Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Abstract

Let $\Gamma <\mathsf{SL}(d,\mathbb{R})$ be a torsion-free, Zariski-dense Borel-Anosov subgroup, and let $\phi$ be a linear functional positive on the limit cone of $\Gamma$. For every $\chi \in \mathrm{Hom}(\Gamma ,\mathbb{R})$, we construct twisted $(\phi ,\chi )$-conformal densities and prove their existence, uniqueness and ergodicity. For every non-trivial normal subgroup ${\Gamma }_0⊲\Gamma$ with nilpotent quotient, we classify the ergodic $\phi$-conformal densities of ${\Gamma }_0$ in terms of characters of $\Gamma /{\Gamma }_0$. Up to scaling, they are naturally parametrized by $\mathrm{Hom}(\Gamma /{\Gamma }_0,\mathbb{R})$, and their dimensions are precisely the critical exponents of the corresponding twisted Poincaré series. We also prove a shadow lemma for twisted densities of normal subgroups, with estimates uniform over densities with fixed parameters, and a twisted Hopf-Tsuji-Sullivan theorem.