Timeline for Orbital integral for matrix coefficients
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2018 at 13:49 | vote | accept | Desiderius Severus | ||
| Apr 11, 2016 at 2:16 | answer | added | LSpice | timeline score: 5 | |
| Apr 10, 2016 at 20:28 | comment | added | Asaf | I don't get that, Peter-Weyl is the fundamental theorem behind rep. theory of compact groups (together with some version of Schur's orthogonality), the theorem is basic and valid in great generality (it doesn't need more than the spectral theorem for compact self-adjoint operator). Somehow the "correct" manner to study those subjects would be start with Peter-Weyl, go on to Weyl's work (conjugation to maximal torus etc.) and then try to study HC's work (this is the approach taken in Varadarajan's). Taking Peter-Weyl away is just like taking the Newton-Leibnitz formula away from calculus. | |
| Apr 10, 2016 at 17:53 | comment | added | Yemon Choi | I think the issue is not so much Peter-Weyl but some form of Schur's lemma (which seems to be the key mechanism behind all orthogonality relations I've come across in the setting of group reps) | |
| Apr 10, 2016 at 17:41 | comment | added | Desiderius Severus | @LSpice I am not at ease with Peter-Weyl and what really hides behind, hence I prefer to avoid its use, never knowing if it would be using a huge machinery for such a simple result. I am more familiar with orthogonality of matrix coefficients, but you seems to say they are not so different, hence it would be ok ;) | |
| Apr 10, 2016 at 16:29 | comment | added | LSpice | Of course 'supercuspidal' is vacuous in this setting! If you don't like to use Peter–Weyl, I guess that you don't want to use orthogonality of matrix coefficients to prove that $\Theta_\pi(f) = \dim(\pi)^{-1}$? | |
| Apr 10, 2016 at 11:04 | comment | added | Desiderius Severus | @PaulBroussous I am in a compact setting, more precisely: I consider $B$ a division quaternion algebra and $G = Z \backslash B^\times$; I am trying to do the work locally for the (compact) unit group $G_p$ in a ramified place. (Otherwise $\dim(\pi)$ should probably be the formal degree of $\pi$ ?) | |
| Apr 10, 2016 at 9:18 | comment | added | Paul Broussous | Could you give make your notation and assumption more precise ? What is your group ? What sort of representation are you considering ? What are the assumptions on the group and representation ? What is ${\rm dim}(\pi )$ if $G$ is not compact ? | |
| Apr 10, 2016 at 8:59 | history | asked | Desiderius Severus | CC BY-SA 3.0 |