Quasifinite Representations of Classical Lie subalgebras of $W_{\mathrm{\infty },p}$

We show that there are exactly two anti-involution ${\sigma }_{\pm }$ of the algebra of differential operators on the circle that are a multiple of $p(t{\mathrm{\partial }}_t)$ preserving the principal gradation ($p\in \text{CC}[x]$ non-constant). We classify the irreducible quasifinite highest weight representations of the central extension ${\widehat{\text{D}}}_p^{\pm }$ of the Lie subalgebra fixed by $-{\sigma }_{\pm }$. The most important cases are the subalgebras ${\widehat{\text{D}}}_x^{\pm }$ of $W_{\mathrm{\infty }}$, that are obtained when $p(x)=x$. In these cases we realize the irreducible quasifinite highest weight modules in terms of highest weight representation of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra $\text{CC}[u]\mathrm{/}(u^{m+1})$ and its classical Lie subalgebras of $C$ and $D$ types.