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GL OH: Frenkel and Etingof - Shared screen with speaker view
Sam Raskin
18:09
Here is the link: https://us02web.zoom.us/j/89917028917?pwd=SVFJSWtzU2dxRWlhTldMbnZjZnRJdz09
Edward Witten
28:08
I think it would be nice to construct Hecke operators in genus 0 and 1. Main problem is to make sense of the question.
Pavel I Etingof
30:58
We construct them explicitly in genus 0, I will tell about it Feb 10 (and today for 4 points where they are in JKontsevich’s paper). Genus 1 can be done similarly but we have not done the calculations yet.
Dennis Gaitsgory
43:42
Are these H_\lambda expected to be Hilbert-Schmidt or something like that?
Pavel I Etingof
44:59
Dennis-no, but some powers of them should be
Pavel I Etingof
45:21
Which is just as good
Dennis Gaitsgory
45:39
Thank you!
Dima ARINKIN
47:32
they are not generally Hilbert-Schmidt just because there are too few modifications, so the kernel is a distribution, not a function, correct?
Pavel I Etingof
48:07
Dima, exactly, the support is not full for the Schwartz kernel
Pavel I Etingof
49:18
But even if it is full, it may not be HS. You may have to raise to a power
Dima ARINKIN
51:19
I see. (There may be something like this geometrically: if we want the kernel of the Hecke transformation on Bun\times Bun to be a local system on some quasi-compact chart, we need to work with high enough power (i.e., large enough Hecke operators)
Pavel I Etingof
52:46
Right, it is just dimension count. But e.g. for 4 points the support is full but H_x is not HS, although |H_x|^{1+\eps} is HS for any \eps>0
Aswin Balasubramanian
01:15:37
In this analytical framework, are Opers somehow special among all local systems ? (or) Is there a similar story for arbitrary irreducible ^\vee G-local systems ?
Pavel I Etingof
01:16:30
yes, they are connections on a particular holomorphic bundle
Dima ARINKIN
01:16:47
I think what Ed just said explains what is special about opers: they give diff equations on Bun_G
Pavel I Etingof
01:17:08
But from analytic point of view of representations of pi_1 it is not easy to describe it, it’s transcendental
Aswin Balasubramanian
01:27:22
Re: Dima's comment : Does that mean that more general eigenvalues could exist for these Hecke operators .. but those eigenvalues will not obey the differential equations on Bun_G that are obeyed by opers ?
Pavel I Etingof
01:28:53
We expect and can prove in some cases that there are no other eigenvalues. But here we get only “real” opers
Aswin Balasubramanian
01:29:52
Ah, I see. thanks!
Roman Travkin
01:40:56
Continuing the theme raised by Vladimir as well as by Sam last time, is it possible to get all locsystems w/real m’my from the global Whittaker D-module, which corresponds to O_{LocSys_{^LG}} ?
Sam Raskin
01:44:03
I would ask Volodya’s question as follows: what (if anything) is the relationship between Laumon-Rothstein and the Fourier theory on a torus the EFK theory considers for G = G_m? It seems to me there’s been an analogy, but it seems very plausible that there’s a more direct relationship.