Basic loci of Coxeter type in Shimura varieties
type: Item
instance of: scholarly article
title: Basic loci of Coxeter type in Shimura varieties
zbMATH Open document ID: 1350.14026
author: Ulrich Görtz
author: Xuhua He
published in: Cambridge Journal of Mathematics
publication date: 2015-10-28T00:00:00Z
review text: This paper is concerned with an explicit description of the basic locus in the reduction modulo \(p\) of Shimura varieties. Recent important works [\textit{I. Vollaard} and \textit{T. Wedhorn}, Invent. Math. 184, No. 3, 591--627 (2011; Zbl 1227.14027); \textit{M. Rapoport} et al., Math. Z. 276, No. 3--4, 1165--1188 (2014; Zbl 1312.14072); \textit{B. Howard} and \textit{G. Pappas}, Algebra Number Theory 8, No. 7, 1659--1699 (2014; Zbl 1315.11049)] show that for some unitary Shimura varieties, the basic locus can be described in terms of the Bruhat-Tits building, and there is a stratification where the strata are isomorphic to a union of classical Deligne-Lusztig varieties. Moreover the closure relation can be also described in a simple manner by group theory. This paper under review gives a rather complete list of group-theoretic data for which such an explicit description can be obtained. Thus, it gives a great contribution to explicit descriptions of basic loci of Shimura varieties. Let \(F\) be a non-Archimedean local field. Let \(L\) be the completion of the maximal unramified extension of \(F\), and let \(\sigma\) denote the Frobenius of the extension \(L/F\). Fix a datum \((G,\mu)\), where \(G\) is a connected quasi-simple semi-simple algebraic group over \(F\) which splits over a finite tamely ramified extension, and \(\mu\) is a minuscule cocharacter. For each standard rational maximal parahoric subgroup \(P\) of \(G(L)\) and \(b\in G(L)\), define the affine Deligne-Lusztig variety (ADLV) \[ X(\mu,b)_P:=\{g\in G(L)/P; g^{-1} b \sigma(b)\in \bigcup_{w\in \text{Adm}(\mu)} P w P \}. \] Let \(B(G,\mu)\subset G(L)\) be the union of \(\sigma\)-conjugacy classes of elements \(b\) for which \(X(\mu,b)_P\) is non-empty. One has a fiber space \(Z_J\) over \(B(G,\mu)\) with fibers \(X(\mu,b)_P\), where \(J\) is the type of \(P\). Inside \(Z_J\) there is the basic locus (the preimage over the basic \(\sigma\)-conjugacy class), and EO strata (subsets of form \(Z_{J,w}\) f…
reviewed by: Chia-Fu Yu
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 6499657
zbMATH Keywords: Shimura varieties
zbMATH Keywords: basic loci
zbMATH Keywords: Deligne-Lusztig varieties
MaRDI profile type: Publication
DOI: 10.4310/CJM.2015.V3.N3.A2
description: scientific article; zbMATH DE number 6499657