Covariants, Invariant Subsets, and First Integrals
Authors: Frank Grosshans ; Hanspeter Kraft
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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.