On representations of the centrally extended Heisenberg double of SL2

Abstract

For the centrally extended Heisenberg double of SL₂, its center is determined, the central factor algebras are described, and classifications of simple Harish-Chandra modules, simple Whittaker modules, and simple quasi-Whittaker modules are obtained. Two classes of simple weight modules with infinite-dimensional weight spaces are given. We also give a classification of simple modules that decompose into a direct sum of simple finite-dimensional ${\mathfrak{s}\mathfrak{l}}_2$ -modules with finite multiplicities.