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Dimensions of Affine Deligne–Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators
 
Elizabeth Milićević Haverford College, Haverford, Pennsylvania, USA
Petra Schwer Karlsruhe Institute of Technology, Karlsruhe, Germany
Anne Thomas University of Sydney, Sydney, Australia
Dimensions of Affine Deligne--Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators
Softcover ISBN:  978-1-4704-3676-6
Product Code:  MEMO/261/1260
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5403-6
Product Code:  MEMO/261/1260.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3676-6
eBook: ISBN:  978-1-4704-5403-6
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
Dimensions of Affine Deligne--Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators
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Dimensions of Affine Deligne–Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators
Elizabeth Milićević Haverford College, Haverford, Pennsylvania, USA
Petra Schwer Karlsruhe Institute of Technology, Karlsruhe, Germany
Anne Thomas University of Sydney, Sydney, Australia
Softcover ISBN:  978-1-4704-3676-6
Product Code:  MEMO/261/1260
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5403-6
Product Code:  MEMO/261/1260.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3676-6
eBook ISBN:  978-1-4704-5403-6
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 Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2612019; 101 pp
    MSC: 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 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.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2612019; 101 pp
MSC: 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 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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.