#### 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.

Done