  This paper is a sequel to "Localization of $\frak{u}$-modules. I",
hep-th/9411050. We are starting here the geometric study of the tensor category
$\cal{C}$ associated with a quantum group (corresponding to a Cartan matrix of
finite type) at a root of unity. The main results establish isomorphisms
between homogeneous components of irreducible objects in $\cal{C}$ and spaces
of vanishing cycles at the origin of certain Goresky-MacPherson sheaves on
configuration spaces; establish isomorphisms of the stalks at the origin of the
above GM sheaves with certain Hochschild complexes (which compute the
Hochschild homology of a certain "triangular" subalgebra of our quantum group
with coefficients in the coresponding irreducible representation); establish
the analogous results for tensor products of irreducibles. In geometry, the
tensor product of representations corresponds to a "fusion" of sheaves on
configuration spaces --- operation defined using the functor of nearby cycles.
