Perverse Sheaves and Applications to Representation Theory (Mathematical Surveys and Monographs, 258) by Pramod N. Achar (PDF)

Description

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 ℓ-adic sheaves, 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.

User’s Reviews

Editorial Reviews: Review …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 About the Author Pramod N. Achar: Louisiana State University, Baton Rouge, LA

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