Orbit Closures and Invariants

Michael Bate; Haralampos Geranios; Benjamin Martin

doi:10.1007/s00209-019-02228-6

Orbit Closures and Invariants

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

^*Corresponding author for this work

Mathematical Science

Research output: Contribution to journal › Article › peer-review

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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 language	English
Pages (from-to)	1121-1159
Number of pages	39
Journal	Mathematische Zeitschrift
Volume	293
Issue number	3-4
Early online date	23 Jan 2019
DOIs	https://doi.org/10.1007/s00209-019-02228-6
Publication status	Published - Dec 2019

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

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10.1007/s00209-019-02228-6Licence: CC BY

Bate2019_MZ_OrbitClosuresAndInvariants_VoR
© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Final published version, 607 KBLicence: CC BY

https://arxiv.org/abs/1604.00924Licence: Unspecified

Cite this

@article{02bb85f24c0f4c2a8309e82b90b356d0,

title = "Orbit Closures and Invariants",

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{\textquoteright}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.",

keywords = "double cosets, etale slice, G-complete reducibility, Geometric invariant theory, quotient variety, Quotient variety, Double cosets, G-Complete reducibility, {\'E}tale slice, TUPLES, COMPLETE REDUCIBILITY, INSTABILITY, Etaleslice, ALGEBRAIC-GROUPS, LIE-ALGEBRAS, REDUCTIVE SUBGROUPS, DOUBLE COSET DENSITY, CLOSED ORBITS, CONJUGACY CLASSES",

author = "Michael Bate and Haralampos Geranios and Benjamin Martin",

note = "The first author would like to thank Sebastian Herpel for the conversations we had which led to the first iteration of some of the ideas in this paper, and also Stephen Donkin for some very helpful nudges towards the right literature. All three authors acknowledge the funding of EPSRC grant EP/L005328/1. We would like to thank the anonymous referee for their very insightful comments and for pointing out a subtle gap in the proof of Theorem 1.1.",

year = "2019",

month = dec,

doi = "10.1007/s00209-019-02228-6",

language = "English",

volume = "293",

pages = "1121--1159",

journal = "Mathematische Zeitschrift",

issn = "0025-5874",

publisher = "Springer New York",

number = "3-4",

}

TY - JOUR

T1 - Orbit Closures and Invariants

AU - Bate, Michael

AU - Geranios, Haralampos

AU - Martin, Benjamin

N1 - The first author would like to thank Sebastian Herpel for the conversations we had which led to the first iteration of some of the ideas in this paper, and also Stephen Donkin for some very helpful nudges towards the right literature. All three authors acknowledge the funding of EPSRC grant EP/L005328/1. We would like to thank the anonymous referee for their very insightful comments and for pointing out a subtle gap in the proof of Theorem 1.1.

PY - 2019/12

Y1 - 2019/12

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.

KW - double cosets

KW - etale slice

KW - G-complete reducibility

KW - Geometric invariant theory

KW - quotient variety

KW - Quotient variety

KW - Double cosets

KW - G-Complete reducibility

KW - Étale slice

KW - TUPLES

KW - COMPLETE REDUCIBILITY

KW - INSTABILITY

KW - Etaleslice

KW - ALGEBRAIC-GROUPS

KW - LIE-ALGEBRAS

KW - REDUCTIVE SUBGROUPS

KW - DOUBLE COSET DENSITY

KW - CLOSED ORBITS

KW - CONJUGACY CLASSES

UR - http://www.scopus.com/inward/record.url?scp=85060604702&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/orbit-closures-invariants

UR - https://abdn.pure.elsevier.com/en/en/researchoutput/orbit-closures-and-invariants(02bb85f2-4c0f-4c2a-8309-e82b90b356d0).html

U2 - 10.1007/s00209-019-02228-6

DO - 10.1007/s00209-019-02228-6

M3 - Article

VL - 293

SP - 1121

EP - 1159

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3-4

ER -

Orbit Closures and Invariants

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Ben Martin

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Orbit Closures and Invariants

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Ben Martin

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