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.