Quasifinite representations of the Lie superalgebra of quantum pseudo differential operators

In this paper we extend general results obtained by V. Kac and J. Liberati, in “Unitary quasifinite representations of $W_{\mathrm{\infty }}$”, (Letters Math. Phys., 53 (2000), 11-27), for quasifinite highest weight representations of $\mathbb{Z}$-graded Lie algebras to $1\mathrm{/}2\mathbb{Z}$-graded Lie superalgebras, and we apply these to classify the irreducible quasifinite highest weight modules of the Lie superalgebra of quantum pseudo-differential operators.