Let $G$ be a group and let $H$ be a subgroup of finite index.

Let $V$ be an irreducible complex representation of $G$ (no topology or anything: $V$ is just a non-zero complex vector space with a linear action of $G$ and no non-trivial invariant subs).

Now consider $V$ as a representation of $H$. Is $V$ a finite direct sum of irreducible $H$-reps?

I am almost embarrassed to ask this question here. It looked to me initially like the answer should be "yes and this question is trivial". If $G$ is finite it is trivial and Clifford theory tells you basically what can happen. Here is another case I can do: if $H$ has index two in $G$ then $V$ is indeed a finite direct sum of irreducibles. For either $V$ is irreducible as an $H$-rep, in which case we're done, or $V$ is reducible, so there's $0\not=W\not=V$ an $H$-stable sub. Say $g\in G$ with $g\not\in H$. One checks easily that $gW$ is $H$-stable, that $W\cap gW$ is $G$-stable, so must be zero, and that $W+gW$ is $G$-stable, so must be $V$. Hence $V$ is the direct sum of $W$ and $gW$. This implies that $W$ is irreducible as an $H$-rep---for if $X$ were a non-trivial sub then the same argument shows $V=X\oplus gX$ but this is strictly smaller than $W\oplus gW=V$.

I thought that this argument should trivially generalise to, say, the case where $H$ is a normal subgroup of prime index. But I can't even do the case where $H$ is normal and $G/H$ has order $3$, because I can't rule out $V$ being the sum of any two of $W$, $gW$ and $g^2W$, and the intersection of any two being trivial.

Either I am missing something silly (most likely!) or there's some daft counterexample. I almost feel that I would be able to prove something if I knew Schur's lemma [edit: by which I mean that if I knew $End_G(V)=\mathbf{C}$ then I might know how to proceed], but in this generality I don't see any reason why it should be true. Perhaps if I knew a concrete example of an irreducible complex representation of a group for which Schur's lemma failed then I might be able to get back on track. [edit: in a deleted response, Qiaochu pointed out that $G=\mathbf{C}(t)^\times$ acting on $\mathbf{C}(t)$ provided a simple example] [final remark that in the context in which this question arose, $G$ was a locally profinite group and $V$ was smooth and I could use Schur's Lemma, but by then I was interested in the general case...]

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