Orbit Closures and Invariants

Michael Bate* (Corresponding Author), Haralampos Geranios, Benjamin Martin

*Corresponding author for this work

Research output: Contribution to journalArticle

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Abstract

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let XH denote the set of fixed points of H in X, and NG(H) the normalizer of H in G. In this paper we study the natural map of quotient varieties ψX,H : XH/NG(H) → X/G induced by the inclusion XH ⊆ X. We show that, given G and H, ψX,H is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by J-P. Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for ψX,H to be an isomorphism. We show how to extend some other results in Luna’s paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then
the double coset HgK is closed for generic g ∈ G if and only if H ∩ gKg−1
is reductive for generic g ∈ G.
Original languageEnglish
Pages (from-to)1121-1159
Number of pages39
JournalMathematische Zeitschrift
Volume293
Issue number3-4
Early online date23 Jan 2019
DOIs
Publication statusPublished - Dec 2019

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Closure
Orbit
Subgroup
Invariant
If and only if
Linear Algebraic Groups
Normalizer
Positive Characteristic
Coset
Morphism
Isomorphism
Quotient
Inclusion
Fixed point
Denote
Closed
Theorem

Keywords

  • double cosets
  • etale slice
  • G-complete reducibility
  • Geometric invariant theory
  • quotient variety
  • Quotient variety
  • Double cosets
  • G-Complete reducibility
  • Étale slice
  • TUPLES
  • COMPLETE REDUCIBILITY
  • INSTABILITY
  • Etaleslice
  • ALGEBRAIC-GROUPS
  • LIE-ALGEBRAS
  • REDUCTIVE SUBGROUPS
  • DOUBLE COSET DENSITY
  • CLOSED ORBITS
  • CONJUGACY CLASSES

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Orbit Closures and Invariants. / Bate, Michael (Corresponding Author); Geranios, Haralampos; Martin, Benjamin.

In: Mathematische Zeitschrift, Vol. 293, No. 3-4, 12.2019, p. 1121-1159.

Research output: Contribution to journalArticle

Bate, Michael ; Geranios, Haralampos ; Martin, Benjamin. / Orbit Closures and Invariants. In: Mathematische Zeitschrift. 2019 ; Vol. 293, No. 3-4. pp. 1121-1159.
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N2 - Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let XH denote the set of fixed points of H in X, and NG(H) the normalizer of H in G. In this paper we study the natural map of quotient varieties ψX,H : XH/NG(H) → X/G induced by the inclusion XH ⊆ X. We show that, given G and H, ψX,H is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by J-P. Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for ψX,H to be an isomorphism. We show how to extend some other results in Luna’s paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; thenthe double coset HgK is closed for generic g ∈ G if and only if H ∩ gKg−1is reductive for generic g ∈ G.

AB - Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let XH denote the set of fixed points of H in X, and NG(H) the normalizer of H in G. In this paper we study the natural map of quotient varieties ψX,H : XH/NG(H) → X/G induced by the inclusion XH ⊆ X. We show that, given G and H, ψX,H is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by J-P. Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for ψX,H to be an isomorphism. We show how to extend some other results in Luna’s paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; thenthe double coset HgK is closed for generic g ∈ G if and only if H ∩ gKg−1is reductive for generic g ∈ G.

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