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Suppose we have a compact group $G$ with continuous $f$ on $G$ that is also $G$-finite.

I am told that then, out of all the irreducible representations $\pi$ of $G$ we must have $\pi(f)\neq 0$ only for finitely many. Here the representation is thought of as a transformation on the space of functions.

Why is this true? I feel like it is very short, and I was just missing something quick. Can someone enlighten me on this one.

Thanks again!!

(Also, by $G$-finite I mean that the vector space of right and left translates is finite.)

  • 1
    possible duplicate of [Compact Group $G$ with a $G$-finite function.](http://math.stackexchange.com/questions/38573/compact-group-g-with-a-g-finite-function)2011-05-11
  • 1
    @Zev: Accident! I deleted the other one so it should be ok.2011-05-11

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