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 $s=0,\frac{1}{2}$. 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.