Logo

GL OH: Frenkel and Etingof - Shared screen with speaker view
Alexander Braverman
03:53:10
Is it also expected that the spectrum is simple?
Pavel I Etingof
03:53:37
Yes, over complex numbers
Alexander Braverman
03:54:14
You mean, as opposed to p-adic numbers?
Pavel I Etingof
03:54:59
for real numbers sometimes there is a double eigenvalue but in general simple. For p-adic I don’t know but likely it is usually simple as well.
Alexander Braverman
03:56:22
Also, is there a reason why Edik considers only ramification of "Iwahori" type?
Dennis Gaitsgory
03:57:09
As opposed to parahoric?
Alexander Braverman
03:57:26
As opposed to arbitrary level structure
Pavel I Etingof
03:57:37
That’s the only case we studied but one can consider the more general case with wild ramification, of course
Dennis Gaitsgory
03:57:43
I guess it would be much harder
Alexander Braverman
03:58:07
Sure, but the setup should make sense, I think
Dennis Gaitsgory
03:58:15
Also we don’t really understand the categorical theory for deeper ramification
Pavel I Etingof
03:58:22
Maybe Stokes data will have to be considered then
dhy
03:58:38
Is there a local version of the conjecture?
Dennis Gaitsgory
03:58:59
A great qstn: will ask
Pavel I Etingof
03:59:09
But this is a very interesting question. One can make ordinary tame points collide and compute confluent limits
Davide Gaiotto
04:00:32
I have been working out some simple wild example, such as sl_2 with spectral curve x^2 = z^3 + z + u
Pavel I Etingof
04:01:29
I’ll talk about the 4-point case, then there are various limits you can take, over C this is confluent limits of Lame equations
Aswin Balasubramanian
04:17:09
Why was it important to restrict to very stable bundles for the statement on the last slide ? (about restriction of \Delta )
Pavel I Etingof
04:18:08
It is not very important, just easier to see integrals which will be defined after break converge
Aswin Balasubramanian
04:18:46
I see, thanks!
Sam Raskin
04:22:41
If I think of a D-module on Bun_G as a system of differential equations, I can imagine taking something like L^2 solutions of that system, and I’d imagine D_{Bun_G} would give L^2(Bun_G). According to geometric Langlands, D_{Bun_G} should correspond to the structure sheaf of opers. So I wonder if there might be a general conjecture taking as input a D-module F on Bun_G, and recovering your conjecture for F = D_{Bun_G}.
Sam Raskin
04:29:17
The fact that eigensheaves have regular singularities is shown in the first AGKRRV paper.
Dennis Gaitsgory
04:29:23
Why is reg. sing. Important?
Alexander Braverman
04:29:24
why does irreducibility imply that the eigenspace is one-dimensional?
Pavel I Etingof
04:30:44
Sasha, because it is an invariant pairing between representations V and V^* of \pi_1
Pavel I Etingof
04:32:37
The space of pairings is 1-dim if V is irreducible
Pavel I Etingof
04:45:12
also eigenvalue of the Hecke operator is a section of \rho_\chi\otimes \bar\rho_\mu, and it has to be single-valued
Roman Bezrukavnikov
04:48:35
There is an elementary construction, grafting, producing many examples of surfaces with an SL(2) oper with real monodromy
Pavel I Etingof
04:49:02
Roma, that’s right, it is described in Goldman’s paper
Roman Bezrukavnikov
04:49:15
yes, thanks.
Aswin Balasubramanian
05:03:25
Why was it important to have BunG^{stable} open dense in Bun_G to have the Hecke operators ? And is this property the same as asking for the stack Bun_G to be "very good" in the sense of Beilinson-Drinfeld ?
Pavel I Etingof
05:05:25
Roughly ecause measure has to be divided by the volume of the automorphism group, which is infinite
Pavel I Etingof
05:06:01
It almost always holds, anyway
Aswin Balasubramanian
05:09:31
Thanks!