Orbits and invariants associated with a pair of commuting involutions
The authors consider: \(\sigma\), \(\theta\), commuting involutions of the connected reductive algebraic group \(G\), where \(\sigma\), \(\theta\), and \(G\) are defined over a (usually algebraically closed) field \(k\), \(\text{char }k\neq 2\). We have fixed point groups \(H:=G^\sigma\) and \(K:=G^\theta\) and an action \((H\times K)\times G\to G\), where \(((h,k),g)\mapsto hgk^{-1}\), \(h\in H\), \(k\in K\), \(g\in G\). Let \(G/ /(H\times K)\) denote \(\text{Spec }{\mathcal O}(G)^{H\times K}\) (the categorical quotient). Let \(A\) be maximal among subtori \(S\) of \(G\) such that \(\theta(s)=\sigma(s)=s^{-1}\) for all \(s\in S\). There is the associated Weyl group \(W:=W_{H\times K}(A)\). The authors show the following: The inclusion \(A\to G\) induces an isomorphism \(A/W\overset\sim{} G/ /(H\times K)\). In particular, the closed \((H\times K)\)-orbits are precisely those which intersect \(A\). The fibers of \(G\to G/ /(H\times K)\) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. The authors investigate the structure of \(W\) and its relation to other naturally occurring Weyl groups and to the action of \(\sigma\theta\) on the \(A\)-weight spaces of \(\mathfrak g\); the relation of the orbit type stratification of \(A/W\) and \(G/ /(H\times K)\). Along the way they simplify some of R. Richardson's proofs for the symmetric case \(\sigma=\theta\), and at the end they quickly recover results of M. Berger, M. Flensted-Jensen, B. Hoogenboom, and T. Matsuki for the case \(k=\mathbb{R}\).
scientific article; zbMATH DE number 1820778
scientific article