Let M be a connected compact pseudoRiemannian manifold acted upontopologically transitively and isometrically by a connected noncompact simpleLie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspacestangent to M and G, respectively, where G carries any bi-invariant metric, thenwe have n_0 \leq m_0. We study G-actions that satisfy the condition n_0 = m_0.With no rank restrictions on G, we prove that M has a finite covering \hat{M}to which the G-action lifts so that \hat{M} is G-equivariantly diffeomorphic toan action on a double coset K\backslash L/\Gamma, as considered in Zimmer'sprogram, with G normal in L (Theorem A). If G has finite center and\rank_\R(G)\geq 2, then we prove that we can choose \hat{M} for which L issemisimple and \Gamma is an irreducible lattice (Theorem B). We also prove thatour condition n_0 = m_0 completely characterizes, up to a finite covering, suchdouble coset G-actions (Theorem C). This describes a large family of doublecoset G-actions and provides a partial positive answer to the conjectureproposed in Zimmer's program.
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机译:令M为一个连通的紧拟伪黎曼流形,它由一个连通的非紧简朴素群G在拓扑上传递和等距地作用。如果m_0,n_0是分别等于M和G的最大光子空间的维数,其中G携带任何双不变度量,则我们有n_0 \ leq m_0。我们研究满足条件n_0 = m_0的G动作。没有对G的等级限制,我们证明M具有有限的覆盖范围\ hat {M},G动作可以举升到\ hat {M},因此\ hat {M}是G-在Zimmer程序中考虑了对双陪集K \反斜杠L / \ Gamma的等变微分变矩定理,L为G法线(定理A)。如果G具有有限的中心和\ rank_ \ R(G)\ geq 2,那么我们证明我们可以选择L为半简单且\ Gamma是不可约格的\ hat {M}(定理B)。我们还证明,在有限覆盖范围内,我们的条件n_0 = m_0完全表征了这种双重陪集G动作(定理C)。这描述了一个双重偶数G动作的大家族,并为Zimmer程序中提出的猜想提供了部分肯定的答案。
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