1
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.