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

Let U_ε(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 U_ε(g) contains a huge commutative subalgebra isomorphic to the algebra Z_G 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 U_ε(g) on which Z_G acts according to a non–trivial character η_g
given by evaluation of regular functions at g ∈ G. Then V is a representation of
the finite–dimensional algebra U_ηg= U_ε(g)/U_ε(g)Ker η_g. We show that in this case,
under certain restrictions on m, Uη_g contains a subalgebra Uη_g (m₋) of dimension m^{1/2 dim O}, where ^O is the conjugacy class of g, and Uη_g (m₋) has
a one–dimensional representation C_χg. We also prove that if V is not trivial then
the space of Whittaker vectors Hom_Uηg (m₋) (Cχ_g ,V) is not trivial and the algebra
W_ηg =End_Uηg (U_ηg ⊗_Uηg (m₋) C_χg) naturally acts on it which gives rise to a Schur–type duality between representations of the algebra U_ηg and of the algebra W_ηg
called a q-W algebra.