Variations on themes of Kostant

TL;DR

This paper describes the cohomology algebra of spherical Schubert varieties in the loop Grassmannian for the Langlands dual group, providing a quotient description and a topological proof of Kostant's theorem.

Contribution

It offers a new quotient description of the cohomology algebra of spherical Schubert varieties and a topological proof of Kostant's polynomial algebra structure.

Findings

Cohomology algebra expressed as Sym(g^e)/J

Description applies to arbitrary spherical Schubert varieties

Provides a topological proof of Kostant's theorem

Abstract

Let g be a complex semisimple Lie algebra and let G' be the Langlands dual group. We give a description of the cohomology algebra of an arbitrary spherical Schubert variety in the loop Grassmannian for G' as a quotient of the form Sym(g^e)/J. Here, J is an appropriate ideal in the symmetric algebra of g^e, the centralizer of a principal nilpotent in g. We also discuss a `topological' proof of Kostant's famous result on the structure of the polynomial algebra on g.