هندسهٔ فضاهای مدولی و نظریهٔ نمایش‌ها (مجموعهٔ ریاضیات پارک سیتی)

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. Cover Title page Preface Introduction Perverse sheaves and the topology of algebraic varieties Introduction Lecture 1: The decomposition theorem Deligne’s theorem in cohomology The global invariant cycle theorem Cohomological decomposition theorem The local invariant cycle theorem Deligne’s theorem The decomposition theorem Exercises for Lecture 1 Lecture 2: The category of perverse sheaves P(Y) Three "Whys", and a brief history of perverse sheaves The constructible derived category D(Y) Definition of perverse sheaves Artin vanishing and Lefschetz hyperplane theorems The perverse t-structure Intersection complexes Exercises for Lecture 2 Lecture 3: Semi-small maps Semi-small maps The decomposition theorem for semi-small maps Hilbert schemes of points on surfaces and Heisenberg algebras The endomorphism algebra End (f*QX) Geometric realization of the representations of the Weyl group Exercises for Lecture 3 Lecture 4: Symmetries: VD, RHL, IC splits off Verdier duality and the decomposition theorem Verdier duality and the decomposition theorem with large fibers The relative hard Lefschetz theorem Application of RHL: Stanley’s theorem Intersection cohomology of the target as a direct summand Pure Hodge structure on intersection cohomology groups Exercises for Lecture 4 Lecture 5: The perverse filtration The perverse spectral sequence and the perverse filtration Geometric description of the perverse filtration Hodge-theoretic consequences Character variety and Higgs moduli: P=W Let us conclude with a motivic question Exercises for Lecture 5 An introduction to affine Grassmannians and the geometric Satake equivalence Introduction Some motivations Scope and contents Conventions and notations Acknowledgement Lecture I: Affine Grassmannians and their first properties The affine Grassmannian of GL(n) Affine Grassmannians of general groups Groups attached to the punctured disc The Beauville-Laszlo theorem The determinant line bundle Affine Grassmannians over the complex numbers Affine Grassmannians for p-adic groups Lecture II: More on the geometry of affine Grassmannians Schubert varieties Digression: Some sub-ind-schemes in Gr Opposite Schubert “varieties” and transversal slices The Picard group Central extensions and affine Kac-Moody algebras Lecture III: Beilinson-Drinfeld Grassmannians and factorisation structures Beilinson-Drinfeld Grassmannians Factorisation property The Ran space Rigidified line bundles on GR_Ran Lecture IV: Applications to the moduli of G-bundles One point uniformization Line bundles and conformal blocks Adèlic uniformization The geometric Satake equivalence The Satake category Sat_G Sat_G as a Tannakian category Langlands dual group Fusion product Bootstraps From the geometric Satake to the classical Satake Complements on sheaf theory Equivariant category of perverse sheaves Universally local acyclicity Lectures on Springer theories and orbital integrals Introduction Topics of these notes What we assume from the reader Lecture I: Springer fibers The setup Springer fibers Examples of Springer fibers Geometric Properties of Springer fibers The Springer correspondence Comments and generalizations Exercises Lecture II: Affine Springer fibers Loop group, parahoric subgroups and the affine flag variety Affine Springer fibers Symmetry on affine Springer fibers Further examples of affine Springer fibers Geometric Properties of affine Springer fibers Affine Springer representations Comments and generalizations Exercises Lecture III: Orbital integrals Integration on a p-adic group Orbital integrals Relation with affine Springer fibers Stable orbital integrals Examples in SL(2) Remarks on the Fundamental Lemma Exercises Lecture IV: Hitchin fibration The Hitchin moduli stack Hitchin fibration Hitchin fibers Relation with affine Springer fibers A global version of the Springer action Exercises Perverse sheaves and fundamental lemmas Grothendieck’s dictionary of sheaves and functions The dictionary Character sums The Swan conductors and Euler-Poincaré characteristics The Hasse-Davenport identity Purity and perversity Deligne’s theorem on weights Perverse sheaves and the decomposition theorem Perverse continuation method The Fourier-Deligne transform Small maps Support theorem for abelian fibrations Double unipotent action Invariant functions The Kloosterman orbital integrals Cohomological interpretation of the Kloosterman integral Arc spaces and families of Kloosterman integrals Global family Coordinate calculation in a special case Action of GL(n-1) on gl(n) by conjugation Invariant functions Untwisted integrals Twisted integrals Global model Adjoint action Invariant theory Stable orbital integrals Waldspurger’s nonstandard fundamental lemma Global model The Hitchin fibration The Langlands-Shelstad fundamental lemma Lectures on K-theoretic computations in enumerative geometry Aims & Scope K-theoretic enumerative geometry Quantum K-theory of Nakajima varieties K-theoretic Donaldson-Thomas theory Old vs. new Acknowledgements Before we begin Symmetric and exterior algebras KG(X) and K o G(X) Localization Rigidity The Hilbert scheme of points of 3-folds Our very first DT moduli space cO\vir and \tO\vir Nekrasov’s formula Tangent bundle and localization Proof of Nekrasov’s formula Nakajima varieties Algebraic symplectic reduction Nakajima quiver varieties Quasimaps to Nakajima varieties Symmetric powers PT theory for smooth curves Proof of Theorem 5.1.16 Hilbert schemes of surfaces and threefolds More on quasimaps Balanced classes and square roots Relative quasimaps in an example Stable reduction for relative quasimaps Moduli of relative quasimaps The degeneration formula and the gluing operator Nuts and bolts The Tube The Vertex The index limit Cap and capping Large framing vanishing Difference equations Shifts of Kähler variables Shifts of equivariant variables Difference equations for vertices Stable envelopes and quantum groups K-theoretic stable envelopes Triangle lemma and braid relations Actions of quantum groups Quantum Knizhnik-Zamolodchikov equations Commuting difference operators from R-matrices Minuscule shifts and qKZ The gluing operator in the stable basis Proof of Theorem 10.2.11 Lectures on perverse sheaves on instanton moduli spaces Introduction Uhlenbeck partial compactification –in brief Heisenberg algebra action on the Gieseker partial compactification Stable envelopes Sheaf theoretic analysis of stable envelopes R-matrix for Gieseker partial compactification Perverse sheaves on instanton moduli spaces W-algebra representation on equivariant intersection cohomology groups Concluding remarks Back Cover