Unitary representations for the Schr\"{o}dinger-Virasoro Lie algebra

In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schr\"{o}dinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schr\"{o}dinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schr\"{o}dinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.