Basic loci of Coxeter type in Shimura varieties
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 \(Z_{J,w}\) for \(w\in EO(G,\mu)^J:=\text{Adm}^J (\mu)\cap {}^J \tilde W \subset \tilde W)\). The authors consider the triple \((G,\mu,J)\) for which the basic locus is a union of EO strata and each EO stratum is a union of classical Deligne-Lusztig varieties. Under a mild condition, they give a complete classification of such triples up to isomorphism (Theorem 5.1.2). They also obtain a explicit description for the basic locus as work of Vollaard-Wedhorn. Moreover, non-basic Newton strata are shown to be zero-dimensional. In the function field case, they prove the closure relation among the Bruhat-Tits strata. The paper is well written and the reader can use the authors' previous papers for necessary background.
scientific article; zbMATH DE number 6499657