episciences.org_5976_1652736325
1652736325
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
Épijournal de Géométrie Algébrique
24916765
09
25
2020
Volume 4
Covariants, Invariant Subsets, and First Integrals
Frank
Grosshans
Hanspeter
Kraft
Let $k$ be an algebraically closed field of characteristic 0, and let $V$ be
a finitedimensional vector space. Let $End(V)$ be the semigroup of all
polynomial endomorphisms of $V$. Let $E$ be a subset of $End(V)$ which is a
linear subspace and also a semisubgroup. Both $End(V)$ and $E$ are
indvarieties which act on $V$ in the obvious way. In this paper, we study
important aspects of such actions. We assign to $E$ a linear subspace $D_{E}$
of the vector fields on $V$. A subvariety $X$ of $V$ is said to $D_{E}$
invariant if $h(x)$ is in the tangent space of $x$ for all $h$ in $D_{E}$ and
$x$ in $X$. We show that $X$ is $D_{E}$ invariant if and only if it is the
union of $E$orbits. For such $X$, we define first integrals and construct a
quotient space for the $E$action. An important case occurs when $G$ is an
algebraic subgroup of $GL(V$) and $E$ consists of the $G$equivariant
polynomial endomorphisms. In this case, the associated $D_{E}$ is the space the
$G$invariant vector fields. A significant question here is whether there are
nonconstant $G$invariant first integrals on $X$. As examples, we study the
adjoint representation, orbit closures of highest weight vectors, and
representations of the additive group. We also look at finitedimensional
irreducible representations of SL2 and its nullcone.
09
25
2020
5976
arXiv:1703.01890
10.48550/arXiv.1703.01890
https://arxiv.org/abs/1703.01890v4
https://arxiv.org/abs/1703.01890v3
10.46298/epiga.2020.volume4.5976
https://epiga.episciences.org/5976

https://epiga.episciences.org/6802/pdf