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).