Localization of $\frak{u}$-modules. III. Tensor categories arising from configuration spaces

This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number $\zeta$ an abelian artinian category $\FS$. We call its objects {\em finite factorizable sheaves}. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility ("factorization") and finiteness conditions. In Chapter 2 the tensor structure on $\FS$ is defined using functors of nearby cycles. It makes $\FS$ a braided tensor category. In Chapter 3 we define, using vanishing cycles functors, an exact tensor functor $$\Phi:\FS\lra\CC$$ to the category $\CC$ connected with the corresponding quantum group. In Chapter 4 we show that $\Phi$ is an equivalence. Some proofs are only sketched.