Highest weight modules and polarized embeddings of shadow spaces

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(F) over the field F. For any non-empty set K ⊂ 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 ∗, which is the shadow geometry of type oppI (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(F) is untwisted. In that case, the point-line geometry Γ has an embedding eK into the Weyl module V (λK)F of highest weight λK = ∑ k∈K λk . We show that this embedding is polarized in the sense described above. We then prove that the minimal polarized embedding relative to eK exists and equals the unique irreducible G(F)-module L(λK) of highest weight λK . More precisely we show that the polar radical of eK (the intersection of all singular hyperplanes) coincides with Dedicated to Karin, Anna, and Sophie. This research was completed in part while visiting the University of Siena on a grant from the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni in the summers of 2007 and 2008. R.J. Blok ( ) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA e-mail: [email protected] 68 J Algebr Comb (2011) 34: 67–113 the radical of the contravariant bilinear form considered by Wong to obtain the irreducible (restricted) representations of G(F) in positive characteristic. This viewpoint allows us to “recognize” the irreducible G(F)-modules of highest weight λK geometrically as minimal polarized embeddings of the appropriate shadow space.