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Dimensions of Affine Deligne–Lusztig Varieties: A New Approach via Labeled Folded Alcove Walks and Root Operators

About this Title

Elizabeth Milićević, Department of Mathematics & Statistics, Haverford College, 370 Lancaster Avenue, Haverford, PA, USA, Petra Schwer, Department of Mathematics, Karlsruhe Institute of Technology, EnglerstraÃe 2, 76133 Karlsruhe, Germany and Anne Thomas, School of Mathematics & Statistics, Carslaw Building F07, University of Sydney NSW 2006, Australia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 261, Number 1260
ISBNs: 978-1-4704-3676-6 (print); 978-1-4704-5403-6 (online)
DOI: https://doi.org/10.1090/memo/1260
Published electronically: November 7, 2019
MSC: Primary 20G25; Secondary 05E10, 20F55, 51E24.

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Preliminaries on Weyl groups, affine buildings, and related notions
  • 3. Labelings and orientations, galleries, and alcove walks
  • 4. Dimensions of galleries and root operators
  • 5. Affine Deligne–Lusztig varieties and folded galleries
  • 6. Explicit constructions of positively folded galleries
  • 7. The varieties $X_x(1)$ in the shrunken dominant Weyl chamber
  • 8. The varieties $X_x(1)$ and $X_x(b)$
  • 9. Conjugating to other Weyl chambers
  • 10. Diagram automorphisms
  • 11. Applications to affine Hecke algebras and affine reflection length

Abstract

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 (2000). Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that $b$ is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman (2010). Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that $b$ is basic, and reveals new patterns. Since we work only in the standard apartment of the building for $G(F)$, our results also hold in the $p$-adic context, where we formulate a definition of the dimension of a $p$-adic Deligne–Lusztig set. We present two immediate applications of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.

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