Generic singularities of certain Schubert varieties

Let G be a connected semisimple algebraic group, a a Borel subgroup, T a ma ximal torus in B with Weyl group W, and Q a subgroup containing B. For w is an element of W, let X-wQ denote the Schubert variety (BwQ) over bar/Q. Fo r y is an element of W such that X-yQ subset of or equal to X-wQ, one knows that ByQ/Q admits a T-stable transversal in X-wQ, which we denote by N-yQ, N-wQ. We prove that, under certain hypotheses, N-yQ,N-wQ is isomorphic to t he orbit closure of a highest weight vector in a certain Weyl module. We al so obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal pa rabolic subgroup corresponding to a minuscule or cominuscule fundamental we ight, and X-yQ is an irreducible component of the boundary of X-wQ (that is , the complement of the open orbit of the stabiliser in G of X-wQ) As a con sequence, we describe the singularity of X-wQ along ByQ/Q and obtain that t he boundary of X-wQ equals its singular locus.