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

Let $U_{\epsilon }(\mathfrak{g})$ be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd primitive m-th root of unity $\epsilon$. The center of $U_{\epsilon }(\mathfrak{g})$ 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 $\mathfrak{g}$. Let $V$ be a finite-dimensional representation of $U_{\epsilon }(\mathfrak{g})$ on which $Z_G$ acts according to a non-trivial character ${\eta }_g$ given by evaluation of regular functions at $g\in G$. Then $V$ is a representation of the finite-dimensional algebra $U_{\eta_g}=U_\varepsilon({\mathfrak g})/U_\varepsilon({\mathfrak g}){\rm Ker}~\eta_g$. We show that in this case, under certain restrictions on $m$, $U_{{\eta }_g}$ contains a subalgebra $U_{{\eta }_g}({\mathfrak{m}}_{-})$ of dimension $m^{{\frac{1}{2}}{\rm dim}~$\mathcal{O}$}$,where$$\mathcal{O}$$istheconjugacyclassof$g$,and$U_{\eta_g}({\mathfrak m}-)$hasaone-dimensionalrepresentation$$\mathbb{C}${\chi_g}$\mathrm{.}Wealsoprovethatif$V$ is not trivial then the space of Whittaker vectors ${\rm Hom}{U{\eta_g}({\mathfrak m}-)}($\mathbb{C}${\chi_g},V)$isnottrivialandthealgebra$W_{\eta_g}={\rm End}{U{\eta_g}}(U_{\eta_g}\otimes_{U_{\eta_g}({\mathfrak m}-)}$\mathbb{C}${\chi_g})$ naturally acts on it which gives rise to a Schur-type duality between representations of the algebra $U_{{\eta }_g}$ and of the algebra $W_{{\eta }_g}$ called a q-W algebra.