
Journal of Lie Theory 18 (2008), No. 2, 433443 Copyright Heldermann Verlag 2008 Linear Maps Preserving Fibers Gerald W. Schwarz Dept. of Mathematics, Brandeis University, MS 050, P.O. Box 549110, Waltham, MA 024549110, U.S.A. schwarz@brandeis.edu [Abstractpdf] Let $G\subset{\rm GL}(V)$ be a complex reductive group where $\dim V<\infty$, and let $\pi\colon V\to V/G$ be the categorical quotient. Let ${\cal N}:=\pi^{1}\pi(0)$ be the null cone of $V$, let $H_0$ be the subgroup of GL$(V)$ which preserves the ideal $\cal I$ of $\cal N$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H = H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of GL$(V)$ preserving a fiber $F$ of $\pi$ when $V$ is an irreducible cofree $G$module. Keywords: Invariants, null cone, cofree representations. MSC: 20G20, 22E46, 22E60 [ Fulltextpdf (192 KB)] for subscribers only. 