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.
Source : ScholeXplorer
IsRelatedTo DOI 10.37236/747
