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

Abstract

In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(\mu ,b{)}_J$ for any tamely ramified group $G$ and its parahoric subgroup $P_J$. We show that $X(\mu ,b{)}_J\ne \varnothing$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $\s$-conjugacy classes of a twisted loop group.

19 pages

References in corpus (3)

• A guide to the reduction modulo p of Shimura varieties
• Basic loci in Shimura varieties of Coxeter type
• $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties

Cited by in corpus (5)

• Langlands-Rapoport Conjecture Over Function Fields
• Hecke algebras and $p$-adic groups
• Vertexwise criteria for admissibility of alcoves
• Cocenters of $p$-adic groups, I: Newton decomposition
• Stratifications in the reduction of Shimura varieties