The Peter-Weyl theorem asserts that for a compact Lie group $G$ every unitary irreducible representation is necessarily finite-dimensional and any unitary representation is a direct sum of irreducibles.

1) Is not any representation of a compact group (finite or infinite-dimensional) representation of a compact group unitarizable and thus completely reducible? Or does this only hold for finite-dimensional?

2) Now if $G$ is not compact, how exactly does the situation change? If I have an infinite-dimensional unitary representation of a non-compact group, is it also always completely reducible? Also, which noncompact group possess (nontrivial) finite-dimensional unitary representations?

  1. In the finite-dimensional case this is clear by averaging. In the infinite-dimensional case, it depends on what you mean by a representation. A very general thing you could ask for is a representation on some topological vector space, but most topological vector spaces aren't Hilbert spaces so it's unclear what it would even mean for such a representation to be unitarizable. If you mean a continuous representation on a Hilbert space, then the same averaging argument works. The content of the Peter-Weyl theorem is not that unitary representations are completely reducible: that's a general fact. It asserts a lot of more interesting facts than this, such as that matrix coefficients of unitary irreducibles separate points.

  2. Noncompact semisimple Lie groups in general have many interesting irreducible infinite-dimensional representations, some but not all of which are unitarizable. It's still true that unitary representations are completely reducible (and the proof is the same), but often there are no nontrivial finite-dimensional ones: for example, if $G$ is a noncompact simple Lie group such as $PSL_2(\mathbb{R})$, it can't embed into any unitary group $U(n)$ (all of which are compact).


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