 It's a pleasure to welcome everybody to the Schubert Seminar for fall 2022. And the first point is we have gotten two new organizers of the seminar, Rebecca Golden from George Mason University and Rikard Rimani from UNC Chapel Hill. They'll join Leonardo and myself. It will be an even richer selection of subjects. Time this semester has changed slightly. It will be Mondays 4.30 to 5.30, again, approximately every two weeks. And yeah, two weeks from now, we will start with a special case to that. We will have a special talk by Andrea Kunckoff, which will be Wednesday, September 28. But still at 4.30. Yeah, another thing we have in the planning for this semester is we plan to have, at some point, lightning talks by graduate students. So we have something in mind, maybe 10 minutes, maybe 15 minutes, something in that area. And maybe we will have one days of that. Maybe we'll have two days. It depends a little bit on how many graduate students we accommodate. But for this, we are very happy to receive nominations of people you think would be an excellent person to give a lightning talk. It's fine if advisors nominate their students. But anybody can nominate anybody. If you're a graduate student, you think your advisor is dragging it, then just nominate yourself. That's perfectly fine. Just send an email to us organizers. If we get a lot of nominations, then we will give priorities to finishing graduate students. And these graduate student talks might be at the time of the semester where postdoc applications are being sent out. Yeah, enough about this. So today, it's a pleasure to introduce our first speaker of the semester, Sravan Kumar from UNC. And right now also at Princeton, he will speak about conformal blocks for Galois covers of algebraic groups. And a special note, please right-click on the window where you can see his notes, then pin that window, then you will get to see it. Yeah, so Sravan, please go ahead. Thanks. Thank you very much, Anders, for the introduction. So what I'm going to talk about eventually, that will be maybe last 10 minutes are at the most 15 minutes, which will be a joint work with Juju Hong of my department. So that each, as the title said, conformal blocks for Galois covers of algebraic curves. But before I introduce that topic, I have to tell you what the classical story is about the Berlin formula in the non-equivariate setting, in the usual setting or in the classical setting. So I will start there. And not only I will start there, but probably I will spend maybe 35 minutes to that topic. And then towards the last 15 or so minutes, I will come to the topic of the equivariate situation or the Galois cover situation. So let's begin with as usual some notation. So we start with a, let G be a simply connected, simple complex algebraic group. So my base field is complex numbers all the time. And we are going to take a sigma, a reduced projective curve. So I'm not assuming that it's a smooth, but it should be reduced. And I'm not assuming that it is irreversible either, but I'm assuming that it is connected. So it's a connected reduced projective curve. So by a S-pointed curve, we mean, first of all, the curve sigma, and then bunch of points. So I'm going to fix one number S, small s, which will be at least one. So I'm not taking it to be zero. So at least one. And then I'm taking bunch of points P1, P2 through PS. And these points are points of sigma, but they must be smooth points. So I am not, I mean, I'm requiring them to be smooth points. So I don't take any of them to be singular point. Now, we also fix a level, or what is called the central charge of the theory. And that level is going to be some positive integer C. I mean, bigger than zero. Now, we are going to define this dominant, dominant chamber for the affine while group, which depends upon the choice of C, small c. So DC is set of all those. Okay, so I am hiding some location. So G was a simply connected simple algebra group. I am making a choice of a boreal there, B. And also I'm making a choice of the carton, H, which is in the boreal. And the early algebra will be the corresponding lowercase underlined letters. So small h underlined denotes the linear algebra of the carton H. So now we are going to take linear forms lambda in H star, such that lambda of alpha i check is non-negative integer, where alpha i check are all the simple coroutes. And also we require that lambda of theta check. Now theta is the highest root. So theta check should be at most C, including C. Okay, so I mean, I'm just wrong. H is the cartons of algebra of Lee G, alpha i check are the simple coroutes, and theta check is the coroutes corresponding to the highest root theta. Now one central object in this whole theory is the affine linear algebra. I'm sure you all know what the affine linear algebra is, but just to set the notation, so I will denote it by G hat, the German G hat, where German G is the algebra of G. So I have deviated here in my notation. I could also have just said a small G bar or underline G, but that's one German letter I can kind of write it. So I wrote in German G. So G hat is G tensor C, Lorentz C double bracket T, plus a central element capital C. And the bracket here is X tensor F, Y tensor G is the bracket X, Y tensor F times G, plus X, Y the inner product of X and Y, and then we take a residue of D F times G, so the derivative of F times G, and residue just means the coefficient of T in worst. Now here X, Y is the normalized invariant form on the linear algebra, so that we normalize it in the manner, so that the highest root is squared, length is squared is two. So pretty good. You push up a little bit.