Bruhat strata for Shimura varieties of PEL type

TL;DR

This paper investigates the structure of Bruhat stratification in PEL type Shimura varieties, establishing smoothness, dimension, and closure relations of the strata, thus enhancing understanding of their geometric properties.

Contribution

It provides a detailed analysis of Bruhat strata, proving their smoothness, calculating their dimensions, and describing their closure relations in PEL type Shimura varieties.

Findings

All Bruhat strata are smooth.

Dimensions of the strata are explicitly determined.

Closures of Bruhat strata are unions of other Bruhat strata.

Abstract

The Bruhat stratification for Shimura varieties of PEL type is studied. In the Siegel case this stratification is a scheme-theoretic variant of the stratification by the a-number. We show that all Bruhat strata are smooth and determine their dimensions. We also prove that the closure of a Bruhat stratum is a union of Bruhat strata and describe which Bruhat strata are contained in the closure of a given Bruhat stratum.