Timeline for Gelfand's trick (Gelfand's lemma) in positive characteristic?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 24, 2022 at 5:02 | comment | added | LSpice | Name of this preprint: Ben Shalom - Weak Gelfand Pair Property And Application To $\operatorname{GL}(n+1)$, $\operatorname{GL}(n)$ Over Finite Fields. | |
| Jul 11, 2019 at 1:35 | vote | accept | ferrari | ||
| Jul 9, 2019 at 15:27 | vote | accept | ferrari | ||
| Jul 9, 2019 at 15:27 | |||||
| Jun 27, 2019 at 9:00 | comment | added | darij grinberg | Algebraic closure is only relevant if you define Gelfand pairs via intertwiners of representations, right? | |
| Jun 27, 2019 at 8:58 | answer | added | Bugs Bunny | timeline score: 5 | |
| Jun 26, 2019 at 11:23 | comment | added | Geoff Robinson | It (the last quesstion) is not immediate ( if even true) as far as I can see. Yes, that is what I mean by Nakayama reciprocity. | |
| Jun 26, 2019 at 10:59 | comment | added | ferrari | I assume that by Nakayama reciprocity, you mean the statement of Frobenius reciprocity for the induction and restriction functors? Then, is the Gelfand pair condition is equivalent to the Hecke algebra being commutative over any characteristic when G is a compact (or locally compact) group? | |
| Jun 26, 2019 at 9:54 | comment | added | Geoff Robinson | In the case of finite groups in characteristic zero or coprime to |G|, it is Frobenius reciprocity which makes the Gelfand pair condition equivalenet to the Hecke algebra being commutative. In other characteristics, Frobenius reciprocity is replaced with Nakayama reciprocity, which is more subtle. | |
| Jun 26, 2019 at 9:45 | comment | added | ferrari | Ah, so it is actually in the equivalence between the Hecke algebra being commutative and the multiplicity one statement? Indeed, the use of semisimplicity (Maschke's theorem) is also present in Theorem 2 in Chapter IV of Lang's $SL_2(\mathbb{R})$ and in Proposition 6 here: mathematics.stanford.edu/wp-content/uploads/2013/08/…. | |
| Jun 26, 2019 at 3:53 | comment | added | tj_ | The proof of Lemma 1 in math.toronto.edu/murnaghan/courses/mat445/GelfandPairs.pdf uses that the group algebra $\mathbb{C}G$ is semisimple. I guess, thatGelfand's lemma works over alg. closed fields those characteristic do not divide the order of $G$ | |
| Jun 26, 2019 at 1:46 | history | asked | ferrari | CC BY-SA 4.0 |