Exercise 28, § 4

Let G be a locally compact group and K a compact subgroup of G.

a) The pair $(G,K)$ is a Gelfand pair if and only if, for every unitary representation $\pi$ of G in a complex Hilbert space E, the dimension of $E^K$ is at most 1. (To prove that the condition is necessary, prove that an irreducible representation of a commutative involutive Banach algebra in a Hilbert space is of dimension 1; to prove that it is sufficient, use the cor. to Th. 4 of V, p. 454 to verify that the characters of the algebra $\mathscr{H}(G,K)$ separate the points.)

b) Suppose that $(G,K)$ is a Gelfand pair. Let $\varphi \in \mathscr{H}(G,K)$. Suppose that $\varphi$ is a function of positive type and that $\varphi (e) = 1$. Prove that $\varphi$ is a K-spherical function if and only if $\varphi$ has a Hilbertian representation $(\varrho , x)$ such that $\varrho$ is irreducible.

c) Suppose that $(G,K)$ is a Gelfand pair. The set of K-spherical functions of positive type on G is in bijection with the set of irreducible representations $\pi \in \widehat{G}$ such that $E^K_{\pi}$ is not zero.