Frank Grosshans ; Hanspeter Kraft - Covariants, Invariant Subsets, and First Integrals

epiga:5976 - Épijournal de Géométrie Algébrique, September 25, 2020, Volume 4 - https://doi.org/10.46298/epiga.2020.volume4.5976
Covariants, Invariant Subsets, and First IntegralsArticle

Authors: 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.

Volume: Volume 4
Published on: September 25, 2020
Accepted on: July 25, 2020
Submitted on: December 16, 2019
Keywords: Mathematics - Representation Theory,14L30, 22E47, 13A50, 34C14