Quasifinite representations of a class of Block type Lie algebras $\BB$

Intrigued by a well-known theorem of Mathieu's on Harish-Chandra modules over the Virasoro algebra, we give an analogous result for a class of Block type Lie algebras $\BB$, where the parameter $q$ is a nonzero complex number. We also classify quasifinite irreducible highest weight $\BB$-modules and irreducible $\BB$-modules of the intermediate series. In particular, we obtain that an irreducible $\BB$-module of the intermediate series may be a nontrivial extension of a $\Vir$-module of the intermediate series if $q$ is half of a negative integer, where $\Vir$ is a subalgebra of $\BB$ isomorphic to the Virasoro algebra.