Schubert Varieties and the Fusion Products

Abstract

For each $A\in N^n$ we define a Schubert variety ${\mathrm{sh}}_A$ as a closure of the ${\mathrm{SL}}_2(C[t])$-orbit in the projectivization of the fusion product $M^A$. We clarify the connection of the geometry of the Schubert varieties with an algebraic structure of $M^A$ as ${\mathfrak{sl}}_2\otimes C[t]$ modules. In the case, when all the entries of $A$ are different, ${\mathrm{sh}}_A$ 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.