Certain Moduli of Algebraic $G$-vector Bundles over Affine $G$-varieties

Abstract

Let $G$ be a reductive complex algebraic group and $P$ a complex $G$-module with algebraic quotient of dimension $\ge 1$. We construct a map from a certain moduli space of algebraic $G$-vector bundles over $P$ to a $\mathbb{C}$-module possibly of infinite dimension, which is an isomorphism under some conditions. We also show non-triviality of moduli of algebraic $G$-vector bundles over a $G$-stable affine hypersurface of some type. In particular, we show that the moduli space of algebraic $G$-vector bundles over a $G$-stable affine quadric with fixpoints and one-dimensional quotient contains $\mathbb{C}^p$.