The non-abelian Leopoldt conjecture and equalities of $\mathcal{L}$-invariants

Let $G$ be a reductive group quasi-split at $p$. Using arguments of Hansen–Thorne, we show that under the non-abelian Leopoldt conjecture (NALC), Hansen’s $p$-adic overconvergent cohomology eigenvariety for $G$ is étale over its image in weight space at any non-critical classical tempered cuspidal point of `cohomological multiplicity one’. This applies to all non-critical classical cuspidal points if $G={\mathrm{R}\mathrm{e}\mathrm{s}}_{F\mathrm{/}\mathbb{Q}}{\mathrm{G}\mathrm{L}}_n$. We then let $\pi$ be a $p$-ordinary regular algebraic cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_n({\mathbb{A}}_{\mathbb{Q}})$ such that ${\pi }_p$ is Steinberg. Combining the above étaleness result for the classical point attached to $\pi$, and a local-global compatibility result from our earlier work, we deduce – under a tangent vector hypothesis that is true for at least half the simple roots – the equality of Fontaine–Mazur and automorphic $\mathcal{L}$-invariants for $\pi$. Where this assumption is satisfied, we deduce the NALC implies a conjecture of Gehrmann: that automorphic $\mathcal{L}$-invariants are independent of cohomological degree. Our approach is inspired by (and generalises) previous work of Gehrmann–Rosso. When $\pi ={\mathrm{Sym}}^{n-1}{\pi }_f$ is the symmetric power lift of a modular form, we verify all assumptions other than the NALC, and deduce a functoriality result for the automorphic $\mathcal{L}$-invariants.