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 $u{D}_1^{m_1}...D_n^{m_n}$ for $u\in F[G]$ (the group algebra), and $m_i\ge 0$ with ${\sum }_{i=1}^n{m}_i\ge 1$, 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 $Z$.

LaTeX, 11 pages. To appear in Israel Journal of Mathematics