Dimensions of Affine Deligne–Lusztig Varieties: A New Approach via Labeled Folded Alcove Walks and Root Operators
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...
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| Main Authors: | , , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2019
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| Series: | Memoirs of the American Mathematical Society |
| ISBN: | 9781470436766, 1470436760 |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
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| Summary: | 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 questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. The authors 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 a conjecture of Gortz, Haines, Kottwitz, and Reuman.The authors' 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 they work only in the standard apartment of the building for $G(F)$, their results also hold in the $p$-adic context, where they formulate a definition of the dimension of a $p$-adic Deligne-Lusztig set. The authors present two immediate applications of their main results, to class polynomials of affine Hecke algebras and to affine reflection length. |
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| Bibliography: | Includes bibliographical reference (p. 99-101) September 2019, volume 261, number 1260 (fourth of 7 numbers) |
| ISBN: | 9781470436766 1470436760 |
| ISSN: | 0065-9266 1947-6221 |
| DOI: | 10.1090/memo/1260 |