$P$-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 Görtz et al. (2010). 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.

arXiv comment21 pages

References in corpus (3)

• A guide to the reduction modulo p of Shimura varieties
• Minimal length elements of extended affine Coxeter groups, II
• On parahoric subgroups

Cited by in corpus (7)

• Towards a theory of local Shimura varieties
• Minimal length elements of extended affine Coxeter groups, II
• Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties
• Note on affine Deligne-Lusztig varieties
• $P$-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras
• Fundamental elements of an affine Weyl group
• Geometric and homological properties of affine Deligne-Lusztig varieties