# Orbit Closures and Invariants

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

## 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 1121-1159 39 Mathematische Zeitschrift 293 3-4 23 Jan 2019 https://doi.org/10.1007/s00209-019-02228-6 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