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1 February 2001 Orbits and invariants associated with a pair of commuting involutions
Aloysius G. Helminck, Gerald W. Schwarz
Duke Math. J. 106(2): 237-279 (1 February 2001). DOI: 10.1215/S0012-7094-01-10622-4


Let σ,θ be commuting involutions of the connected reductive algebraic group G, where σ,θ, and G are defined over a (usually algebraically closed) field k, char k≠2. We have fixed point groups H:≠Gσ and K:≠Gθ and an action (H×KGG, where ((h, k), g)↦hgk−1, hH, kK, gG. Let G//(H×K) denote Spec $\mathscr{O}$(G)H×K (the categorical quotient).

Let A be maximal among subtori S of G such that θ(s)=σ(s)=s−1 for all sS. There is the associated Weyl group W:=WH×K(A). We show the following.

· The inclusion AG induces an isomorphism A/W$\widetilde{\to}$G//(H×K). In particular, the closed (H×K)-orbits are precisely those which intersect A.

· The fibers of GG//(H×K) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits.

We investigate

· the structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of $\mathfrak {g}$;

· the relation of the orbit type stratifications of A/W and G//(H×K).

Along the way we simplify some of R. Richardson's proofs for the symmetric case σ=θ, and at the end we quickly recover results of M. Berger, M. Flensted-Jensen, B. Hoogenboom, and T. Matsuki [Ber], [FJ1], [Hoo], [Mat] for the case k=ℝ.


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Aloysius G. Helminck. Gerald W. Schwarz. "Orbits and invariants associated with a pair of commuting involutions." Duke Math. J. 106 (2) 237 - 279, 1 February 2001.


Published: 1 February 2001
First available in Project Euclid: 13 August 2004

zbMATH: 1015.20031
MathSciNet: MR1813432
Digital Object Identifier: 10.1215/S0012-7094-01-10622-4

Primary: 20G15
Secondary: 14L30 , 20G20 , 22E46

Rights: Copyright © 2001 Duke University Press


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Vol.106 • No. 2 • 1 February 2001
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