Schubert varieties and the fusion products

For each $A\in\N^n$ we define a Schubert variety $\sh_A$ as a closure of the \Slt(\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 \slt\otimes\C[t] modules. In the case when all the entries of $A$ are different $\sh_A$ is smooth projective 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.Comment: 34 page

Two dimensional current algebras and affine fusion product

In this paper we study a family of commutative algebras generated by two infinite sets of generators. These algebras are parametrized by Young diagrams. We explain a connection of these algebras with the fusion product of integrable irreducible representations of the affine $sl_2$ Lie algebra. As an application we derive a fermionic formula for the character of the affine fusion product of two modules. These fusion products can be considered as a simplest example of the double affine Demazure modules.Comment: 22 page