Journal of Lie Theory

no. 2
Linear Maps Preserving Fibers
Journal of Lie Theory, Volume 18 (2008) no. 2, pp. 433-443
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.
DOI: 10.5802/jolt.504
Classification: 20G20, 22E46, 22E60
Keywords: Invariants, null cone, cofree representations 
@article{JOLT_2008_18_2_a11,
     author = {G. W. Schwarz},
     title = {Linear {Maps} {Preserving} {Fibers}},
     journal = {Journal of Lie Theory},
     pages = {433--443},
     year = {2008},
     volume = {18},
     number = {2},
     doi = {10.5802/jolt.504},
     zbl = {1157.14034},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.504/}
}
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%T Linear Maps Preserving Fibers
%J Journal of Lie Theory
%D 2008
%P 433-443
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%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.504/
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G. W. Schwarz. Linear Maps Preserving Fibers. Journal of Lie Theory, Volume 18 (2008) no. 2, pp. 433-443. doi: 10.5802/jolt.504

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