Timeline for Decomposition of Regular Representation of Non-compact Lie group
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2013 at 23:15 | comment | added | Victor Protsak | Rather than adding yet another answer, let me just highlight the relevant term "Plancherel measure". Unitary representations appearing in the decomposition of the regular representation, i.e. those in the support of the Plancherel measure, are known as "tempered representations" and form an important part of the unitary dual. See e.g. Wallach's book. As Paul Garrett explained, it is known explicitly for complex semisimple Lie groups after Gelfand-Naimark's pioneering work and in many other cases, including connected nilpotent groups via Kirillov's orbit method. | |
| Feb 4, 2013 at 1:36 | answer | added | Jim Humphreys | timeline score: 1 | |
| Feb 3, 2013 at 23:44 | answer | added | paul garrett | timeline score: 6 | |
| Feb 3, 2013 at 23:21 | comment | added | Yemon Choi | (For more detail on what can be done when we have Type I groups, see e.g. Dixmier's book on C-star algebras) | |
| Feb 3, 2013 at 23:20 | comment | added | Yemon Choi | Also, as Will Sawin points out, thinking about the most basic non-compact Lie group will immediately show that you need to figure out just what you want as "an analogue" of the Peter-Weyl theorem... | |
| Feb 3, 2013 at 23:20 | comment | added | Yemon Choi | An immediate issue is that one needs to distinguish between Type I groups and groups which are, erm, not of Type I. Semisimple connected Lie groups are Type I (Harish Chandra/Godement/Stinespring) so this may be enough for what you really want - I have a feeling that you probably aren't thinking of solvable Type II examples like the Mautner group when you asked this question... | |
| Feb 3, 2013 at 23:17 | answer | added | Will Sawin | timeline score: 7 | |
| Feb 3, 2013 at 22:51 | history | asked | 7-adic | CC BY-SA 3.0 |