Unitary representations of SL(2, R)

Question

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)?

Answer 1

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.

Answer 2

I just want to elaborate more on questions 3. and 4. I'll consider the locally isomorphic groups SU(1,1) of SL2(R) and SU(2) of SO(3)

There is an analogy between the discrete series of SU(1,1) and the unitary irreps of SO(3). Both have holomorphic representations on the group's orbit on the flag manifold S^2 = SL(2,C)/B (B is a Borel subgroup). In the case of SU(2), the orbit is the whole of SU(2) while for SU(1,1) its is a noncomapct supspace: The Poicare disc. In both cases the representation space is a reproducing kernel Hilbert space and the group action is throug a Mobius transformation. This analogy generalizes to other non-compact groups having a holomorphic discrete series and it can be considered as a generalization of the Borel-Weil construction for compact groups.

Concerning question 4. I think that you are talking about Wigner's theory of Lie group contraction, in which a Lie group with the same dimension and with more "flat" directions is associated to the original Lie group. For example there is a contraction of SU(2) to Eucledian group in two dimensions and SU(1,1) to the Poncare group in two dimensions. There are interseting connections to the group representations of the contracted versions, and also of the Casimirs.

Answer 3

I think that Rob H.'s answer is probably best; but, for (1) and (2), if you are interested in small general and special linear groups particularly, you could do worse than to consult Lang's $\operatorname{SL}_2(\mathbb R)$, whose subject I will leave it to you to guess. Bump's Automorphic forms and representations also covers the $\operatorname{GL}_2$ picture nicely, albeit possibly from a different point of view from the one you have in mind.

Answer 4

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.