De Concini-Kac-Procesi conjecture gives a good estimate for the dimensions of finite--dimensional non-restricted representations of quantum groups at m-th root of unity. According to De Concini, Kac and Procesi such representations can be split into families parametrized by conjugacy classes in an algebraic group G, and the dimensions of representations corresponding to a conjugacy class O are divisible by m^{dim O/2}. The talk will consist of two parts. In the first part I shall present an approach to the proof of De Concini-Kac-Procesi conjecture based on the use of q-W algebras and Bruhat decomposition in G. It turns out that properties of representations corresponding to a conjugacy class O depend on the properties of intersection of O with certain Bruhat cells. In the second part, which is more technical, I shall discuss q-W algebras and some related results in detail.
