Here is the link: https://us02web.zoom.us/j/89917028917?pwd=SVFJSWtzU2dxRWlhTldMbnZjZnRJdz09

I think it would be nice to construct Hecke operators in genus 0 and 1. Main problem is to make sense of the question.

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.

Are these H_\lambda expected to be Hilbert-Schmidt or something like that?

Dennis-no, but some powers of them should be

Which is just as good

Thank you!

they are not generally Hilbert-Schmidt just because there are too few modifications, so the kernel is a distribution, not a function, correct?

Dima, exactly, the support is not full for the Schwartz kernel

But even if it is full, it may not be HS. You may have to raise to a power

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)

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

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 ?

yes, they are connections on a particular holomorphic bundle

I think what Ed just said explains what is special about opers: they give diff equations on Bun_G

But from analytic point of view of representations of pi_1 it is not easy to describe it, it’s transcendental

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 ?

We expect and can prove in some cases that there are no other eigenvalues. But here we get only “real” opers

Ah, I see. thanks!

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}} ?

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.