Dimensions of affine Deligne-Lusztig varieties : a new approach via labeled folded alcove walks and root operators /

Dimensions of affine Deligne-Lusztig varieties : a new approach via labeled folded alcove walks and root operators / Elizabeth Milićević, Petra Schwer, Anne Thomas.

Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne-Lusztig varieties X_x(b), which are indexed by elements b \in G(F) and x \in W, were introduced by Rapoport. Basic...

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Bibliographic Details
Online Access: Full Text (via ProQuest)
Main Authors: Milićević, Elizabeth (Author), Schwer, Petra (Author), Thomas, Anne (Mathematician) (Author)
Format: eBook
Language:English
Published: Providence : American Mathematical Society, 2019.
Series:Memoirs of the American Mathematical Society ; no. 1260.
Subjects:
Representations of Lie groups.
Table of Contents:
  • Cover
  • Title page
  • Chapter 1. Introduction
  • 1.1. History of the problem
  • 1.2. Key ideas in this approach
  • 1.3. Summary of main results
  • 1.4. Outline of proof and organization of the paper
  • 1.5. Applications
  • 1.6. Acknowledgements
  • Chapter 2. Preliminaries on Weyl groups, affine buildings, and related notions
  • 2.1. Weyl groups and root systems
  • 2.2. Hyperplanes, alcoves, and Weyl chambers
  • Chapter 3. Labelings and orientations, galleries, and alcove walks
  • 3.1. Labelings and orientations of hyperplanes
  • 3.2. Combinatorial galleries.
  • 3.3. Labeled folded alcove walks
  • Chapter 4. Dimensions of galleries and root operators
  • 4.1. The dimension of a folded gallery
  • 4.2. Root operators
  • 4.3. Counting folds and crossings
  • 4.4. Independence of minimal gallery
  • Chapter 5. Affine Deligne-Lusztig varieties and folded galleries
  • 5.1. Dimensions of affine Deligne-Lusztig varieties
  • 5.2. Connection to folded galleries
  • 5.3. Dimension of a -adic Deligne-Lusztig set
  • 5.4. Deligne-Lusztig galleries
  • Chapter 6. Explicit constructions of positively folded galleries
  • 6.1. Motivation: the shrunken Weyl chambers.
  • 6.2. Constructing one positively folded gallery
  • 6.3. An infinite family of positively folded galleries
  • Chapter 7. The varieties ₃ 1) in the shrunken dominant Weyl chamber
  • 7.1. The ₀ position
  • 7.2. Arbitrary spherical directions
  • 7.3. Dependence upon Theorem 7.5 and comparison with Reuman's criterion
  • Chapter 8. The varieties ₃ 1) and ₃)
  • 8.1. Forward-shifting galleries
  • 8.2. Nonemptiness and dimension for arbitrary alcoves
  • 8.3. The ₀ position in the shrunken dominant Weyl chamber
  • 8.4. Dimension in the shrunken dominant Weyl chamber.
  • 8.5. Obstructions to further constructive proofs
  • 8.6. Galleries, root operators, crystals, and MV-cycles
  • Chapter 9. Conjugating to other Weyl chambers
  • 9.1. Conjugating galleries
  • 9.2. Conjugating by simple reflections
  • 9.3. Conjugate affine Deligne-Lusztig varieties
  • Chapter 10. Diagram automorphisms
  • Chapter 11. Applications to affine Hecke algebras and affine reflection length
  • 11.1. Class polynomials of the affine Hecke algebra
  • 11.2. Reflection length in affine Weyl groups
  • Bibliography
  • Back Cover.

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