Tensor product varieties and crystals: The ADE case

Let $\mathfrak {g}$ be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of $\mathfrak {g}$ are described: tensor product and multiplicity varieties. These varieties are closely related to Nakajima's quiver varieties and should play an important role in the geometric constructions of tensor products and intertwining operators. In particular, it is shown that the set of irreducible components of a tensor product variety can be equipped with the structure of a $\mathfrak {g}$-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finitedimensional representations of $\mathfrak {g}$, and that the number of irreducible components of a multiplicity variety is equal to the multiplicity of a certain representation in the tensor product of several others. Moreover, the decomposition of a tensor product into a direct sum is described geometrically.