Nondivergence of Reductive group action on Homogeneous Spaces
Han Zhang, Runlin Zhang
TL;DR
This work characterizes when a reductive subgroup $H$ acts non-divergently on $X=G/\Gamma$ with $\Gamma$ arithmetic by providing equivalent algebraic criteria tied to parabolic subgroups, Weyl-group weights, and instability in rational representations. The authors develop a novel covering-theory approach, integrating Dahl–Margulis non-divergence for unipotent flows with a topological argument to reduce the problem to explicit algebraic obstructions such as the existence of a $g$-conjugate placing $H$ in a Q-parabolic with linearly dependent fundamental weights, or the non-$\mathbb{Q}$-anisotropy of centralizers. They further treat the real rank-one case, proving the criterion holds in the non-arithmetic setting via Garland–Raghunathan reduction theory and obtaining a compact-core lemma useful in related geometric analyses. The results extend classical nondivergence phenomena for unipotent and torus actions to broader reductive settings, offering concrete, checkable conditions for uniform non-divergence with potential applications to counting, equidistribution, and rigidity phenomena on homogeneous spaces.
Abstract
Let $X=G/Γ$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence of a fixed compact set in $X$ intersecting \textit{every} $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting. When $G$ is of real rank one, $Γ$ is a non-cocompact lattice of $G$ and $H<G$ is an algebraic group, we also obtain an algebraic condition on $H$ which is equivalent to the return of \textit{every} $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice.