Highest weight modules and polarized embeddings of shadow spaces

TL;DR

This paper explores the structure of shadow spaces in spherical buildings, demonstrating that certain embeddings are polarized and identifying the minimal polarized embedding with irreducible modules, with applications to specific geometries.

Contribution

It proves that Weyl modules of highest weight are polarized embeddings of shadow spaces and characterizes the minimal polarized embedding as an irreducible G-module, including general results and specific cases.

Findings

Weyl modules are polarized embeddings of shadow spaces.

Minimal polarized embedding is the irreducible G-module of highest weight.

Applications to minuscule weight geometries and polar grassmannians.

Abstract

Let Gamma be the K-shadow space of a spherical building Delta. An embedding V of Gamma is called polarized if it affords all "singular" hyperplanes of Gamma. Suppose that Delta is associated to a Chevalley group G. Then Gamma can be embedded into what we call the Weyl module for G of highest weight lambda_K. It is proved that this module is polarized and that the associated minimal polarized embedding is precisely the irreducible G-module of highest weight lambda_K. In addition a number of general results on polarized embeddings of shadow spaces are proved. The last few sections are devoted to the study of specific shadow spaces, notably minuscule weight geometries, polar grassmannians, and projective flag-grassmannians. The paper is in part expository in nature so as to make this material accessible to a wide audience.