It is well known that if two (irreducible) admissible representations have the same characters, then they are isomorphic. To my knowledge, this is true for both Lie groups and p-adic groups.

In the case of finite groups, this follows from the orthogonality relations of characters. I would like to know what are the main ingredients in the proof of its generalization in infinite dimensional case. In particular, is it simple if we assume both representations to be irreducible.

I checked the books by Knapp, Wallach, but couldn't find any hint(I'm not an expert in this field, and not familiar with these books). I appreciate a lot for any explanations of the proof, or suggestions on references containing it.