Simple Harish-Chandra modules, intermediate series modules, and Verma modules over the loop-Virasoro algebra

Abstract

This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also establish an irreducibility criterion for Verma modules over the loop Virasoro algebra and determine necessary and sufficient conditions for the simple quotient of a Verma module to be a Harish-Chandra module.