**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.

#### Reviews & Endorsements

...Pramod Achar provides a very nice and comprehensive introduction to the theory of perverse sheaves with an emphasis on their applications to representation theory.

...In the author's opinion, perverse sheaves are easy, in the sense that most arguments come down to a rather short list of tools, such as proper base change, smooth pullback, and open-closed distinguished triangles. The author tries to emphasize this perspective with computational exercises and with the Quick Reference. This is the main feature of this book. I believe this book is a valuable reference for algebraists who want to learn the theory of perverse sheaves. Readers can profit tremendously from attempting the hundreds of exercises scattered throughout the book.

-- Jun Hu, Beijing Institute of Technology

#### Table of Contents

# Table of Contents

## Perverse Sheaves and Applications to Representation Theory

- Preface ix10
- Chapter 1. Sheaf theory 114
- 1.1. Sheaves 114
- 1.2. Pullback, push-forward, and base change 720
- 1.3. Open and closed embeddings 1629
- 1.4. Tensor product and sheaf Hom 2336
- 1.5. The right adjoint to proper push-forward 3043
- 1.6. Relations among natural transformations 3649
- 1.7. Local systems 4154
- 1.8. Homotopy 5164
- 1.9. More base change theorems 5669
- 1.10. Additional notes and exercises 6275

- Chapter 2. Constructible sheaves on complex algebraic varieties 6780
- 2.1. Preliminaries from complex algebraic geometry 6780
- 2.2. Smooth pullback and smooth base change 7487
- 2.3. Stratifications and constructible sheaves 8093
- 2.4. Divisors with simple normal crossings 8699
- 2.5. Base change and the affine line 90103
- 2.6. Artin’s vanishing theorem 94107
- 2.7. Sheaf functors and constructibility 97110
- 2.8. Verdier duality 102115
- 2.9. More compatibilities of functors 105118
- 2.10. Localization with respect to a \Gm-action 111124
- 2.11. Homology and fundamental classes 117130
- 2.12. Additional notes and exercises 124137

- Chapter 3. Perverse sheaves 129142
- 3.1. The perverse 𝑡-structure 129142
- 3.2. Tensor product and sheaf Hom for perverse sheaves 135148
- 3.3. Intersection cohomology complexes 138151
- 3.4. The noetherian property for perverse sheaves 144157
- 3.5. Affine open subsets and affine morphisms 148161
- 3.6. Smooth pullback 155168
- 3.7. Smooth descent 160173
- 3.8. Semismall maps 168181
- 3.9. The decomposition theorem and the hard Lefschetz theorem 172185
- 3.10. Additional notes and exercises 177190

- Chapter 4. Nearby and vanishing cycles 181194
- Chapter 5. Mixed sheaves 217230
- 5.1. Étale and ℓ-adic sheaves 217230
- 5.2. Local systems and the étale fundamental group 225238
- 5.3. Passage to the algebraic closure 230243
- 5.4. Mixed ℓ-adic sheaves 235248
- 5.5. \scD-modules and the Riemann–Hilbert correspondence 242255
- 5.6. Mixed Hodge modules 249262
- 5.7. Further topics around purity 255268

- Chapter 6. Equivariant derived categories 261274
- 6.1. Preliminaries on algebraic groups, actions, and quotients 261274
- 6.2. Equivariant sheaves and perverse sheaves 268281
- 6.3. Twisted equivariance 277290
- 6.4. Equivariant derived categories 283296
- 6.5. Equivariant sheaf functors 290303
- 6.6. Averaging, invariants, and applications 296309
- 6.7. Equivariant cohomology 304317
- 6.8. The language of stacks 310323
- 6.9. Fourier–Laumon transform 315328
- 6.10. Additional exercises 323336

- Chapter 7. Kazhdan–Lusztig theory 325338
- Chapter 8. Springer theory 361374
- Chapter 9. The geometric Satake equivalence 387400
- 9.1. The affine flag variety and the affine Grassmannian 387400
- 9.2. Convolution 394407
- 9.3. Categorification of the affine and spherical Hecke algebras 398411
- 9.4. The Satake isomorphism 402415
- 9.5. Exactness and commutativity 404417
- 9.6. Weight functors 410423
- 9.7. Standard sheaves and Mirković–Vilonen cycles 416429
- 9.8. Hypercohomology as a fiber functor 424437
- 9.9. The geometric Satake equivalence 428441
- 9.10. Additional exercises 430443

- Chapter 10. Quiver representations and quantum groups 433446
- Appendix A. Category theory and homological algebra 467480
- A.1. Categories and functors 467480
- A.2. Monoidal categories 472485
- A.3. Additive and abelian categories 474487
- A.4. Triangulated categories 480493
- A.5. Chain complexes and the derived category 486499
- A.6. Derived functors 490503
- A.7. 𝑡-structures 497510
- A.8. Karoubian and Krull–Schmidt categories 504517
- A.9. Grothendieck groups 508521
- A.10. Duality for rings of finite global dimension 509522

- Appendix B. Calculations on \Cⁿ 513526
- Quick reference 541554
- Bibliography 545558
- Index of notation 557570
- Index 559572