Exercise 8, § 21

Let H and F be subgroups of G, and let S be a subset of G such that G is the disjoint union of the double classes $FsH$. Let $(N, \sigma )$ be a representation of H. For $s\in G$, set $H_s= F\cap sHs^{-1}$, and denote by $\sigma_s$ the representation of $H_s$ in N defined by $\sigma_s(x) =\sigma (s^{-1}xs)$. a) Prove that the representation Res$^G_F$(Ind$^G_H(\sigma )$) is isomorphic to the direct sum of the representations Ind$^F_{H_s}(\sigma_s)$ for $s\in S$. (Denote the representation Ind$^G_H(\sigma )$ by $(M, \pi )$. The space M can be identified with the direct sum of the $\pi (x)N$ for $x\in G/H$. For $s\in S$, let $M(s)$ be the subspace of M generated by the $\pi (x)N$ with $x\in FsH$. Observe that $M(s)$ is the direct sum of the subspaces $\pi (xs)N$ for $x\in F/H_s$, and deduce from Exercise 5, b) that as a K[F]-module, $M(s)$ is isomorphic to Ind$^F_{H_s}(\sigma_s).)$ b) Take F = H (so that we have $H_s= H\cap sHs^{-1}$). Prove that the representation Ind$^G_H(\sigma )$ is irreducible if and only if $\sigma$ is irreducible and for every $s\in G$- H, the supports (VIII, p. 66) of the representations $\sigma_s$ and Res$^H_{H_s}(\sigma )$ are disjoint (“Mackey’s criterion”: apply a) and Frobenius reciprocity).

c) Assume that H is normal; then Ind$^G_H(\sigma )$ is irreducible if and only if $\sigma$ is irreducible and is not isomorphic to any of its conjugates $\sigma \circ$ int($g$) for $g\in G$- H.