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

Abstract

Let $U_\varepsilon({\mathfrak g})$ be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd primitive m-th root of unity $\varepsilon$. The center of $U_\varepsilon({\mathfrak g})$ 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 $U_\varepsilon({\mathfrak g})$ 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 $\mathcal{O}$ is the conjugacy class of $g$, and $U_{\eta_g}({\mathfrak m}_-)$ has a one-dimensional representation $\mathbb{C}_{\chi_g}$. We also prove that if $V$ is not trivial then the space of Whittaker vectors ${\rm Hom}_{U_{\eta_g}({\mathfrak m}_-)}(\mathbb{C}_{\chi_g},V)$ is not trivial and the algebra $W_{\eta_g}={\rm End}_{U_{\eta_g}}(U_{\eta_g}\otimes_{U_{\eta_g}({\mathfrak m}_-)}\mathbb{C}_{\chi_g})$ 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.