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

Released on by Elizabeth Milićević
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  • Dimensions of Affine Deligne–Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators Book Detail

  • Author : Elizabeth Milićević
  • Release Date : 2019-12-02
  • Publisher : American Mathematical Soc.
  • Genre : Education
  • Pages : 101
  • ISBN 13 : 1470436760
  • File Size : 63,63 MB
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Dimensions of Affine Deligne–Lusztig Varieties: A New Approach Via Labeled Folded Alcove Walks and Root Operators by Elizabeth Milićević PDF Summary

Book Description: 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 Xx(b), which are indexed by elements b∈G(F) and x∈W, were introduced by Rapoport. Basic questions about the varieties Xx(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.

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Hodge Ideals

Hodge Ideals

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The authors use methods from birational geometry to study the Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of