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Softcover ISBN:  9781470436766 
Product Code:  MEMO/261/1260 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470454036 
Product Code:  MEMO/261/1260.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Softcover ISBN:  9781470436766 
eBook ISBN:  9781470454036 
Product Code:  MEMO/261/1260.B 
List Price:  $162.00 $121.50 
MAA Member Price:  $145.80 $109.35 
AMS Member Price:  $97.20 $72.90 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 261; 2019; 101 ppMSC: Primary 20; Secondary 05; 51
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 Görtz, Haines, Kottwitz, and Reuman.
The authors' approach is constructive and typefree, 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.

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


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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 Görtz, Haines, Kottwitz, and Reuman.
The authors' approach is constructive and typefree, 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.

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