Why does Schur's lemma (that there are no intertwining operators between two non-isomorphic, finite-dimensional, and irreducible representations of a group) require that the reps be finite-dimensional? Is this true in general for any group and any pair of irreps, or is there a counter-example when the representations are infinite-dimensional?
EDIT: Thank you for the responses so far! The reason I was asking was that I've been reading Terrance Tao's blogpost on the proof of the Peter-Weyl theorem. He uses Schur's lemma to prove it, and uses functional analysis to prove Schur's lemma. But from what I can tell, his proof only seems to use the version stated above (so not the fact that a $G$-stable map $T: V\to V$ must be a scalar multiple of the identity). Am I mistaken in thinking that, or did T. Tao just prove more than was necessary?
Best Answer
For the record, there is a version of Schur's lemma due to Dixmier initially and then to Quillen, which applies to infinite dimensional modules. You can find Quillen's paper here.