Classification of Harish-Chandra modules over some Lie algebras related to the Virasoro algebra

In this paper, we provide a uniform method to thoroughly classify all Harish-Chandra modules over some Lie algebras related to the Virasoro algebras. We first classify such modules over the Lie algebra $W(\varrho)[s]$ for $s=0,\frac12$. With this result and method, we can also do such works for some Lie algebras and superconformal algebras related to the Virasoro algebra, including the several kinds of Schr\"odinger-Virasoro Lie algebras, which are open up to now.