2025/09/30 by Caprace, Pierre-Emmanuel, Carter, Max
#22D25 #43A65 #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Primary: 22D10 #Representation Theory (math.RT) #Secondary: 20G05
paper · doi:10.48550/arxiv.2509.26212
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.