Vector Bundles Associated to Lie Algebras

Abstract

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra $\mathfrak{g}$. These are sheaves on locally closed subvarieties of the projective variety $\mathbb{E}(r,\mathfrak{g})$ of elementary subalgebras of $\mathfrak{g}$ of dimension $r$. We show that representations of constant radical or socle rank studied in \cite{CFP3} which generalize modules of constant Jordan type lead to algebraic vector bundles on $\mathbb{E}(r,\mathfrak{g})$. For $\mathfrak{g}=Lie(G)$, the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $\mathbb{E}(r,\mathfrak{g})$.

Replaces second half of arXiv:1207.5898 . To appear in Crelle