I found the following claim in some lecture notes:
irreducible representations of a compact Lie group must be finite-dimensional
Is it true or not? What about compact connected Lie groups? What about complex rather than real representations?
2026-04-07 14:16:18.1775571378
Irreducible representations of compact Lie groups
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A more general assertion holds for compact topological groups, without the assumption of Lie-ness, due to the compactness of Hilbert-Schmidt operators. There are many sources for this, and many on-line. My old notes at http://www.math.umn.edu/~garrett/m/repns/notes_2014-15/06a_unitary_of_top.pdf include this sort of result and various further related.