### 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.

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 | |

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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## Profiles

### Ben Martin

- School of Natural & Computing Sciences, Mathematical Science - Personal Chair
- Mathematical Sciences (Research Theme)

Person: Academic

## Cite this

Bate, M., Geranios, H., & Martin, B. (2019). Orbit Closures and Invariants.

*Mathematische Zeitschrift*,*293*(3-4), 1121-1159. https://doi.org/10.1007/s00209-019-02228-6