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

Abstract

We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in [3]. The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.Proof. By Proposition 2.4.1, any σ-conjugacy class of M J (L) is represented by some element inW J . Let x = ǫ λ w, x ′ = ǫ λ ′ w ′ ∈W J such that x and x ′ are in the same σ-conjugacy class of G(L). By Kottwitz [8] and [9],. In other words,

$$P$$ P -alcoves, parabolic subalgebras and cocenters of affine Hecke algebras

“…Let P be a semistandard parabolic subgroup of G. The notion of P-alcove elements (inW ) was introduced by Görtz, Haines, Kottwitz, and Reuman in [5] and generalized in [4] for non-split groups. Roughly speaking, w is a P-alcove element if the finite part of w lies in the finite Weyl group of P, and w sends the fundamental alcove to a certain region of the apartment.…”

Fundamental elements of an affine Weyl group

“…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 tow 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.…”

“…Thanks to [GHN,Lemma 4.4.2], w is either a v ′ -alcove element or a v ′′ -alcove element as desired.…”

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

Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties

“…If P J is an Iwahori subgroup and b is basic, a conjecture on the non-emptiness pattern (for split groups) is given by Görtz, Haines, Kottwitz, and Reuman in [3] in terms of P -alcoves in [3] and the generalization of this conjecture to any tamely ramified groups is proved in [4]. The non-emptiness pattern for basic b and other parahoric subgroups can then be deduced from Iwahori case easily.…”