Spherical orbits and representations of $U_{\epsilon }(\mathfrak{g})$

Abstract

Let $U_{\epsilon }(\mathfrak{g})$ be the simply connected quantized enveloping algebra associated to a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of $U_{\epsilon }(\mathfrak{g})$ is proved for representations corresponding to the spherical conjugacy classes of the simply connected algebraic group $G$ with Lie algebra $\mathfrak{g}$. We achieve this result by means of a new characterization of the spherical conjugacy classes of $G$ in terms of elements of the Weyl group.

This new version includes part I and part II. Some typos are corrected and some remarks are added