`Conormal Varieties on the Cominuscule Grassmannian'

Let G be a reductive group, LG its loop group, and P a co-minuscule parabolic subgroup of G.  Lakshmibai, Ravikumar, and Slofstra have constructed an embedding \phi of the cotangent bundle T^*G/P as an open subset of a Schubert variety of the loop group LG.  This raises the following question: When is the conormal variety C_w of a Schubert variety X(w) in G/P itself a Schubert variety? We classify the 'good' w for which \phi(C_w) is a Schubert varieties.  In particular, the conormal varieties of determinantal varieties are given by Schubert conditions.

This allows us various consequences: The identification of the ideal sheaf of C_w in T^*G/P for 'good' w; The conormal fibre at 0 of the rank k (usual, symmetric resp.) determinantal variety is the co-rank k (usual, symmetric resp.) determinantal variety; The conormal varieties and conormal fibres at identity for 'good' w are compatibly Frobenius split in T^*G/P. The Frobenius splitting of T^*G/P was first shown by Kumar, Lauritzen, and Thomsen.

Joint work with Lakshmibai.