# Covariants, Invariant Subsets, and First Integrals

## TL;DR

This paper investigates the structure of polynomial endomorphisms acting on vector spaces, focusing on invariant subsets, first integrals, and quotient constructions, especially in the context of algebraic group actions and their orbit structures.

## Contribution

It introduces a framework linking $D_E$-invariant subvarieties with $E$-orbits and constructs quotient spaces, extending the understanding of invariants in polynomial endomorphism actions.

## Key findings

- Characterization of $D_E$-invariant subvarieties as unions of $E$-orbits
- Construction of quotient spaces for $E$-actions
- Analysis of $G$-invariant first integrals in specific group representations

## Abstract

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.