FUNDAMENTAL REPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF 1

To every finite-dimensional irreducible representation V of the quantum group U(E)(g underbar) where E is a primitive lth root of unity (l odd) and g underbar is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C(V) in the adjoint group G of g underbar. We describe explicitly, when g underbar is of type A(n), B(n), C(n), or D4, the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.