Timeline for When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 9, 2010 at 4:13 | comment | added | Marty | Thanks for the update - I'll check out those references. You certainly need a Hilbert space in the definition of a unitary rep. But you can consider completely continuous representations, in the category of bounded reps on Banach spaces. | |
| Jun 9, 2010 at 3:43 | comment | added | Victor Protsak | Correction: they proved it for the space of the cusp forms (which they introduced), I am not sure who was the first to do the cocompact case. I read the proof in their book (Generalized functions, vol 6 "Representation theory and automorphic functions"). From MathSciNet it appears that first there was a note in Doklady, MR0142691, with full proofs published in Trudy Mosc. Mat. Ob., MR0159899. Also, don't you need Hilbert space (not just Banach) to get a unitary representation? | |
| Jun 9, 2010 at 3:04 | comment | added | Marty | Thanks for the comment - do you a specific reference for a paper by Gelfand and P-S? I'm curious about the history of these things. | |
| Jun 9, 2010 at 0:32 | comment | added | Victor Protsak | I just wanted to point out that the direct decomposition theorem for automorphic representations (cocompact $\Gamma$) is due to Gelfand and Piatetskii-Shapiro and their proof establishes the general version. | |
| Jun 8, 2010 at 23:24 | history | answered | Marty | CC BY-SA 2.5 |