Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over the Twisted Schr\"{o}dinger-Virasoro Lie algebra

TL;DR

This paper classifies irreducible weight modules over the Schrödinger-Virasoro Lie algebra, showing their support covers the entire weight lattice and all nontrivial weight spaces are infinite-dimensional, with implications for finite-dimensional cases.

Contribution

It provides a complete classification of irreducible weight modules with infinite-dimensional weight spaces and characterizes finite-dimensional weight modules as Harish-Chandra modules.

Findings

Support of irreducible modules coincides with the weight lattice

All nontrivial weight spaces are infinite-dimensional

Finite-dimensional weight modules are Harish-Chandra modules

Abstract

It is shown that the support of an irreducible weight module over the Schr\"{o}dinger-Virasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schr\"{o}dinger-Virasoro Lie algebra with a nontrivial finite-dimensional weight space, is a Harish-Chandra module.