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

Let $U_q$ be the quantum group corresponding to a complex simple Lie algebra $\mathfrak g$ with root system $R$. Assume the quantum parameter $q\in \C$ 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 $U_q$. We first prove that this problem is equivalent to finding the extensions between the finitely many simple modules for the small quantum group $u_q$ in $U_q$. Then we show that the extension groups in question are determined by a finite subset with small highest weights. When the order of $q^2$ 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.