A theorem of Lakshmibai states that the cotangent bundle of a
Grassmannian variety has a compactification which is a smooth Schubert variety
in an affine partial flag variety. I will explain how to extend this theorem to
any cominuscule Grassmannian, using a fairly natural construction that involves
looking primarily at the Dynkin diagram, the root system, and the Weyl group of
the corresponding simple algebraic group.
In particular, this construction provides an sample application of
Billey-Postnikov decompositions, a type of factorization in the Weyl
group corresponding to fibre bundle structures on Schubert varieties. Time
permitting, I will talk about the existence problem for Billey-Postnikov
decompositions, and explain why Billey-Postnikov decompositions are useful for
understanding the structure and classification of smooth and rationally smooth
Schubert varieties.
This talk concerns joint work with Lakshmibai and Ravikumar (first part of
the talk) and with Ed Richmond (second part).