Embeddings of the complex ball into Siegel space

TL;DR

This paper investigates a map between unitary groups related to Shimura varieties, demonstrating its extension to the ordinary locus and extending Satake's results to positive characteristics.

Contribution

It introduces a new map between unitary groups and shows its extension to integral models and positive characteristic settings.

Findings

The map extends to the ordinary locus of the integral model.

It generalizes Satake's results on endomorphism rings to positive characteristics.

Provides insights into the structure of Shimura varieties and their models.

Abstract

We study properties of a map from a certain unitary group in $n$ variables to a related unitary group in $\binom{n}{k}$ variables. We explain how it gives rise to a map between canonical models of Shimura varieties and we prove that it extends to the ordinary locus of the integral model. Finally we extend results of Satake on the endomorphism ring of a generic image point to positive characteristics.