I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group `G` being `SL(2, R)`, can be completely described and that there is a discrete and continuous part of the spectrum of `L^2(G)`.


1. How are those representations described?
2. Do all unitary representations come from `L^2(G)`?
3. How are those related to representation of compact `SO(3, R)`?
4. What happens in the flat limit between `SL(2, R)` and `SO(3, R)`?


Also, is it possible to answer the questions above simultaneously for all Lie groups, not just `SL(2, R)`?



Unitary representations of SL(2, R)
I strongly recommend you read the article "Representations of semisimple Lie groups" by Knapp and Trapa in the park city/ias proceedings "Representation theory of Lie groups". It's a very nice introduction to the problem of describing the "unitary dual" (which is what you are asking about) which focusses on SL(2,R). For example, page 9 says "the irreducible unitary representations that appear in L^2(G) do not nearly exhaust the unitary dual" for general semisimple Lie groups (thus answering you question 2). For more info, you can check out knapp's book "Representation theory of semisimple groups: an overview based on examples". For example, sections II.4 and II.5 describe the unitary duals of SL(2,C) and SL(2,R) respectively. The unitary duals of GL(n,C) and GL(n,R) were described by Vogan. Some other unitary duals are known, but in general, I don't think anything else is known. One approach is via Langlands' parametrization of irreducible admissible representations of reductive groups. This result is known for all groups and unitary representations are admissible, so the problem would be to identify which admissible representations are unitary (the knapp-trapa article talks about this). As for 3), every irreducible unitary representation of a compact group is finite-dimensional, so you don't get any of the infinite dimensional representations you get for SL(2,R). I don't know what you mean by 4).


For a full answer to 1) you can check out [link text](http://en.wikipedia.org/wiki/Representation_theory_of_SL2%28R%29#Unitary_representations).


