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

Format:

Book

Author/Creator:

Milicevic, Elizabeth, author.

Schwer, Petra, author.

Thomas, Anne, author.

Series:

Memoirs of the American Mathematical Society ; vol. 261, no. 1260.

Memoirs of the American Mathematical Society, 0065-9266 ; September 2019, volume 261, number 1260

Language:

English

Subjects (All):

Automorphic forms.

Geometry, Algebraic.

Physical Description:

1 online resource (114 pages) : color illustrations.

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2019]

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 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.

Contents:

Preliminaries on Weyl groups, affine buildings, and related notions

Labelings and orientations, galleries, and alcove walks

Dimensions of galleries and root operators

Affine Deligne-Lusztig varieties and folded galleries

Explicit constructions of positively folded galleries

The varieties Xx(1) in the shrunken dominant Weyl chamber

The varieties Xx(1) and Xx(b)

Conjugating to other Weyl chambers

Diagram automorphisms

Applications to affine Hecke algebras and affine reflection length.

Notes:

Description based on online resource; title from PDF title page (ebrary, viewed January 6, 2020).

Includes bibliographical references.

ISBN:

1-4704-5403-3

9781470454043

OCLC:

1140791032