Exercise 5, § 1

Let G be a topological group and H an open subgroup of G. Denote by $p$ the canonical projection $G\rightarrow G/H$. a) Let $\sigma : G/H\rightarrow G$ be a mapping such that $p\circ \sigma =$ Id$_{G/H}$. Denote by $\beta : G\times G/H\rightarrow H$ the mapping defined by $\beta (g, x) =\sigma (gx)^{-1}g$. The mapping $\beta$ is continuous.

b) Let $\pi$ be a bounded representation of H in a Hilbert space E. Denote by F the Hilbert space $\ell^2(G/H; E)$. The formula

defines a continuous bounded representation of G in F.

c) Suppose that there exists a Hilbert norm $q$ on F such that the representation $\varrho$ is a unitary representation of G in $(F, q)$. Then there exists a Hilbert norm $\widetilde{q}$ on E such that $\pi$ is a unitary representation of H in $(E,\widetilde{q})$.