Recently in a paper we get the following result:
Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of $G\rtimes\Gamma$ on a Hilbert space $H$ satisfies $\dim{H_\Gamma}\leq 1$. Here $H_\Gamma$ is the space of $\Gamma$-invariant vectors in $H$.
We are not experts in representation theory.
My question is :
Does this result appear in the literature before?
I would think that this result would go all the way back to Frobenius. Anyway, the proof seems easy enough:
Let $V$ be the trivial $\Gamma$-module. We want to show that $\dim H_\Gamma=\dim\mathrm{Hom}_{\Gamma}(V,H)\leq 1$. To this end, note that by Frobenius reciprocity, $$ \mathrm{Hom}_{\Gamma}(V,H)\cong\mathrm{Hom}_{G\rtimes\Gamma}(\mathrm{Ind}_{\Gamma}^{G\rtimes\Gamma}V,H) $$ so it suffice to show that the right-hand side is at most 1-dimensional. As $H$ is irreducible, this amounts to showing that every irreducible summand of $\widehat{V}:=\mathrm{Ind}_{\Gamma}^{G\rtimes\Gamma}V$ appears with multiplicity 1 (by Schur's lemma).
Well, this is easy since $\widehat{V}\cong\mathbb{C}G$ as a $G$-module and $\mathbb{C}G$ decomposes as a direct sum of non-isomorphic $G$-modules. On the other hand, if $U\oplus U\leq\widehat{V}$ for some irreducible $G\rtimes\Gamma$-module $U$, then decomposing each $U$ as a $G$-module shows that there are multiple copies of the same irreducible $G$-module in $\widehat{V}$ upon restriction to $G$. This is a contradiction.
Hence, $\dim H_\Gamma\leq 1$.
