Highest weight modules and polarized embeddings of shadow spaces

• Authors: Rieuwert J. Blok
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2011

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 Δ be a spherical Moufang building with diagram M over some index set I, whose strongly transitive automorphism group is a Chevalley group $G(\mathbb{F})$ over the field $\mathbb{F}$ . For any non-empty set K驴I let Γ be the K-shadow space of Δ. Extending the notion in to this situation, we say that an embedding of Γ is polarized if it induces all singular hyperplanes. Here a singular hyperplane is the collection of points of Γ not opposite to a point of the dual geometry Γ 驴, 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.We assume that $G(\mathbb{F})$ is untwisted. In that case, the point-line geometry Γ has an embedding e K into the Weyl module $V(\lambda_{K})_{\mathbb{F}}^{0}$ of highest weight 驴 K =驴 k驴K 驴 k . We show that this embedding is polarized in the sense described above. We then prove that the minimal polarized embedding relative to e K exists and equals the unique irreducible $G(\mathbb{F})$ -module L(驴 K ) of highest weight 驴 K . More precisely we show that the polar radical of e K (the intersection of all singular hyperplanes) coincides with the radical of the contravariant bilinear form considered by Wong to obtain the irreducible (restricted) representations of $G(\mathbb{F})$ in positive characteristic.This viewpoint allows us to "recognize" the irreducible $G(\mathbb{F})$ -modules of highest weight 驴 K geometrically as minimal polarized embeddings of the appropriate shadow space.