Characters of the discrete Heisenberg group and of its completion

Abstract

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

Induced characters, Mackey analysis and primitive ideal spaces of nilpotent discrete groups

“…Proposition 1.5 below, which generalizes Theorem 4.7 of [28], in particular shows that the two definitions coincide when G is 2-step nilpotent. Following [8], we call a nilpotent group centrally inductive if every character φ of G vanishes outside of G f (φ).…”

“…The following lemma generalizes [28,Lemma 5.4]. N be a normal subgroup of G which is contained in Z 2 (G) and let φ be a character of G. Then there exist ψ ∈ Ch(N ) and ϕ ∈ Ch(S ψ ) such that ϕ| N = ψ and ind G S ψ ϕ = φ.…”

“…Induced traces were first introduced, using invariant means on the space of bounded functions, for 2-step nilpotent groups in [28]. Proposition 1.5 below, which generalizes Theorem 4.7 of [28], in particular shows that the two definitions coincide when G is 2-step nilpotent.…”

Spectrum of a homogeneous graph

“…The characters of G are known (cf. [1,11,15]). Let R/Z be the real numbers mod Z and denote an element of G by (m, n, p).…”

“…Let R/Z be the real numbers mod Z and denote an element of G by (m, n, p). As in [11,Corollary 6.5] or [15], G contains, among others, the one-dimensional unitary representations We have …”