Vector fields and Luna strata

Schwarz, Gerald W.

doi:10.1016/j.jpaa.2012.04.008

Cited by 5 publications

(6 citation statements)

References 9 publications

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“…It is then clear that every G-invariant vector field on X induces a vector field on the quotient X//G. Schwarz shows in [Sch13] that if the induced map Vec(X) G → Vec(X//G) is surjective, then the Luna strata of the quotient X//G are intrinsic, i.e. they are permuted by all automorphisms of X//G.…”

Section: Linear Semigroupsmentioning

confidence: 99%

Covariants, Invariant Subsets, and First Integrals

Grosshans¹,

Kraft²

2020

Épijournal De Géométrie Algébrique

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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.

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Section: Linear Semigroupsmentioning

confidence: 99%

Covariants, Invariant Subsets, and First Integrals

Grosshans¹,

Kraft²

2020

Épijournal De Géométrie Algébrique

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“…There is a holomorphic (complex) time-dependent vector field A(t, z) such that the integral of A from 0 to t gives back ϕ t . By Schwarz [24…”

Section: Lemma 33 ([4]mentioning

confidence: 99%

“…Proof. We know that Z has no codimension 1 strata (see [24,Proposition 3.1]). If S is of codimension 2, then it is subprincipal.…”

Section: Multiples Of the Adjoint Representationmentioning

confidence: 99%

“…Let ϕ ∈ Aut(Z). Then, by Schwarz [24] we know that ϕ permutes the strata of the same codimension. The strata of Z are products S 1 × · · · × S d where S i is a stratum of Z i .…”

Section: Multiples Of the Adjoint Representationmentioning

confidence: 99%

“…If V is 2-principal, then the principal isotropy group is the kernel of G → GL(V ) [24,Remark 2.5]. Thus, if V is admissible, then we can assume that it has TPIG by just dividing out by the ineffective part of the action.…”

Section: Introductionmentioning

confidence: 99%

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Quotients, automorphisms and differential operators

Schwarz

2013

Journal of the London Mathematical Society

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Abstract. Let V be a G-module where G is a complex reductive group. Let Z := V / /G denote the categorical quotient and let π : V → Z be the morphism dual to the inclusion O(V ) G ⊂ O(V ). Let ϕ : Z → Z be an algebraic automorphism. Then one can ask if there is an algebraic map Φ : V → V which lifts ϕ, i.e., π(Φ(v)) = ϕ(π(v)) for all v ∈ V . In Kuttler [Kut11] the case is treated where V = rg is a multiple of the adjoint representation of G. It is shown that, for r sufficiently large (often r ≥ 2 will do), any ϕ has a lift.We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of Z, and we show that if a holomorphic ϕ and its inverse lift holomorphically, then ϕ has a lift Φ which is an automorphism such that Φ(gv) = σ(g)Φ(v), v ∈ V , g ∈ G where σ is an automorphism of G. We reduce the lifting problem to the group of automorphisms of Z which preserve the natural grading of O(Z) ≃ O(V ) G . Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler's methods to show lifting in case V contains a copy of g.

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Sufficient Conditions for Holomorphic Linearisation

Kutzschebauch

Lárusson

Schwarz

2016

Transformation Groups

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Abstract. Let G be a reductive complex Lie group acting holomorphically on X = C n . The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on C n such that the G-action becomes linear. Equivalently, is there a Gequivariant biholomorphism Φ : X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient Q X , called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism ϕ : Q X → Q V which is stratified, i.e., the stratum of Q X with a given label is sent isomorphically to the stratum of Q V with the same label.The counterexamples to the Linearisation Problem construct an action of G such that Q X is not stratified biholomorphic to any Q V . Our main theorem shows that, for most X, a stratified biholomorphism of Q X to some Q V is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to C n , only that X is a Stein manifold.

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Vector fields and Luna strata

Cited by 5 publications

References 9 publications

Covariants, Invariant Subsets, and First Integrals

Covariants, Invariant Subsets, and First Integrals

Quotients, automorphisms and differential operators

Sufficient Conditions for Holomorphic Linearisation

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