Orthogonality relations on certain homogeneous spaces.

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. [ABSTRACT FROM AUTHOR]