Highest weight modules and polarized embeddings of shadow spaces

Blok, Rieuwert J.

Journal of Algebraic Combinatorics, Tome 34 (2011) no. 1, pp. 67-113.

Résumé

Summary: The present paper was inspired by the work on polarized embeddings by Cardinali et al. (J. Algebr. Comb. $25(1)$:7-23, 2007) although some of our results in it date back to 1999. They study polarized embeddings of certain dual polar spaces, and identify the minimal polarized embeddings for several such geometries. We extend some of their results to arbitrary shadow spaces of spherical buildings, and make a connection to work of Burgoyne, Wong, Verma, and Humphreys on highest weight representations for Chevalley groups. Let $\Delta $ be a spherical Moufang building with diagram M over some index set $I$, whose strongly transitive automorphism group is a Chevalley group $G(\mathbb F) G(\mathbb{F})$ over the field $\mathbb F$ mathbbF. For any non-empty set $K\subset I$ let $\Gamma $ be the $K$-shadow space of $\Delta $. Extending the notion in to this situation, we say that an embedding of $\Gamma $ is polarized if it induces all singular hyperplanes. Here a singular hyperplane is the collection of points of $\Gamma $ not opposite to a point of the dual geometry $\Gamma ^{\ast }$, which is the shadow geometry of type opp $_{ I }( K)$ opposite to $K$. We prove a number of results on polarized embeddings, among others the existence of (relatively) minimal polarized embeddings.

Détail

Mots-clés : keywords building, shadow space, Grassmannian, polarized embedding, Chevalley group, highest weight module, representation theory