  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.
