On the Howe--Moore property for automorphism groups of buildings

Let $G<Aut(X)$ be a totally disconnected locally compact group acting strongly transitively on a locally finite building $X$ of finite-rank and minimal non-spherical type. For sufficiently large thickness, every weakly mixing strongly continuous unitary representation of $G$ is $C_0$. Consequently, if $G$ has no non-trivial finite-dimensional unitary representations, then $G$ has the Howe--Moore property. More concretely, this applies to rank-three compact-hyperbolic crystallographic types of thickness $q+1$ for $q\geq 19379$, if there are no compact quotients. As an application, we prove that the corresponding Caprace--R\'emy Kac--Moody lattices in these types, which are known to be finitely presented simple and Kazhdan, are character-rigid: their extremal characters are only the regular and the trivial character. Consequently they also have no non-trivial invariant random subgroups.