TITLE: Unitary representations of SL(2, R)
QUESTION [19 upvotes]: 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).

How are those representations described?
Do all unitary representations come from L^2(G)?
How are those related to representation of compact SO(3, R)? 
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)?

REPLY [2 votes]: Regarding classification of irreducible unitary representations of arbitrary Lie groups, Michel Duflo showed (I guess in the early 80's) that at least for algebraic Lie groups the classification can be reduced to the case of reductive Lie groups. See Vogan's Unitary representations of reductive Lie groups for more information.