Orthogonality relations on certain homogeneous spaces
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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.
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Additional Information
- Chi-Wai Leung
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 306606
- Email: cwleung@math.cuhk.edu.hk
- Received by editor(s): June 1, 2020
- Received by editor(s) in revised form: June 2, 2021
- Published electronically: November 15, 2021
- Additional Notes: This work was supported by Hong Kong RGC Research Grant (2130501).
- Communicated by: Matthew A. Papanikolas
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 1115-1126
- MSC (2020): Primary 43A05, 43A30, 43A65, 43A85; Secondary 22E45
- DOI: https://doi.org/10.1090/proc/15690
- MathSciNet review: 4375707