Alladi Ramakrishnan Hall

Conormal Varieties on the Cominuscule Grassmannian

Rahul Singh

Northeastern University

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.