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Covariants, Invariant Subsets, and First Integrals

Frank Grosshans, Hanspeter Kraft

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.

Covariants, Invariant Subsets, and First Integrals

Abstract

Let be an algebraically closed field of characteristic 0, and let be a finite-dimensional vector space. Let be the semigroup of all polynomial endomorphisms of . Let be a subset of which is a linear subspace and also a semi-subgroup. Both and are ind-varieties which act on in the obvious way. In this paper, we study important aspects of such actions. We assign to a linear subspace of the vector fields on . A subvariety of is said to -invariant if is in the tangent space of for all in and in . We show that is -invariant if and only if it is the union of -orbits. For such , we define first integrals and construct a quotient space for the -action. An important case occurs when is an algebraic subgroup of ) and consists of the -equivariant polynomial endomorphisms. In this case, the associated is the space the -invariant vector fields. A significant question here is whether there are non-constant -invariant first integrals on . 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.

Paper Structure

This paper contains 30 sections, 32 theorems, 69 equations.

Table of Contents

  1. Introduction and main results
  2. Basic material
  3. Vector fields and $\mathcal{D}$-invariant subsets
  4. $\mathcal{D}$-invariant ideals
  5. Linear spaces of vector fields
  6. Endomorphisms
  7. The semigroup of endomorphisms
  8. The case of a vector space
  9. Linear semigroups
  10. $\mathop{\mathrm{Vec}}\nolimits_G$-symmetry
  11. $G$-equivariant endomorphisms
  12. $\mathop{\mathrm{Vec}}\nolimits_G$-symmetric subvarieties
  13. Stabilizers
  14. Reductive groups
  15. Dense orbits
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $X$ be a smooth complex variety, and let $\xi \in \mathop{\mathrm{Vec}}\nolimits(X)$ be an algebraic vector field. Then a Zariski-closed subvariety $Y \subseteq X$ is invariant with respect to the flow defined by the differential equation $\dot x = \xi(x)$ if and only if $\xi(y) \in T_{y}Y$ for

Theorems & Definitions (93)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 83 more