Some Results on Affine Deligne–lusztig Varieties

He, Xuhua

doi:10.1142/9789813272880_0100

Cited by 11 publications

(9 citation statements)

References 76 publications

(139 reference statements)

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“…In notes based on his ICCM 2013 talk, He announced a result which has some overlap with Theorem B and thus provided some more evidence for Conjecture 1.1 in the case of split b; compare Theorem 6.3 in [He13].…”

Section: Introductionmentioning

confidence: 98%

“…A series of several papers throughout the last decade established the converse for b basic. In [He13], He proved a nonemptiness pattern for X x (1) if the translation part of x is quasi-regular, and in [Bea12] the first author proved a nonemptiness statement under a length additivity hypothesis on the pair of finite Weyl group elements associated to x. In [GH10], Görtz and He then proved the nonemptiness conjecture in [GHKR10], although still under Reuman's original hypothesis that the alcoves lie in the shrunken Weyl chambers; i.e.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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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 Xx(b), which are indexed by elements b ∈ G(F ) and x ∈ W , were introduced by Rapoport [Rap00]. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. We 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 Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman [GHKR10]. Our 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 we work only in the standard apartment of the building for G(F ), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne-Lusztig set. We present two immediate applications of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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

2019

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“…By Theorem A of [GHN], N x, [b0,x] = ∅ if and only if there is a pair (J, w) such that xa is a (J, w, δ)-alcove and κ MJ (w −1 xδ(w)) / ∈ κ MJ ([b 0,x ] ∩ M J (L)). In [He1], Corollary 12.2 (see also [He2], Theorem 5.3 and the corresponding footnote for the generalization to non-split G), He proves that if x is in the shrunken Weyl chambers and N x, [b0,x]…”

Section: Comparing Sets Of σ-Conjugacy Classesmentioning

confidence: 99%

Minimal Newton Strata in Iwahori Double Cosets

Viehmann

2019

International Mathematics Research Notices

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The set of Newton strata in a given Iwahori double coset in the loop group of a reductive group G is indexed by a finite subset of the set B(G) of Frobenius-conjugacy classes. For unramified G, we show that it has a unique minimal element and determine this element. Under a regularity assumption we also compute the dimension of the corresponding Newton stratum. We derive corresponding results for affine Deligne-Lusztig varieties.

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“…In general, there is a large body of literature on emptiness and dimensions of affine Deligne Lusztig varieties, as many basic questions still remain open. For example, see [15,16,17,24,33] for some closely related and more general results discovered using different techniques. Proof.…”

Section: Connections With Demazure Modules and Affine Deligne-lusztigmentioning

confidence: 99%

Convex polytopes for the central degeneration of the affine Grassmannian

Zhou

2019

Advances in Mathematics

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We study the algebraic geometry and combinatorics of the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) in type A. More specifically, we elucidate the central degeneration of semi-infinite orbits and explain its relations with Levi restriction. Also, we discuss the central degeneration of Mirković-Vilonen cycles in the affine Grassmannian, and the corresponding transformations of Mirković-Vilonen polytopes. In addition, we shed some light on the geometry of Iwahori MV cycles in the affine Grassmannian and generalized MV cycles in the affine flag variety, which are closely related to Demazure modules and affine Deligne-Lusztig varieties respectively.

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Some Results on Affine Deligne–lusztig Varieties

Cited by 11 publications

References 76 publications

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

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

Minimal Newton Strata in Iwahori Double Cosets

Convex polytopes for the central degeneration of the affine Grassmannian

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