Harish-Chandra modules for divergence zero vector fields on a torus

Zhiqiang Li, Shaobin Tan and Qing Wang

Vol. 301 (2019), No. 1, 243–265

DOI: 10.2140/pjm.2019.301.243

Abstract

The Lie algebra of divergence zero vector fields on a torus is an infinite-dimensional Lie algebra of skew derivations over the ring of Laurent polynomials. We consider the semidirect product of the Lie algebra of divergence zero vector fields on a torus with the algebra of Laurent polynomials. In this paper, we prove that a Harish-Chandra module of the universal central extension of the derived Lie subalgebra of this semidirect product is either a uniformly bounded module or a generalized highest weight module. We also classify all the generalized highest weight Harish-Chandra modules.

Keywords

Harish-Chandra, divergence zero vector fields, generalized highest weight module

Mathematical Subject Classification 2010

Primary: 17B10, 17B66

Secondary: 17B65, 17B68

Milestones

Received: 8 July 2018

Revised: 26 October 2018

Accepted: 29 November 2018

Published: 16 September 2019