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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 ?

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    I'm not sure exactly what you're asking in the second bullet point, but concerning the first I can tell you that it's obviously not true: How about the two-dimensional representation of $\mathbb{R}$ given by $t \mapsto \begin{bmatrix} \cos{t} &-\sin{t} \\\ \sin{t} & \cos{t}\end{bmatrix}$ or any character representation?2011-07-04
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    Maybe the question is whether evry inf. dimensional representation of a noncompact Lie-group is induced from a maximal compact subgroup?2011-07-04

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