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 w ∈ W , let XwQ denote the Schubert variety BwQ/Q. For y ∈ W such that XyQ ⊆ XwQ, one knows that ByQ/Q admits a T -stable transversal in XwQ, which we denote by NyQ,wQ. We prove that, under certain hypotheses, NyQ,wQ 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 XyQ is an irreducible component of the boundary of XwQ (that is, the complement of the open orbit of the stabiliser in G of XwQ). As a consequence, we describe the singularity of XwQ along ByQ/Q and obtain that the boundary of XwQ equals its singular locus.