Classification of quasifinite $W_\infty$-modules

Abstract

It is proved that an irreducible quasifinite $W_\infty$-module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight $W_\infty$-module is a module of the intermediate series. For a nondegenerate additive subgroup $G$ of $F^n$, where $F$ is a field of characteristic zero, there is a simple Lie or associative algebra $W(G,n)^{(1)}$ spanned by differential operators $uD_1^{m_1}... D_n^{m_n}$ for $u\in F[G]$ (the group algebra), and $m_i\ge0$ with $\sum_{i=1}^n m_i\ge1$, where $D_i$ are degree operators. It is also proved that an indecomposable quasifinite weight $W(G,n)^{(1)}$-module is a module of the intermediate series if $G$ is not isomorphic to $Z$.