Whittaker modules for $U_q(\mathfrak{sl}_3)$

Abstract

In this paper, we study the Whittaker modules for the quantum enveloping algebra $U_q(\sl_3)$ with respect to a fixed Whittaker function. We construct the universal Whittaker module, find all its Whittaker vectors and investigate the submodules generated by subsets of Whittaker vectors and corresponding quotient modules. We also find Whittaker vectors and determine the irreducibility of these quotient modules and show that they exhaust all irreducible Whittaker modules. Finally, we can determine all maximal submodules of the universal Whittaker module. The Whittaker model of $U_q(\sl_3)$ are quite different from that of $U_q(\sl_2)$ and finite-dimensional simple Lie algebras, since the center of our algebra is not a polynomial algebra.