episciences.org_5976_1652736325 1652736325 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Épijournal de Géométrie Algébrique 2491-6765 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 finite-dimensional 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 semi-subgroup. Both \$End(V)\$ and \$E\$ are ind-varieties 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 non-constant \$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 finite-dimensional 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