Timeline for Questions about the Bernstein center of a $p$-adic reductive group
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2012 at 8:20 | answer | added | Marc Palm | timeline score: 2 | |
| Apr 30, 2012 at 20:56 | history | edited | user4245 | CC BY-SA 3.0 |
edited title
|
| Apr 30, 2012 at 20:51 | vote | accept | user4245 | ||
| Apr 30, 2012 at 20:03 | comment | added | Mariano Suárez-Álvarez | Minor nitpick: the why part of your title does not make sense... maybe it wanted to be «What is the Bernstein center of a reductive group and why is it that?» or something along those lines. | |
| Apr 30, 2012 at 17:58 | answer | added | Jef | timeline score: 50 | |
| Apr 28, 2012 at 14:28 | history | edited | user4245 | CC BY-SA 3.0 |
added 60 characters in body
|
| Apr 27, 2012 at 22:43 | comment | added | user4245 | @Alexander: "the key sentence" is only my guess and still need to be convinced by experts. Thank you for your helpful comments. | |
| Apr 27, 2012 at 20:12 | comment | added | Alexander Chervov | Role is apparantly the same as role of ZU(g) - it allows to parametrize irreps by some managable set of params (similar to highest weights) - center acts as scalar irreps, hopefully center is finitely generated so essentially you take values of generators z_1...z_n in V and get numbers l1...ln if these numbers are different irreps are different, most probably converse is not formally true, but "informally" true, up to some "details". | |
| Apr 27, 2012 at 20:09 | comment | added | Alexander Chervov | I think questions "What is the original motivation to introduce it ? What role does it play in the theory of automorphic forms ?" are easy to answer from the key sentence " some analogy of "the centre of enveloping algebra"" (even without knowledge what B.'s center is :). I think: motivation is to have something analogous to ZU(g). U(g) is not good thing in p-adic case - so you want some substitute. continued... | |
| Apr 27, 2012 at 15:28 | comment | added | Qiaochu Yuan | (Agh. Above when I say "equivalences" I mean endo-natural transformations.) | |
| Apr 27, 2012 at 14:24 | history | edited | user4245 | CC BY-SA 3.0 |
added 13 characters in body
|
| Apr 27, 2012 at 14:03 | history | edited | user4245 | CC BY-SA 3.0 |
added 2 characters in body
|
| Apr 27, 2012 at 8:09 | comment | added | Alexander Chervov | Is the Berntein's center finitely generated algebra ? Is it polynomial algebra for semi-simple groups ? (These properties holds true for Z(U(g)) ). | |
| Apr 27, 2012 at 1:12 | history | edited | user4245 | CC BY-SA 3.0 |
added 252 characters in body
|
| Apr 27, 2012 at 0:58 | history | edited | user4245 | CC BY-SA 3.0 |
added 52 characters in body; edited title
|
| Apr 26, 2012 at 20:23 | comment | added | Marc Palm | +1 nice question, even $GL(2)$ would be nice with some explanation, why this point of view is useful. | |
| Apr 26, 2012 at 16:44 | answer | added | SGP | timeline score: 3 | |
| Apr 26, 2012 at 16:32 | history | edited | user4245 | CC BY-SA 3.0 |
added 11 characters in body; edited title
|
| Apr 26, 2012 at 16:28 | comment | added | user4245 | @BR :Thanks for your references ! It will be nice if someone could explain this in some concrete example, say for GL(n). | |
| Apr 26, 2012 at 16:19 | comment | added | B R | Qiaochu is correct. The Bernstein center of a $p$-adic group $G$ is the algebra of endomorphisms of the identity functor on the category of smooth representations of $G$. It can also be realized as a space of distributions. See, e.g., hazu.hr/~tadic/41-Moy-T-Howe-proc.pdf or ams.org/journals/ert/2002-006-11/S1088-4165-02-00181-4/… | |
| Apr 26, 2012 at 16:12 | comment | added | Qiaochu Yuan | The meaning I know is that the Bernstein center of a category $C$ is the (commutative) monoid of self-equivalences of the identity functor $\text{id}_C$. If that category is $R\text{-Mod}$ then this is the center of $R$ in the usual sense. I don't know what category $C$ is meant here though. | |
| Apr 26, 2012 at 16:06 | history | edited | user4245 |
edited tags
|
|
| Apr 26, 2012 at 15:21 | history | edited | user4245 |
edited tags
|
|
| Apr 26, 2012 at 15:08 | history | asked | user4245 | CC BY-SA 3.0 |