Schubert Varieties and the Fusion Products

Abstract

For each A ∈ ℕ_n_ we define a Schubert variety sh_A_ as a closure of the SL2(ℂ[t])orbit in the projectivization of the fusion product MA. We clarify the connection of the geometry of the Schubert varieties with an algebraic structure of MA as sl2 ⊗ ℂ[t] modules. In the case, when all the entries of A are different, sh_A_ is a smooth projective complex algebraic variety. We study its geometric properties: the Lie algebra of the vector fields, the coordinate ring, the cohomologies of the line bundles. We also prove that the fusion products can be realized as the dual spaces of the sections of these bundles.