On Whittaker modules for a Lie algebra arising from the 2-dimensional torus

Shaobin Tan, Qing Wang and Chengkang Xu

Vol. 273 (2015), No. 1, 147–167

DOI: 10.2140/pjm.2015.273.147

Abstract

Let $A$ be the ring of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. We denote by $\tilde{L}$ the semidirect product of $A$ and $B$, and by $L$ the universal central extension of the derived Lie algebra of $\tilde{L}$. We study the Whittaker modules for the Lie algebra $L$. The irreducibilities for the universal Whittaker modules are given. Moreover, a $\mathbb{Z}$-gradation is defined on the universal Whittaker modules and we determine all $\mathbb{Z}$-graded irreducible quotients of the reducible universal Whittaker modules.

Keywords

Whittaker module, infinite-dimensional Lie algebra, torus

Mathematical Subject Classification 2010

Primary: 06B15

Secondary: 17B65, 17B66

Milestones

Received: 7 December 2013

Accepted: 6 May 2014

Published: 6 December 2014