Isometric actions of simple Lie groups on pseudoRiemannian manifolds | Annals of Mathematics

Abstract Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m0,n0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n0≤m0. We study G-actions that satisfy the condition n0=m0. With no rank restrictions on G, we prove that M has a finite covering Mˆ to which the G-action lifts so that Mˆ is G-equivariantly diffeomorphic to an action on a double coset K∖L/Γ, as considered in Zimmer’s program, with G normal in L (Theorem A). If G has finite center and rankℝ(G)≥2, then we prove that we can choose Mˆ for which L is semisimple and Γ is an irreducible lattice (Theorem B). We also prove that our condition n0=m0 completely characterizes, up to a finite covering, such double coset G-actions (Theorem C). This describes a large family of double coset G-actions and provides a partial positive answer to the conjecture proposed in Zimmer’s program.

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数学年刊(Annals of Mathematics)

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