Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Abstract

Let $Γ<\mathsf{SL}(d,\mathbb R)$ be a Zariski-dense Borel-Anosov subgroup and let $φ$ be positive on its limit cone. For every $χ\in \mathrm{Hom}(Γ, \mathbb R)$, we construct twisted $(φ,χ)$-conformal densities and prove their uniqueness, atomlessness, and ergodicity. For every normal subgroup $Γ_0\lhdΓ$ with nilpotent quotient, we classify the ergodic $φ$-conformal densities of $Γ_0$ in terms of characters of $Γ/Γ_0$.