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

Let $G$ be a closed type-preserving subgroup of the automorphism group of a thick locally finite building $X$ of finite rank, and assume that $G$ acts Weyl-transitively. We prove that every unitary representation of $G$ is mixing, unless its restriction to a parabolic subgroup of minimal non-spherical type is amenable in the sense of Bekka. It follows that every unitary representation of $G$ that is weakly contained in the regular representation, is mixing. In case $X$ is of minimal non-spherical type and its thickness satisfies some modest lower bound, we deduce that $G$ has the Howe–Moore property provided its only compact quotient is trivial. We also obtain results on rigidity of invariant random subgroups for Kac–Moody lattices of compact hyperbolic type, yielding examples of infinite finitely presented Kazhdan groups with exactly two ergodic invariant random subgroups.