Abstract
Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine Gvariety. 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 Gvarieties X if and only if H is a Gcompletely reducible subgroup of G (in the sense defined by JP. 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 (fromto)  11211159 
Number of pages  39 
Journal  Mathematische Zeitschrift 
Volume  293 
Issue number  34 
Early online date  23 Jan 2019 
DOIs  
Publication status  Published  Dec 2019 
Bibliographical 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.Keywords
 double cosets
 etale slice
 Gcomplete reducibility
 Geometric invariant theory
 quotient variety
 Quotient variety
 Double cosets
 GComplete reducibility
 Étale slice
 TUPLES
 COMPLETE REDUCIBILITY
 INSTABILITY
 Etaleslice
 ALGEBRAICGROUPS
 LIEALGEBRAS
 REDUCTIVE SUBGROUPS
 DOUBLE COSET DENSITY
 CLOSED ORBITS
 CONJUGACY CLASSES
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Ben Martin
 School of Natural & Computing Sciences, Mathematical Science  Personal Chair
Person: Academic