# D(BunG) Seminar

Goals: Last quarter, we saw that BunG was (very) good. In the Spring 2023 quarter, we will see BunG is in fact miraculous. More specifically, the main goal is to prove the miraculous duality theorem for D-mod(BunG). In doing so, we will develop the theory of D-modules on schemes and stacks, and then transition to the higher categorical setting, which will allow us to formulate the main results of compact generation and miraculous duality for D-mod(BunG). We intend to make the seminar accessible and useful to any graduate student with a broad interest in BunG.

Setup: There will be weekly talks Wednesdays 4-6pm at Eckhart 202. There is also an option to attend via Zoom -- contact one of the organizers to receive the Zoom link. Talks will be broken into two periods of 45 minutes each, with a 10 minute break in between for questions. As UChicago consists of leading experts in the broad area of BunG, we have invited faculty and postdocs to give (and attend!) a majority of the talks.

Prerequisites: Basic algebraic geometry: elementary knowledge of projective varieties, line bundles, (quasi-)coherent sheaves, morphisms (smooth, etale, flat), D-modules on varieties. Basic representation theory: finite-dimensional representation theory for reductive or semi-simple Lie groups and Lie algebras.

Organizers: Justin Campbell (cjcampbell AT uchicago.edu), Nikolay Grantcharov (nikolayg AT uchicago.edu), Aaron Slipper (aaronslipper AT uchicago.edu),

### Spring 2023

Title and Recording Speaker Notes Date
$$\text{Bun}_G$$ at infinity $$\text{Bun}_G$$ is not quasi-compact, and its behavior at infinity is controlled by the deeper strata of the Harder-Narasimhan stratification. I will discuss the remarkable geometric contractiveness property of this stratification. As will be explained later in the semester, the contractiveness implies that the category of D-modules on $$\text{Bun}_G$$ behaves as if $$\text{Bun}_G$$ would be quasi-compact. The talk follows the work of V.Drinfeld and D.Gaitsgory "compact generation of the category of D-modules on the stack of G-bundles on a curve", Drinfeld-Gaitsgory Recording
Sasha Beilinson

3/22/23
Infinitesimal Structure of $$\text{Bun}_G$$ Reference: Beilinson-Drinfeld and Beilinson-Ginzburg
Nikolay Grantcharov

3/29/23
Crystals and D-modules Reference: Gaitsgory seminar
Zhilin Luo

4/5/23
Higher Category Theory Reference: Chapter 1 of Gaitsgory-Rozenblyum's DAG book.
Justin Campbell

4/12/23
Derived Categories of Sheaves on Schemes Reference: Chapter 1 of Gaitsgory-Rozenblyum's DAG book.
Justin Campbell

4/19/23
Compact Generation of D-mod(BunG) Reference: Drinfeld-Gaitsgory

4/26/23
Conformal Blocks Reference: Rozenblyum
Nick Rozenblyum

5/3/23
Geometric Eisenstein Series and Constant Term Functors Reference: Gaistgory
Andreas Hayash

5/10/23
Miraculous Duality and Strange Functional Equation Reference: Gaistgory
Nick Rozenblyum

5/17/23

### Winter 2023

Title and Recording Speaker Notes Date
Introduction to stacks and $$\text{Bun}_G$$ The moduli stack of principal G-bundles on an algebraic curve is an important object in numerous contexts. I will give a motivated introduction to the theory of algebraic stacks in general and BunG in particular. Of particular interest will be relations with geometric representation theory. Recording
Nick Rozenblyum Notes

1/4/23
Functor of Points and Descent As a warm-up, I'll define schemes from the functor of points perspective. After upgrading to functors valued in groupoids, stacks will emerge as a natural 2-categorical upgrade of sheaves. We'll see some examples that demonstrate why the Zariski topology is too coarse for many purposes, and should be replaced by the étale or fppf topology. Finally, I'll discuss a couple of approaches to defining Artin or algebraic stacks, which are locally isomorphic to affine schemes in a suitable sense. Recording
Justin Campbell Notes 1/11/23
Algebraicity of $$\text{Bun}_G$$ We will first review the definition of stacks and algebraic stacks, then introduce the classifying stack BG and show that it is an algebraic stack. We will also introduce the stack $$\text{Bun}_{G,X}$$ and focus on the case when $$G = GL_n$$ and X is a smooth projective curve over an algebraically closed field k. We will recall Weil’s automorphic interpretation of $$\text{Bun}_{n,X}(k)$$. Finally, we will sketch a proof for the algebraicity of $$\text{Bun}_{n,X}$$. Recording
Zhilin Luo Notes 1/18/23
$$\text{Bun}_G$$ and Higgs Bundles We will start with a discussion of moduli stacks that parametrize objects of a (sheaf of) categories and how the stack Bun_G fits into this general framework. Next, we define the tangent, cotangent, and also the inertia stack of a stack X. We interpret these stacks as "vector bundle stacks" on X. I'll give a description of the tangent/cotangent stack of a moduli stack and show that the cotangent stack of $$\text{Bun}_G$$ is isomorphic to the stack of Higgs bundles. Time permitting, I'll also discuss some more concrete results related to the geometry of Higgs bundles, in particular, the 'global nilpotent cone' that plays an important role in Geometric Langlands. Recording
Victor Ginzburg Notes 1/25/23
Harder-Narasimhan Stratification The stability condition for vector bundles was first introduced by D. Mumford using a mysterious notion called "slope". Though seemingly lacking in intuition, this notion is nonetheless very important in GIT and more general moduli problems, due to being the condition to obtain a "good" moduli space in "linearizable situations". Stemming from stability is the Harder-Narasimhan filtration, which provides a canonical way to describe any vector bundle on a curve as successive extensions of stable bundles. Such theory is generalized to any reductive group G by A. Ramananthan and later also by K. Behrend using so-called "complementary polyhedra". We will go through the main results of this theory and use GL2 over P^1 as an example to see the equivalence between Ramananthan-Behrend stability and slope stability. If time allows, we will also briefly state S. Schieder's "slope map" as a third description of stability which also makes the equivalence between the previous two stabilities in GLn case more transparent. Recording
Griffin Wang 2/1/23
Moduli of G-bundles as a quotient by the gauge group Firstly we will provide a complex geometric description of holomorphic G-bundles as principal bundles together with a connection. We will use this to interpret the moduli space of G-bundles, of a fixed topological type, with a quotient by the (infinite dimensional) group of gauge transformations. We will reinterpret this as a (infinite dimensional) symplectic quotient in an analogue of the Kempf--Ness Theorem. The Yang--Mills functional appears as the norm of the moment map. The Harder--Narasimhan filtration can be reinterpreted in terms of the equivariant Morse theory of this functional in an analogue of the HKKN stratification. Time permitting, we will describe how Atiyah--Bott used this to compute the cohomology of the moduli space of semistable G-bundles. Recording
Benedict Morrissey 2/8/23
Affine Grassmannian I will explain the construction of the affine Grassmannian and its basic geometric properties. Recording
Ngô Bảo Châu Zhu's Notes 2/15/23
Uniformization of Principal Bundles Given a simply-connected, complex semisimple algebraic group G, a complex smooth projective curve X, and a choice of basepoint on X, there exists a smooth surjective map—called the uniformization map—from the affine Grassmannian of G onto the moduli stack of principal G-bundles over X. The proof has two steps. The first uses a result of Beauville–Laszlo to replace the affine Grassmannian with a global analogue. The second uses a theorem of Drinfeld–Simpson to ensure that any principal G-bundle over the complement of the basepoint admits a Borel reduction, and hence, admits a trivialization over the complement. The force of these statements is that they hold not just on complex points, but on R-points for any C-algebra R. Recording
Minh-Tâm Trinh Notes 2/22/23
A Mosaic of Examples and Basic Properties of BunG This talk will explore in detail many of the aspects of the geometry of BunG that we have encountered so far this quarter. But we shall try to accomplish this through explicit examination of concrete examples. We hope to make this talk ostentatiously low-tech, utilizing what Shreeram Abhyankar calls “high-school algebra” (ie, explicit polynomial computations) when possible. (But don’t worry — cohomology will still make an appearance!) Of particular interest will be the case where the underlying curve is P^1, since here the coarse moduli space is essentially trivial and all the geometry is quintessentially “stacky.” Some of the things we hope to cover include Grothendieck’s theory of vector bundles over P^1; Grothendieck-Harder’s classification of general reductive G-torsors over P^1; and Ramanathan’s description of the deformations and automorphisms of reductive G-torsors over P^1. This will give us a fairly complete picture of what the (stacky) k-points of BunG(P^1) "look like” and how they are topologized. Along the way, we will hope to offer a number of digressions relating these basic examples to the general properties we have encountered/will encounter in future quarters; including, potentially, the general theory of the connected components of BunG, the Harder-Narasimhan filtration, Tamagawa Numbers, one-point and Adelic uniformization, the Tannakian formalism, and the Narasimhan-Seshadri theorem. If time permits, we will discuss some geometry in the case where the underlying curve is elliptic. Recording
Aaron Slipper 3/1/23

References: The winter quarter covered an amalgam of the relevant material from the following three references:

Background on Stacks and Algebraic Geometry can be found in the following books:
Background on Geometric Representation Theory and Affine Flag Varieties can be found in:
• PCMI: Geometry of Moduli Spaces and Representation Theory (see Xinwen Zhu's lecture)
• Schmitt: Affine Flag Manifolds and Principal Bundles (see Görtz’s lecture)