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?

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    $\begingroup$ It is true, essentially due to H. Weyl, about 100 years ago. Also certainly for compact and connected. Also for complex vector space repns. The basic point is that the integral_operators associated to repns of a compact group are Hilbert-Schmidt, so compact, so have finite-dimensional eigenspaces... $\endgroup$ – paul garrett Sep 27 '16 at 0:41
  • $\begingroup$ @paulgarrett: Many thanks! Why don't you write it as a formal answer and polish it a bit by some link to reference? $\endgroup$ – PhysicsMath Sep 27 '16 at 0:52

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.

  • $\begingroup$ some of your notes are great (I'm working on the Rankin-Selberg convolution one) $\endgroup$ – reuns Sep 27 '16 at 2:17
  • $\begingroup$ @user1952009 ... :) $\endgroup$ – paul garrett Sep 27 '16 at 12:19
  • $\begingroup$ Many thanks Prof Garrett! I should also explore your other notes : ) $\endgroup$ – PhysicsMath Sep 27 '16 at 14:50

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