Exercise 25, § 4

Let G be a locally compact group endowed with a left Haar measure $\nu$ and let $K\subset G$ be a compact subgroup of G endowed with the normalized Haar measure $\mu$.

We denote by $\mathscr{H}(G,K)$ the space of functions $\varphi \in \mathscr{K}(G)$ such that $\varphi (k_1gk_2) =$ $\varphi (g)$ for every $g\in G$ and every $(k_1, k_2)\in K\times K$.

The space $\mathscr{H}(G,K)$ is a unital subalgebra of the algebra $\mathscr{M}_c(G)$; one says that $(G,K)$ is a Gelfand pair if the algebra $\mathscr{H}(G,K)$ is commutative. a) The mapping $u^K$ which associates with $f$ the function $f^K$ defined by

is a projector in $\mathscr{K}(G)$ whose image is $\mathscr{H}(G,K)$.

b) Let $f\in \mathscr{K}(G)$. One has

and $(f \varphi )^K=f^K\varphi$ for every function $\varphi \in \mathscr{H}(G,K)$.

c) Let $C\subset G$ be a compact subset. There exists a function $\varphi \in \mathscr{H}(G,K)$ such that $\varphi (x) = 1$ for every $x\in C$.

d) Let $(G,K)$ be a Gelfand pair. The group G is unimodular. (Note that the module of G belongs to $\mathscr{H}(G,K)$ and interpret the integral of $\varphi \in \mathscr{H}(G,K)$ as the value at $e$ of a suitably chosen convolution.)

e) Suppose that there exists a continuous involution $\theta : G\rightarrow$ G such that $K\theta (x)K = Kx^{-1}K$ for every $x\in G$. Then $(G,K)$ is a Gelfand pair. (Verify that the mapping $f\mapsto \nu (f\circ \theta )$ on $\mathscr{K}(G)$ is a left Haar measure on G, then that it is equal to $\nu$; deduce that $(f*g)\circ \theta = (f\circ \theta )*(g\circ \theta )$ and compare with the formula $f\widecheck{*}g=\check{g}*\check{f}.$)

f ) If G is compact, then $(G\times G,\Delta )$ is a Gelfand pair, where Δ is the diagonal subgroup of $G\times G$ image of the homomorphism $x\mapsto (x, x)$ of G into $G\times G$.