$P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties

Goertz, Ulrich; He, Xuhua; Nie, Sian

doi:10.48550/arxiv.1211.3784

Cited by 6 publications

(9 citation statements)

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“…Moreover, any straight element can be regarded as a basic element in the extended affine Weyl group of some Levi subgroup of G. Thus, using the P -alcove introduced in [10] and its generalization in [11], we can show that any element in I ẇI is σ-conjugate to ẇ for some straight w. This is step (3), which completes the reduction.…”

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confidence: 77%

“…0.7. In the remaining sections (9)(10)(11), we study X w(b) for the case where b is basic and w is in the lowest two sided cell of W , and we prove a main conjecture of Görtz-Haines-Kottwitz-Reuman [10] and its generalization. This is achieved by combining the "dimension=degree" theorem with the partial conjugation method developed in [14] and the dimension formula for affine Deligne-Lusztig varieties in the affine Grassmannian in [9] and [37] We also give an upper bound for the dimension of X w(b) for arbitrary w and b.…”

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confidence: 99%

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Geometric and homological properties of affine Deligne-Lusztig varieties

2014

Ann. Math.

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This paper studies affine Deligne-Lusztig varieties Xw(b) in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of Xw(b) for a minimal length elementw in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in [18] to the affine case. We then provide a reduction method that relates the structure of Xw(b) for arbitrary elementsw in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of Görtz, Haines, Kottwitz and Reuman in [10].2000 Mathematics Subject Classification. 14L05, 20G25.By the definition of defect, def(b) = def(τ ) = n − ℓ(c). Therefore dim Xw(b) dw(b).By combining Theorem 12.1 and Corollary 10.4, we have that Corollary 12.2. If G is simple and δ = id. Letw ∈W ′ and b ∈ G(L) be a basic element with supp(η(w)) = S and κ(w) = κ(b). Then dim Xw(b) = dw(b).Remark. The split case was first conjectured in [10, Conjecture 1.1.3]. A weaker result for some split classical groups was proved in [8].

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confidence: 77%

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confidence: 99%

Geometric and homological properties of affine Deligne-Lusztig varieties

2014

Ann. Math.

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150

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“…Let w ∈ W such that w(v) is dominant and let J = {s ∈ S; s(w(v)) = w(v)}. Then w is a P v -alcove element if and only if w is a (J, w −1 , σ)-alcove element defined in [GHN,§3.3].…”

Section: Letmentioning

confidence: 99%

“…We also assume that wσs(v ′ ) = v ′ and s wσs(v ′′ ) = v ′′ . By [GHN,Lemma 4.4.4], either s(v ′ ) = v ′ or s(v ′′ ) = v ′′ . Thanks to [GHN,Lemma 4.4.2], w is either a v ′ -alcove element or a v ′′ -alcove element as desired.…”

Section: Proof Of Theorem12mentioning

confidence: 99%

“…The theorem was first established in [GHKR,Theorem 1.1.5] for split groups, where part (a) is referred to as a version of Hodge-Newton decomposition relating affine Deligne-Lusztig varieties associated to w for G and those for L. Its generalization for quasi-split groups was obtained in [GHN,Theorem 3.3.1]. In this paper, we adopt the (generalized) notion of P -alcove element introduced in loc.cit, but in a slight different form.…”

Section: Introductionmentioning

confidence: 99%

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Fundamental elements of an affine Weyl group

Nie

2013

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Fundamental elements are certain special elements of affine Weyl groups introduced by Görtz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne-Lusztig varieties. In this paper, we obtain characterizations of the fundamental elements and their natural generalizations. We also derive an inverse to a version of "Newton-Hodge decomposition" in affine flag varieties. As an application, we obtain a group-theoretic generalization of Oort's results on minimal p-divisible groups, and we show that, in certain good reduction reduction of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl-Oort stratum. This generalizes a result of Viehmann and Wedhorn.

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Class polynomials for some affine Hecke algebras

Yang

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Abstract. Class polynomials attached to affine Hecke algebras were first introduced by X. He in [He1]. They play an important role in the study of affine Deligne-Lusztig varieties. Motivated by [He2], we compute the class polynomials attached to an affine Hecke algebra of type (twisted) A 2 . Using these class polynomials we prove a conjecture of Görtz-Haines-Kottwitz-Reuman for the general linear group, unitary group and division algebra of semisimple rank 2. Furthermore, we discuss some interesting patterns on affine Deligne-Lusztig varieties.

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$P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties

Cited by 6 publications

References 2 publications

Geometric and homological properties of affine Deligne-Lusztig varieties

Geometric and homological properties of affine Deligne-Lusztig varieties

Fundamental elements of an affine Weyl group

Class polynomials for some affine Hecke algebras

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