**Mathematical Surveys and Monographs**

Volume: 258;
2021;
562 pp;
Softcover

MSC: Primary 32; 14; 20; 17;

**Print ISBN: 978-1-4704-5597-2
Product Code: SURV/258**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

**Electronic ISBN: 978-1-4704-6668-8
Product Code: SURV/258.E**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

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#### Supplemental Materials

# Perverse Sheaves and Applications to Representation Theory

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*Pramod N. Achar*

Since its inception around 1980, the theory of perverse sheaves has
been a vital tool of fundamental importance in geometric
representation theory. This book, which aims to make this theory
accessible to students and researchers, is divided into two parts. The
first six chapters give a comprehensive account of constructible and
perverse sheaves on complex algebraic varieties, including such topics
as Artin's vanishing theorem, smooth descent, and the nearby cycles
functor. This part of the book also has a chapter on the equivariant
derived category, and brief surveys of side topics including
étale and \(\ell\)-adic sheaves,
\(\mathcal{D}\)-modules, and algebraic stacks.

The last four chapters of the book show how to put this machinery
to work in the context of selected topics in geometric representation
theory: Kazhdan-Lusztig theory; Springer theory; the geometric Satake
equivalence; and canonical bases for quantum groups. Recent
developments such as the \(p\)-canonical basis are also discussed.

The book has more than 250 exercises, many of which focus on
explicit calculations with concrete examples. It also features a
4-page “Quick Reference” that summarizes the most commonly
used facts for computations, similar to a table of integrals in a
calculus textbook.

#### Readership

Graduate students and researchers interested in representation theory, derived categories, and perverse sheaves.

#### Table of Contents

# Table of Contents

## Perverse Sheaves and Applications to Representation Theory

- Preface 1010
- Chapter 1. Sheaf theory 1414
- 1.1. Sheaves 1414
- 1.2. Pullback, push-forward, and base change 2020
- 1.3. Open and closed embeddings 2929
- 1.4. Tensor product and sheaf Hom 3636
- 1.5. The right adjoint to proper push-forward 4343
- 1.6. Relations among natural transformations 4949
- 1.7. Local systems 5454
- 1.8. Homotopy 6464
- 1.9. More base change theorems 6969
- 1.10. Additional notes and exercises 7575

- Chapter 2. Constructible sheaves on complex algebraic varieties 8080
- 2.1. Preliminaries from complex algebraic geometry 8080
- 2.2. Smooth pullback and smooth base change 8787
- 2.3. Stratifications and constructible sheaves 9393
- 2.4. Divisors with simple normal crossings 9999
- 2.5. Base change and the affine line 103103
- 2.6. Artin’s vanishing theorem 107107
- 2.7. Sheaf functors and constructibility 110110
- 2.8. Verdier duality 115115
- 2.9. More compatibilities of functors 118118
- 2.10. Localization with respect to a \Gm-action 124124
- 2.11. Homology and fundamental classes 130130
- 2.12. Additional notes and exercises 137137

- Chapter 3. Perverse sheaves 142142
- 3.1. The perverse 𝑡-structure 142142
- 3.2. Tensor product and sheaf Hom for perverse sheaves 148148
- 3.3. Intersection cohomology complexes 151151
- 3.4. The noetherian property for perverse sheaves 157157
- 3.5. Affine open subsets and affine morphisms 161161
- 3.6. Smooth pullback 168168
- 3.7. Smooth descent 173173
- 3.8. Semismall maps 181181
- 3.9. The decomposition theorem and the hard Lefschetz theorem 185185
- 3.10. Additional notes and exercises 190190

- Chapter 4. Nearby and vanishing cycles 194194
- Chapter 5. Mixed sheaves 230230
- 5.1. Étale and ℓ-adic sheaves 230230
- 5.2. Local systems and the étale fundamental group 238238
- 5.3. Passage to the algebraic closure 243243
- 5.4. Mixed ℓ-adic sheaves 248248
- 5.5. \scD-modules and the Riemann–Hilbert correspondence 255255
- 5.6. Mixed Hodge modules 262262
- 5.7. Further topics around purity 268268

- Chapter 6. Equivariant derived categories 274274
- 6.1. Preliminaries on algebraic groups, actions, and quotients 274274
- 6.2. Equivariant sheaves and perverse sheaves 281281
- 6.3. Twisted equivariance 290290
- 6.4. Equivariant derived categories 296296
- 6.5. Equivariant sheaf functors 303303
- 6.6. Averaging, invariants, and applications 309309
- 6.7. Equivariant cohomology 317317
- 6.8. The language of stacks 323323
- 6.9. Fourier–Laumon transform 328328
- 6.10. Additional exercises 336336

- Chapter 7. Kazhdan–Lusztig theory 338338
- Chapter 8. Springer theory 374374
- Chapter 9. The geometric Satake equivalence 400400
- 9.1. The affine flag variety and the affine Grassmannian 400400
- 9.2. Convolution 407407
- 9.3. Categorification of the affine and spherical Hecke algebras 411411
- 9.4. The Satake isomorphism 415415
- 9.5. Exactness and commutativity 417417
- 9.6. Weight functors 423423
- 9.7. Standard sheaves and Mirković–Vilonen cycles 429429
- 9.8. Hypercohomology as a fiber functor 437437
- 9.9. The geometric Satake equivalence 441441
- 9.10. Additional exercises 443443

- Chapter 10. Quiver representations and quantum groups 446446
- Appendix A. Category theory and homological algebra 480480
- A.1. Categories and functors 480480
- A.2. Monoidal categories 485485
- A.3. Additive and abelian categories 487487
- A.4. Triangulated categories 493493
- A.5. Chain complexes and the derived category 499499
- A.6. Derived functors 503503
- A.7. 𝑡-structures 510510
- A.8. Karoubian and Krull–Schmidt categories 517517
- A.9. Grothendieck groups 521521
- A.10. Duality for rings of finite global dimension 522522

- Appendix B. Calculations on \Cⁿ 526526
- Quick reference 554554
- Bibliography 558558
- Index of notation 570570
- Index 572572