Let Gamma be the K-shadow space of a spherical building Delta. An embedding Vof 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 beembedded into what we call the Weyl module for G of highest weight lambda_K. Itis proved that this module is polarized and that the associated minimalpolarized embedding is precisely the irreducible G-module of highest weightlambda_K. In addition a number of general results on polarized embeddings ofshadow spaces are proved. The last few sections are devoted to the study ofspecific shadow spaces, notably minuscule weight geometries, polargrassmannians, and projective flag-grassmannians. The paper is in partexpository in nature so as to make this material accessible to a wide audience.
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