Exercise 32, § 4

Let G be a locally compact group. Suppose that there exists a subgroup H of G which is compact and open in G. Let $\mu$ denote the unique left Haar measure on G such that $\mu(H) = 1$ and $\varphi$ the characteristic function of H.

a) Let $\pi$ be a square-integrable representation of G in a complex Hilbert space E. Now prove that the endomorphism $\pi (\varphi )$ is a Hilbert–Schmidt endomorphism.

b) The space $E^H$ is of finite dimension and satisfies

dim(E$^H$)$\leqslant \frac{1}{c(\pi)}$

where $c(\pi )$ is the formal degree of $\pi$ relative to $\{e\}$. (Use Exercise 6 of V, p. 487.)

c) An infinite discrete group has no square-integrable representation; if $n\geqslant 2$ is an integer, the group $\mathbf{S}\mathbf{L}(n,\mathbf{Z})$ has no square-integrable representation modulo the center.