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!