| Description |
1 online resource (xii, 289 pages) : illustrations |
| Series |
Trends in mathematics |
|
Trends in mathematics.
|
| Contents |
Cover13; -- Table of Contents -- Preface -- Affine Springer Fibers and Affine Deligne8211;Lusztig Varieties -- 1. Introduction -- 1.1. Notation -- 2. The affine Grassmannian and the affine flag manifold -- 2.1. Ind-schemes -- 2.2. The loop group -- 2.3. Lattices -- 2.4. The affine Grassmannian for GLn -- 2.5. The affine Grassmannian for an arbitrary linear algebraic group -- 2.6. Decompositions -- 2.7. Connected components -- 2.8. The Bruhat8211;Tits building -- 3. Affine Springer fibers -- 3.1. Springer fibers -- 3.2. Affine Springer fibers -- 3.3. General properties -- 3.4. Examples -- 3.5. Purity -- 3.6. (Co- )Homology of Ind-schemes -- 3.7. Equivariant cohomology -- 3.8. The fundamental lemma -- 4. Affine Deligne8211;Lusztig varieties -- 4.1. Deligne8211;Lusztig varieties -- 4.2. 963;-conjugacy classes -- 4.3. Affine Deligne8211;Lusztig varieties: the hyperspecial case -- 4.4. Affine Deligne8211;Lusztig varieties: the Iwahori case -- 4.5. The rank 2 case -- 4.6. Type A2 -- 4.7. Type C2 -- 4.8. Type G2 -- 4.9. Relationship to Shimura varieties -- 4.10. Local Shtuka -- 4.11. Cohomology of affine Deligne8211;Lusztig varieties -- References -- Quantization of Hitchins Integrable System and the Geometric Langlands Conjecture -- 1. D-modules on stacks -- 2. Chiral algebras -- 3. Geometry of the affine Grassmannian -- 4. Hecke eigenproperty -- 4.1. Convolution product -- 4.2. Hecke stacks and Hecke functors -- 4.3. Statement of Hecke eigenproperty -- 5. Opers -- 6. Constructing D-modules -- 7. Hitchin integrable system I: definition -- 8. Localization functor -- 9. Quantum integrable system h -- 10. Hitchin integrable system II: D-algebras -- 11. Quantization condition -- 12. Proof of the Hecke eigenproperty -- References -- Faltings Construction of the Moduli Space of Vector Bundles on a Smooth Projective Curve -- 1. Outline of the construction -- 2. Background and notation -- 2.1. Notation -- 2.2. The Picard torus and the Poincar180;e line bundle -- 2.3. Stability -- 2.4. Properties of vector bundles on algebraic curves -- 3. A nice over-parameterizing family -- 4. The generalized 920;-divisor -- 4.1. The line bundle OP(V)(R 183; 920;) -- 4.2. The invariant sections -- 4.3. The multiplicative structure -- 5. Raynauds vanishing result for rank two bundles -- 5.1. The case of genus zero and one -- 5.2. Preparations for the proof of 5.1 -- 5.3. A proof for genus two using the rigidity theorem -- 5.4. A proof based on Cliffords theorem -- 5.5. Generalizations and consequences -- 6. Semistable limits -- 6.1. Limits of vector bundles -- 6.2. Changing limits 8211; elementary transformations -- 6.3. Example: Limits are not uniquely determined -- 6.4. Semistable limits exist -- 6.5. Semistable limits are almost uniquely determined -- 7. Positivity -- 7.1. Notation and preliminaries -- 7.2. Positivity 8211; global sections of O(920;) -- 7.3. The case of deg(OC(920;C)) = 0 -- 8. The construction -- 8.1. Constructing the moduli space of vector bundles -- 8.2. Consequences from the construction -- 8.3. Generalization to the case of arbitrary rank and degree -- 9. Prospect to higher dimension -- References -- Lectures on the Moduli Stack of Vector Bundles on a Curve -- I |
| Summary |
Affine flag manifolds are infinite dimensional versions of familiar objects such as Gramann varieties. The book features lecture notes, survey articles, and research notes - based on workshops held in Berlin, Essen, and Madrid - explaining the significance of these and related objects (such as double affine Hecke algebras and affine Springer fibers) in representation theory (e.g., the theory of symmetric polynomials), arithmetic geometry (e.g., the fundamental lemma in the Langlands program), and algebraic geometry (e.g., affine flag manifolds as parameter spaces for principal bundles). Novel |
| Bibliography |
Includes bibliographical references |
| Notes |
Print version record |
| In |
Springer eBooks |
| Subject |
Flag manifolds.
|
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MATHEMATICS -- Geometry -- Algebraic.
|
|
Geometría
|
|
Flag manifolds
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| Form |
Electronic book
|
| Author |
Schmitt, Alexander H. W
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| LC no. |
2010928342 |
| ISBN |
9783034602884 |
|
303460288X |
|
