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

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 L the semidirect product of A and B, and by L the universal central extension of the derived Lie algebra of L. We study the Whittaker modules for the Lie algebra L. The irreducibilities for the universal Whittaker modules are given.Moreover, a ‫-ޚ‬gradation is defined on the universal Whittaker modules and we determine all ‫-ޚ‬graded irreducible quotients of the reducible universal Whittaker modules.