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Journal of Lie Theory 18 (2008), No. 2, 433--443
Copyright Heldermann Verlag 2008

Linear Maps Preserving Fibers

Gerald W. Schwarz
Dept. of Mathematics, Brandeis University, MS 050, P.O. Box 549110, Waltham, MA 02454-9110, U.S.A.


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

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