The Nevo-Zimmer intermediate factor theorem over local fields

TL;DR

This paper extends the Nevo-Zimmer intermediate factor theorem and the Stuck-Zimmer theorem from higher rank semisimple Lie groups over real numbers to linear groups over arbitrary local fields, broadening their applicability.

Contribution

It provides a new proof of the Nevo-Zimmer theorem that enables the extension of these classification results to linear groups over any local field.

Findings

Extension of the Nevo-Zimmer theorem to local fields

Extension of the Stuck-Zimmer theorem to local fields

New proof technique for intermediate factor classification

Abstract

The Nevo-Zimmer theorem classifies the possible intermediate $G$-factors $Y$ in $X \times G/P \to Y \to X$, 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.