**IAS/Park City Mathematics Series**

Volume: 24;
2017;
436 pp;
Hardcover

MSC: Primary 14; 22;

Print ISBN: 978-1-4704-3574-5

Product Code: PCMS/24

List Price: $104.00

AMS Member Price: $83.20

MAA member Price: $93.60

**Electronic ISBN: 978-1-4704-4234-7
Product Code: PCMS/24.E**

List Price: $104.00

AMS Member Price: $83.20

MAA member Price: $93.60

# Geometry of Moduli Spaces and Representation Theory

Share this page *Edited by *
*Roman Bezrukavnikov; Alexander Braverman; Zhiwei Yun*

A co-publication of the AMS and IAS/Park City Mathematics Institute

This book is based on lectures given at the
Graduate Summer School of the 2015 Park City Mathematics Institute
program “Geometry of moduli spaces and representation
theory”, and is devoted to several interrelated topics in
algebraic geometry, topology of algebraic varieties, and
representation theory.

Geometric representation theory is a young but fast developing
research area at the intersection of these subjects. An early profound
achievement was the famous conjecture by Kazhdan–Lusztig about
characters of highest weight modules over a complex semi-simple Lie
algebra, and its subsequent proof by Beilinson-Bernstein and
Brylinski-Kashiwara. Two remarkable features of this proof have
inspired much of subsequent development: intricate algebraic data
turned out to be encoded in topological invariants of singular
geometric spaces, while proving this fact required deep general
theorems from algebraic geometry.

Another focus of the program was enumerative algebraic
geometry. Recent progress showed the role of Lie theoretic structures
in problems such as calculation of quantum cohomology, K-theory,
etc. Although the motivation and technical background of these
constructions is quite different from that of geometric Langlands
duality, both theories deal with topological invariants of moduli
spaces of maps from a target of complex dimension one. Thus they are
at least heuristically related, while several recent works indicate
possible strong technical connections.

The main goal of this collection of notes is to provide young
researchers and experts alike with an introduction to these areas of
active research and promote interaction between the two related
directions.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.

# Table of Contents

## Geometry of Moduli Spaces and Representation Theory

Table of Contents pages: 1 2

- Cover Cover11
- Title page iii4
- Preface vii8
- Introduction ix10
- Perverse sheaves and the topology of algebraic varieties 112
- Introduction 112
- Lecture 1: The decomposition theorem 314
- Lecture 2: The category of perverse sheaves P(Y) 1728
- Lecture 3: Semi-small maps 2839
- Lecture 4: Symmetries: VD, RHL, IC splits off 3748
- Verdier duality and the decomposition theorem 3748
- Verdier duality and the decomposition theorem with large fibers 3849
- The relative hard Lefschetz theorem 3950
- Application of RHL: Stanley’s theorem 4051
- Intersection cohomology of the target as a direct summand 4152
- Pure Hodge structure on intersection cohomology groups 4354
- Exercises for Lecture 4 4455

- Lecture 5: The perverse filtration 4657

- An introduction to affine Grassmannians and the geometric Satake equivalence 5970
- Introduction 5970
- Lecture I: Affine Grassmannians and their first properties 6677
- Lecture II: More on the geometry of affine Grassmannians 8192
- Lecture III: Beilinson-Drinfeld Grassmannians and factorisation structures 96107
- Lecture IV: Applications to the moduli of G-bundles 108119
- The geometric Satake equivalence 116127
- Complements on sheaf theory 145156

- Lectures on Springer theories and orbital integrals 155166
- Introduction 156167
- Lecture I: Springer fibers 157168
- Lecture II: Affine Springer fibers 170181
- Loop group, parahoric subgroups and the affine flag variety 170181
- Affine Springer fibers 174185
- Symmetry on affine Springer fibers 176187
- Further examples of affine Springer fibers 178189
- Geometric Properties of affine Springer fibers 181192
- Affine Springer representations 186197
- Comments and generalizations 188199
- Exercises 189200

- Lecture III: Orbital integrals 190201
- Lecture IV: Hitchin fibration 203214

- Perverse sheaves and fundamental lemmas 217228
- Lectures on K-theoretic computations in enumerative geometry 251262
- Aims & Scope 252263
- Before we begin 263274
- The Hilbert scheme of points of 3-folds 273284
- Nakajima varieties 298309
- Symmetric powers 304315
- More on quasimaps 314325
- Nuts and bolts 335346
- Difference equations 346357
- Stable envelopes and quantum groups 355366
- Quantum Knizhnik-Zamolodchikov equations 367378

Table of Contents pages: 1 2