The Nevo–Zimmer intermediate factor theorem over local fields

Abstract

The Nevo-Zimmer theorem classifies the possible intermediate Gfactors Y in X × G /P → Y → X, where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure.An important corollary is the Stuck-Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set.We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.1 Subsection 2.2 contains definitions concerning algebraic k-groups, as well as some examples. 2 The reader is referred to Subsection 2.1 for a discussion of measurable group actions, including the definitions of a G-space and a G-map. 1 THE NEVO-ZIMMER INTERMEDIATE FACTOR THEOREM OVER LOCAL FIELDS 2of Furstenberg such as the interplay between stationary and P -invariant measures, and diverges from the original approach of Margulis in [17].We mention that a corrected proof along the lines of [25] was given by Nevo and Zimmer in the unpublished work [19].Our chief motivation is to present a proof of Theorem 1.1 that is as close as possible to the lines of [25] and [17], while extending the result to local fields. Indeed, the factor theorem [17] is already given in this (indeed, even greater) generality. We remark that our approach differs from that of [19].Indeed, the real case of Theorem 1.1 implies the real Lie group version at least for center-free semisimple Lie groups, and the same remark applies to Theorem 1.2. This reduction is based on the fact that every connected semisimple real Lie group with trivial center is isomorphic to G 0 R for some connected but possibly non simply-connected algebraic R-group. See Subsection 7.3 for further discussion.An important corollary of the intermediate factor theorem is the theorem of Stuck and Zimmer discussed in Subsection 1.2 below. Moreover the factor theorem is a consequence of the intermediate factor theorem, and the normal subgroup theorem for groups having property (T ) is a consequence of the Stuck-Zimmer theorem. So these four results are related, implying and generalizing each other.We mention that the Nevo-Zimmer intermediate factor theorem has been extended in several directions since it first appeared; see for example the works of Bader-Shalom [4], Creutz-Peterson [7] and the above mentioned [21].1.2. The Stuck-Zimmer theorem. The following is a formulation of this theorem in the setting of linear algebraic groups over local fields.Theorem 1.2. Assume that rank k (G) ≥ 2 and G k has property (T). Then every faithful, properly ergodic, irreducible and finite measure preserving action of G k is essentially free.More generally, if the action has central kernel (but is possibly not faithful) then µ-almost every point has central stabilizer.For example as G = SL n is simply-connected the above theorem holds for the groups SL n (R), SL n (Q p ) and SL n (F q ((t)))...

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