Seminar

To participate in this seminar, please register here: https://www.msri.org/seminars/25205

This is one of the research seminars for the RAS program, that distinguishes itself from the postdocs and program associates seminars in that speakers are chosen among Research Members, Research Professors with occasional outside speakers.

Abstract:

Let $G$ be a connected semisimple real algebraic group and $\Gamma<G$ be a Zariski dense discrete subgroup. When the homogeneous space $\Gamma\backslash G$ has finite volume, horospherical invariant ergodic measures on $\Gamma\backslash G$ are completely classified by Dani. In the infinite volume case, the complete classification is known only when $G$ has rank one, and $\Gamma$ is either geometrically finite or co-abelian subgroups of uniform lattices. For example, when $\Gamma$ is convex cocompact, the Burger-Roblin measure is the unique non-trivial horospherical invariant “ergodic” measure. In our lecture, we discuss horospherical invariant ergodic measures when $\Gamma<G$ is an Anosov subgroup, which is a higher rank generalization of a convex cocompact subgroup. We propose to call the associated space $\Gamma\backslash G$ an Anosov homogeneous space. Letting $N$ a maximal horospherical subgroup of $G$ and $P=MAN$ its normalizer subgroup with Langlands decomposition, we show that the space of non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures, up to proportionality, is homeomorphic to ${\mathbb R}^{\rm{rank} G -1}$. The homeomorphism is established by assigning a Burger-Roblin measure to every direction in the interior of the limit cone of $\Gamma$. The main key point is to show that each Burger-Roblin measure is ergodic for the $NM$-action. This is joint work with Minju Lee.

Invariant Measures For Horospherical Actions On Anosov Homogeneous Spaces