Exercise 23, § 4

We retain the hypotheses and notation of the preceding exercise, and we suppose that H is a connected semisimple real Lie group.

Let G be a compact subgroup of H. Suppose that there exists a closed connected solvable subgroup N of H such that H = GN. (For every semisimple real Lie group, this condition is satisfied when G is a maximal compact subgroup of H; see for example INT, VII, p. 91, § 3, No.$^o3$, Prop. 7 when G = SL($n,\mathbf{R}$)$.$)

Let $\pi \in \widehat{G}$ and $e_{\pi}=$ dim($\pi$ )$\overline{\chi}_{\pi}$. Let $\varrho$ be the irreducible representation of H given by the last question of the preceding exercise; denote by E the space of $\varrho$.

a) There exists a cyclic vector for the restriction of $\varrho$ to G. (Use Lie's theorem, LIE, III, p. 241, cor., to find a nonzero $x\in E$ such that $\mathbf{C}x$ is invariant under the action of N.) The multiplicity of $\pi$ in the restriction of $\varrho$ to G is at most dim($\pi$ ).

b) Let $r$ be the integer $r$(dim($\pi$ )$^2$) defined in exercise 22; the identity

holds for every family $a_i=e_{\pi}*g_i*e_{\pi}$ of elements of the algebra $\mathscr{M}_c(H)$ of compactly supported measures on H, where $g_i\in \mathscr{K}(H)$ for $1\leqslant i\leqslant r$.

c) Let $\varpi$ be an irreducible unitary representation of H in a Hilbert space F. Let $F_{\pi}$ be the $\pi$-isotypical component of the restriction of $\varpi$ to G and let $p$ be the orthogonal projection of F with image $F_{\pi}$. For every family $(u_i)_{1\leqslant i\leqslant r}$ in $\mathscr{L}(F_{\pi})$, one has

(use exercise 1 of V, p. 486).

d) The multiplicity of $\pi$ in the restriction of $\varpi$ to G is at most dim($\pi$ ). (Use the result of exercise 22 of V, p. 510.)