The type I dichotomy for two-step nilpotent locally compact groups

Abstract

We address the type I dichotomy for two-step nilpotent locally compact groups. Invoking work of Baggett-Kleppner, we characterize the closed points of the unitary dual of such a group $G$ purely in terms of the group structure. An algebraic criterion characterizing when $G$ is a type I group is derived. We show that this criterion automatically holds if $G$ is a central extension of vector groups over a non-discrete locally compact field $k$ such that the commutator map is $k$-bilinear. As an application, we show that the unipotent radicals of minimal parabolics in simple algebraic groups of $k$-rank one are type I groups. We also discuss the type I dichotomy for $p$-torsion contraction groups, and exhibit, for each prime $p$, uncountably many pairwise non-isomorphic such groups that are not type I. This answers a recently posed question by the second author. Finally, we adapt a recent construction of Chirvasitu to obtain numerous examples of two-step nilpotent torsion locally compact groups that are not type I, but that embed as closed cocompact normal subgroups in two-step nilpotent groups that are type I.