# Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric quantization and coadjoint orbits. As regards the present question, I use the terminology of Witten, "Coadjoint orbits of the Virasoro group".

Specifically, my questions are the following (questions 2 and 3 are the most important ones to me; even a partial answer may be satisfactory!):

1. What is the definition of "unipotent representation" for SL(2,R), regardless of geometric quantization?
2. Are all unipotent representations of SL(2,R) known and classified?
3. If yes, is it known how to reproduce these representations by geometric quantization of the "cone-like" coadjoint orbits of SL(2,R)?
4. Same questions for any non-compact, semi-simple Lie group.

I've tried to find an answer by Googling "unipotent representations SL(2,R)" and similar keywords, but the only search results I got were research papers that were too advanced for me :-( In particular, the paper "Unipotent Representations of Complex Semisimple Groups" by Barbasch and Vogan provides a definition of "unipotent representation" (definition 1.17 of the paper), but it is a bit obscure to me... I hope there exists a simpler reformulation of this definition.

(By the way, I've posted the same question on the math stack exchange, but I don't know how the stack exchange and overflow communities communicate.)

• It would be helpful here to say something more explicit about the meaning of "unipotent representation" for a real Lie group. This arises in the work of Barbasch-Vogan and others, suggested in part by the earlier work of Deligne-Lusztig on finite groups of Lie type. (By the way, your questions cover a lot of ground and are rather sophisticated for stack exchange.) – Jim Humphreys Oct 13 '14 at 13:57
• P.S. Concerning the notion "unipotent representation" in your setting, the article by Witten you link to is unhelpful about understanding it. So you need to provide more background and/or references. – Jim Humphreys Oct 13 '14 at 17:43
• @JimHumphreys: You're right; in fact, I don't know what is the definition of "unipotent representation" in general, so I've edited my question accordingly. The paper "Unipotent Representations of Complex Semisimple Groups" by Barbasch and Vogan seems to contain a lot of information on this topic, but it is too advanced for me... – Blagoje Oblak Oct 13 '14 at 19:13