}$’s $(\pi _{v_{i}, \overline {\xi }_{j}}(gK)≔\langle v_{i}, \pi (g)\xi _{j}\rangle )$ be the matrix coefficeints on $G/K$ induced by fixed orthonormal bases $\{v_{i}\}$ and $\{\xi _{j}\}$ for $V_{\pi }$ and $V_{\pi }^{K}$ respectively. A probability measure $\mu$ on $G/K$ is called a spectral measure if there is a subset $\Gamma$ of $\widehat {G}_{K}$ such that the set of all such matrix coefficients $\pi _{v_{i}, \overline {\xi }_{j}},\ \pi \in \Gamma ,$ constitutes an orthonormal basis for $L^{2}(G/K, \mu )$ with some suitable normalization of these matrix coordinate functions.

In this paper, we shall give a characterization of a spectral measure for an admissible pair $(G,K)$ by using the Fourier transform on $G/K$. Also, from this we show that there is a “local translation” (we call it locally regular representation in the sequel) of $G$ on $L^{2}(G/K, \mu )$ under a mild condition. This will give us some necessary conditions for the existence of spectral measures. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied.