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

Abstract

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 $U_{{\eta }_g}=U_{\epsilon }(\mathfrak{g})/U_{\epsilon }(\mathfrak{g})\mathrm{Ker}{\eta }_g$. We show that in this case, under certain restrictions on $m$, $U_{{\eta }_g}$ contains a subalgebra $U_{{\eta }_g}({\mathfrak{m}}_{-})$ of dimension $m^{\frac{1}{2}\dim \mathcal{O}}$, 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 ${\mathrm{Hom}}_{U_{{\eta }_g}({\mathfrak{m}}_{-})}({\mathbb{C}}_{{\chi }_g},V)$ is not trivial and the algebra $W_{{\eta }_g}={\mathrm{End}}_{U_{{\eta }_g}}(U_{{\eta }_g}{\otimes }_{U_{{\eta }_g}({\mathfrak{m}}_{-})}{\mathbb{C}}_{{\chi }_g})$ 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.

arXiv comment25 pages; this paper contains the part of the previous version of the text covering the results on Whittaker vectors in modules over quantum groups at roots of unity; the proof of the De Concini-Kac-Procesi conjecture and the related results on q-W-algebras at roots of unity can be found in arXiv:2102.03208. arXiv admin note: text overlap with arXiv:1011.2431