A proof of De Concini-Kac-Procesi conjecture I. Representations of quantum groups at roots of unity and q-W algebras
Let U_q(g) be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd m-th root of unity q. De Concini, Kac and Proicesi observed that isomorphism classes of irreducible representations of U_q(g) are parameterized by the conjugacy classes in the connected simply connected algebraic group G corresponding to the simple complex Lie algebra g. They also conjectured that the dimension of a representation corresponding to a conjugacy class O is divisible by m^{1/2 dim O}. We show that if O intersects one of special transversal slices S to the set of conjugacy classes in G defined in arXiv:0809.0205 then the dimension of every finite-dimensional irreducible representation of U_q(g) corresponding to O is divisible by m^{1/2 codim S}. In the second part of this paper arXiv:1403.4108 is shown that for every conjugacy class O in G one can find a transversal slice S such that O intersects S and dim O = codim S. This proves the De Concini-Kac-Procesi conjecture. Our result also implies an equivalence between a category of finite-dimensional U_q(g)-modules and a category of finite-dimensional representations of a q-W algebra which can be regarded as a truncation of the quantized algebra of regular functions on S.