Variations on themes of Kostant

Let g be a complex semisimple Lie algebra, and let G be a complex semisimple group with trivial center whose root system is dual to that of g. We establish a graded algebra isomorphism H q (Xλ,C) ∼= Sg e/Iλ, where Xλ is an arbitrary spherical Schubert variety in the loop Grassmannian for G, and Iλ is an appropriate ideal in the symmetric algebra of g, the centralizer of a principal nilpotent in g. We also discuss a ‘topological’ proof of Kostant’s result on the structure of C[g]. 1 Cohomology of ‘spherical’ Schubert varieties In this paper, we discuss a few geometric results which were, to a great extent, inspired by three fundamental papers of Bertram Kostant [Ko1]-[Ko3]. 1.1 Let g be a complex semisimple Lie algebra and write gx⊂g for the centralizer of an element x ∈ g. We fix a principal sl2-triple 〈h, e, f〉 ⊂ g. Thus, h := g h is a Cartan subalgebra of g. The element e ∈ g is a principal nilpotent and we make the choice of positive roots of (g, h) so that e is contained in the span of simple root vectors. We write h Z ⊂h for the weight lattice of (g, h). Given an finite dimensional g-module V and a weight μ ∈ h Z , let V (μ)⊂V denote the corresponding μ-weight space, let SpecV⊂h Z be the set formed by the weights which occur in V with nonzero multiplicity, ie. such that V (μ) 6= 0. We ignore weight multiplicities and let I(V )⊂C[h] denote the ideal of polynomials vanishing at the set SpecVλ, viewed as a finite reduced subscheme in h . Thus, the coordinate ring C[SpecVλ] = C[h ]/I(V ) is a finite dimensional algebra. Write Uk, resp. Sk, for the universal enveloping, resp. symmetric, algebra of a Lie algebra k. A filtration on k gives rise to a filtration on Uk, resp. on Sk, not to be confused with the standard increasing filtration on an enveloping algebra. Following B. Kostant and R. Brylinski, on h, one introduces an increasing filtration F qh, where Fkh := {x ∈ h | ad k+1 e(x) = 0}, k = 0, 1, . . .. The induced filtration gives Uh the structure of a nonnegatively filtered algebra. Clearly, one has Uh = Sh = C[h]. In particular, we may (and will) view C[SpecVλ] as a quotient of the algebra Uh. In this way, the algebra C[SpecVλ] acquires an increasing filtration that descends from the KostantBrylinski filtration on Uh. For the associated graded algebras, we have grF C[SpecVλ] = grF Uh/ grF I(V ).