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Geometry of Moduli Spaces and Representation Theory

Roman Bezrukavnikov, Alexander Braverman, and Zhiwei Yun, editors
American Mathematical Society
Publication Date: 
Number of Pages: 
IAS/PARK CITY Mathematics Series 24
[Reviewed by
Felipe Zaldivar
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The Fundamental Lemma of the Langlands program is a set of combinatorial formulas equating (up to a transfer factor) orbital integrals for a reductive algebraic group \(G\) over a non-archimedean local field on one side and stable orbital integrals for a related endoscopic group \(H\) on the other side.

Ngô’s proof of the general case of the Fundamental Lemma, building on previous work of Laumon and Waldspurger, opened a wide road leading to advances on a long list of arithmetic problems on the theory of automorphic forms and the arithmetic of Shimura varieties, in particular on the stabilization of the Arthur-Selberg trace formula.

A (very) rough summary of the proof of the Fundamental Lemma, in the unitary case, is this: first translate the local orbital integrals that appear in this lemma in terms of Springer fibers arising in the Hitchin fibration; then use the theory of perverse sheaves to finish the proof.

The wording in the previous paragraphs emphasizes the need of one or (more likely) several expositions on the background to this important advance in arithmetic geometry in order to (in principle) reach a wider audience than just the specialists.

The book under review has as its main goal to provide an introduction to geometric representation theory, a relatively young area of mathematics at the crossroads of algebraic geometry and representation theory and with strong ties to the topology and arithmetic of some of these algebraic varieties. As such, the expositions in the book are directly related to some parts of proof of the fundamental lemma.

Ngô’s proof requires a cohomological interpretation of orbit integrals, which then allows the use of the many deep methods of algebraic geometry to render the proof. The first four articles in the book summarize some of these geometric and topological tools.

The first article is an introduction to the use of perverse sheaves in complex algebraic geometry. As such, it touches on several topics, from derived categories, perverse complexes, Verdier duality, and the hard Lefschetz theorem, to Hodge structures on intersection cohomology groups. The lectures included in this first article follow a similar pattern: introduce first the main results or ideas and then follow with some carefully chosen applications or examples to illustrate the general situation; the proofs of the results are just outlines and the reader is invited to follow up them in the existing literature.

The second article, on affine Grassmannians, starts with their construction and basic properties. In particular, here we find the interpretation of these affine Grassmannians as certain moduli of curves. The generalizations by Beilinson and Drinfeld of these Grassmannians are also introduced. The last section of the article is devoted to the geometric Satake equivalence. This includes the construction of the Tannakian category of perverse sheaves on the affine Grassmannian, a construction that relies on properties of the Beilinson-Drinfeld Grassmannian.

The third article’s goals are twofold. The first part focuses on the theory of Springer fibers, geometric objects whose cohomology groups realize representations of Weyl groups. Since Springer fibers are singular, their cohomological study is done via the theory of perverse sheaves reviewed in the first article. In this part we also find a generalization of the classical Springer fibers, the affine ones, where the affine Grassmannian of the second article comes in. The second part of the article focuses on the close relationship between Springer fibers and orbital integrals. These integrals play a role for \(p\)-adic groups analogous to the role that conjugacy classes in finite groups play in the study of their representations. For example, if \(G\) is a unitary group, for \(\gamma\in G\) with centralizer \(G_{\gamma}\), given a smooth function with compact support \(f\) on \(G\), choosing a Haar measure on \(G\) and \(G_{\gamma}\), the orbital integral looks like \[O_{\gamma}(f)=\int_{G_{\gamma}\backslash G} f(x^{-1}\gamma x)dx.\]

A fundamental fact is that, in some cases, orbital integrals count points in affine Springer fibers. The theory of orbital integrals and stable orbital integrals over a non-archimedean local field is developed with some detail, culminating with its cohomological interpretation by Goresky-Kottwitz-MacPherson and Ngô. The last section reviews Ngô’s celebrated discovery that views Hitchin fibers, coming from certain integrable systems, as global affine Springer fibers, and Springer fibers as local versions of Hitchin fibers.

The fourth article harvests the theory developed on the previous three articles, outlining the rich interaction of the theory of global moduli spaces and some arithmetic problems in the Langlands program, where the thread is provided by a still-vague notion of perverse continuation. The article focuses on some specific global moduli spaces where the construction seems to follow that pattern. Instances of these moduli spaces are the ones related to some versions of the fundamental lemma.

Lastly, since representation theory is a mathematical formulation of the notion of symmetry, the last two articles are devoted to some aspects of enumerative algebraic geometry that are touched upon by geometric representation theory. In its modern reincarnation, enumerative geometry is more about moduli spaces than about the numbers related to some of the related computations (we still love these numbers, 3264 and all that, but now as part of a different narrative: Intersection theory). These last articles include some aspects of modern enumerative geometry, from K-theory computations to Nakajima’s varieties associated to quivers and the theory of instanton moduli spaces. Some of the prerequisites are covered in the first chapter but for other ones the reader is advised to review either before beginning or as needed.

As usual with the IAS/Park City publications, this is timely and well-structured set of lectures focused on a mathematical subject whose impact and interest is durable, as evidenced by re-printings of earlier volumes.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is