Exercise 29, § 4

Let G be a compact group and K a closed subgroup of G. Suppose that $(G,K)$ is a Gelfand pair. Denote by $\nu$ (resp. $\mu$) the normalized Haar measure on G (resp. on K).

Let $\varphi$ be a K-spherical function on G.

a) Denote by $E_{\varphi}$ the subspace of $\mathscr{C}(G)$ consisting of functions of the form $\alpha *\varphi$, where $\alpha$ is a measure on G with finite support. For every $\psi \in \mathscr{H}(G,K)$, the endomorphism $v_{\psi}:f\mapsto f*\psi$ of $\mathscr{C}(G))$ is compact and $E_{\varphi}$ is contained in an eigenspace of $v_{\psi}$.

b) The space $E_{\varphi}$ is finite-dimensional. It may be identified with a subspace of G-finite vectors of the regular representation $\boldsymbol{\gamma }_G$ containing $\varphi$.

c) The space of K-invariant vectors of $E_{\varphi}$ is of dimension 1 and the representation of G in $E_{\varphi}$ is irreducible.

d) Let $(f_i)_{i\in I}$ be an orthonormal basis of $E_{\varphi}$ and let $k: G\times G\rightarrow \mathbf{C}$ be the function defined by

One has

for all $f\in E_{\varphi}$ and $x\in G$.

e) One has $k(x, y) =$ dim(E$_{\varphi}$)$\varphi (y^{-1}x)$ for every $(x, y)\in G\times G$.

f) The function $\varphi$ is of positive type.

g) Let $\varphi$ be a K-spherical function on G and let $(\pi , x)$ be a Hilbertian realization of $\varphi$. The representation $\pi$ is irreducible; one has

for every $x\in G$, and $\int_G|\varphi |^2d\nu =$ dim($\pi$ )$^{-1}$.