This question is somewhat of a continuation of this question that I had asked earlier -
Representations of a non-compact group are labeled by its maximal compact subgroup?
I want to know when or is it always true that an unitary representation of a non-compact Lie group is infinite dimensional? If yes then why? If no then kindly give examples.
Also when can one be sure that an infinite dimensional representation of some non-compact Lie group is labeled by a single representation of its maximal compact subgroup ?