Quiver varieties and tensor products

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety \Zl in a quiver variety, and show the following results: (1) The homology group of \Zl is a representation of a symmetric Kac-Moody Lie algebra $\mathfrak g$, isomorphic to the tensor product $V(\lambda_1)\otimes...\otimes V(\lambda_N)$ of integrable highest weight modules. (2) The set of irreducible components of \Zl has a structure of a crystal, isomorphic to that of the $q$-analogue of $V(\lambda_1)\otimes...\otimes V(\lambda_N)$. (3) The equivariant $K$-homology group of \Zl is isomorphic to the tensor product of universal standard modules of the quantum loop algebra \Ul, when $\mathfrak g$ is of type $ADE$. We also give a purely combinatorial description of the crystal of (2). This result is new even when N=1.Comment: 39 pages, no figures; Several references are adde