Representations of quantum groups at roots of unity, Whittaker vectors and q-W algebras

Let Uε(g) be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd m-th root of unity ε. The center of Uε(g) contains a huge commutative subalgebra isomorphic to the algebra ZG of regular functions on (a finite covering of a big cell in) a complex connected, simply connected algebraic group G with Lie algebra g. Let V be a finite-dimensional representation of Uε(g) on which ZG acts according to a non-trivial character ηg given by evaluation of regular functions at g∈G. Then V is a representation of the finite-dimensional algebra Uηg=Uε(g)/Uε(g)Kerηg. We show that in this case, under certain restrictions on m, Uηg contains a subalgebra Uηg(m−) of dimension m12dimO, where O is the conjugacy class of g, and Uηg(m−) has a one-dimensional representation Cχg. We also prove that if V is not trivial then the space of Whittaker vectors HomUηg(m−)(Cχg,V) is not trivial and the algebra Wηg=EndUηg(Uηg⊗Uηg(m−)Cχg) naturally acts on it which gives rise to a Schur-type duality between representations of the algebra Uηg and of the algebra Wηg called a q-W algebra.