The type I dichotomy for two-step nilpotent locally compact groups
Pierre-Emmanuel Caprace, Max Carter
TL;DR
The paper develops a purely algebraic criterion for when a two-step nilpotent locally compact group $G$ is of type I, tying this property to closed images of the central-character maps $\omega_\chi: G/Z \to \widehat{G/Z}$. Building on Poguntke’s parametrization and Baggett–Kleppner’s results, it shows that $G$ is type I (equivalently CCR) exactly when these images are closed for all central characters $\chi$, with the criterion automatically satisfied in several natural bilinear-commutator settings over non-discrete locally compact fields. It applies this framework to a variety of contexts, including Heisenberg-type groups, unipotent radicals in rank-one groups, and a broad class of contraction groups, providing both type I and non-type I examples (via monomial commutation relations and the Glöckner–Willis construction). A notable contribution is a method to embed non-type I groups as closed cocompact normal subgroups inside type I groups through explicit extensions, following Chirvasitu’s approach. The results illuminate the interaction between algebraic structure and representation theory across characteristic, geometry of groups, and dynamics, with concrete implications for algebraic groups over local fields and $p$-torsion phenomena.
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.