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

TL;DR

The paper addresses the classification of Whittaker modules for $U_q(\mathfrak{sl}_3)$ using a fixed singular Whittaker function, since non-singular functions do not exist due to Serre relations. It constructs the universal Whittaker module $M(\eta)$ and introduces two-parameter quotient families $V(\eta;\kappa,c)$, with irreducibility characterized by a non-critical condition and a separate critical-case quotients by maximal submodules; it further analyzes the submodule lattice, composition series, and endomorphism structures, yielding a complete description of irreducible Whittaker modules as quotients of $M(\eta)$. The center of $U_q(\mathfrak{sl}_3)$ plays a key role in the scalar actions that distinguish modules, and the results are shown to be independent of the particular singular Whittaker function chosen. Overall, the work reveals a richer Whittaker theory for $U_q(\mathfrak{sl}_3)$ compared to $U_q(\mathfrak{sl}_2)$, with precise criteria distinguishing non-critical and critical cases and a full classification of irreducibles via central characters.

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.

Theorems & Definitions (39)

• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 4.1
• proof
• Remark 4.2
• Lemma 4.3