Vol. 301, No. 1, 2019

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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
Authors
Zhiqiang Li
School of Mathematical Sciences
Xiamen University
Xiamen,
China
Shaobin Tan
School of Mathematical Sciences
Xiamen University
Xiamen,
China
Qing Wang
School of Mathematical Sciences
Xiamen University
Xiamen,
China