Extensions of simple modules for quantum groups at complex roots of 1

Let Uq be the quantum group corresponding to a complex simple Lie algebra g with root system R. Assume the quantum parameter q is a root of unity. In this paper we study the extensions between simple modules in the category consisting of the finite dimensional modules for Uq. We first prove that this problem is equivalent to finding the extensions between the finitely many simple modules for the small quantum group uq in Uq. Then we show that the extension groups in question are determined by a finite subset with small highest weights. When the order of q² is at least the Coxeter number for R we prove that the dimensions of such extension groups equal the top degree coefficients of some associated Kazhdan-Lusztig polynomial for the affine Weyl group for R. We relate all this to similar (old) results for almost simple algebraic groups and their Frobenius subgroup schemes over fields of large prime characteristics.