Mod-pReduction for Quantum Groups

Let ε( ) be the quantized enveloping algebra associated to the Lie algebra = sl(n + 1) at apth-root of unity and assume thatpis a prime which does not dividen + 1. It is known that the irreducible, finite dimensional representations of ( ) are parametrized, up to isomorphisms, by the conjugacy classes of SL(n + 1). In the paper we prove that the dimension of any ( )-moduleMparametrized by a conjugacy class is divided byp1/2 dim( ). This result was conjectured by C. De Concini, V. G. Kac, and C. Procesi (J. Amer. Math. Soc.5, 1992, 151–190).