Duke Mathematical Journal

Orbits and invariants associated with a pair of commuting involutions

Aloysius G. Helminck and Gerald W. Schwarz

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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.

Article information

Duke Math. J., Volume 106, Number 2 (2001), 237-279.

First available in Project Euclid: 13 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 20G20: Linear algebraic groups over the reals, the complexes, the quaternions 22E46: Semisimple Lie groups and their representations


Helminck, Aloysius G.; Schwarz, Gerald W. Orbits and invariants associated with a pair of commuting involutions. Duke Math. J. 106 (2001), no. 2, 237--279. doi:10.1215/S0012-7094-01-10622-4. https://projecteuclid.org/euclid.dmj/1092403916

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