Perverse sheaves and applications to representation theory
Achar, Pramod N.
Perverse sheaves and applications to representation theory - Providence, Rhode Island : American Mathematical Society, 2021. - xii, 562 pages : illustrations ; 26 cm. - Mathematical surveys and monographs, volume 258 0076-5376 ; .
Includes bibliographical references (pages 545-556) and index.
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 compre hensive 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 equi variant derived category, and brief surveys of side topics including étale and -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 calcula tions 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.
9781470455972
Sheaf theory
Representations of groups
Linear algebraic groups
Several complex variables and analytic spaces--Complex singularities--Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
Group theory and generalizations -- Linear algebraic groups and related topics -- Representation theory for linear algebraic groups
Algebraic geometry -- Special varieties -- Grassmannians, Schubert varieties, flag manifolds
Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Coadjoint orbits; nilpotent varieties
Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Quantum groups (quantized enveloping algebras) and related deformations
512.54 / ACH-P
Perverse sheaves and applications to representation theory - Providence, Rhode Island : American Mathematical Society, 2021. - xii, 562 pages : illustrations ; 26 cm. - Mathematical surveys and monographs, volume 258 0076-5376 ; .
Includes bibliographical references (pages 545-556) and index.
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 compre hensive 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 equi variant derived category, and brief surveys of side topics including étale and -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 calcula tions 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.
9781470455972
Sheaf theory
Representations of groups
Linear algebraic groups
Several complex variables and analytic spaces--Complex singularities--Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
Group theory and generalizations -- Linear algebraic groups and related topics -- Representation theory for linear algebraic groups
Algebraic geometry -- Special varieties -- Grassmannians, Schubert varieties, flag manifolds
Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Coadjoint orbits; nilpotent varieties
Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Quantum groups (quantized enveloping algebras) and related deformations
512.54 / ACH-P
