Generic singularities of certain Schubert varieties

Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For , let denote the Schubert variety . For such that , one knows that ByQ / Q admits a T-stable transversal in , which we denote by . We prove that, under certain hypotheses, is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and is an irreducible component of the boundary of (that is, the complement of the open orbit of the stabiliser in G of ). As a consequence, we describe the singularity of along ByQ / Q and obtain that the boundary of equals its singular locus. Received October 9, 1997; in final form February 19, 1998