A Kirillov theory for divisible nilpotent groups
Let \(N\) be a connected and simply connected nilpotent Lie group with Lie algebra \(\mathfrak n\). \(N\) acts on the linear dual \({\mathfrak n}^*\) of \(\mathfrak n\) by the coadjoint action and \textit{A. A. Kirillov} [Usp. Mat. Nauk 17, 57-110 (1962; Zbl 0106.250)] constructed a bijection between the orbit space \({\mathfrak n}^*/N\) and the unitary dual \(\widehat {N}\) of \(N\). It turned out [cf. \textit{I. Brown}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 6(1973), 407-411 (1974; Zbl 0284.57026)] that this bijection is a homeomorphism between quotient topology and hull-kernel topology. Attempts to extend this Kirillov theory to non-Lie groups encounter two main problems: non type I-ness and finding a suitable substitute for the Lie algebra. Here the authors present a Kirillov theory for divisible nilpotent groups. A group \(G\) is divisible if \(x^n = a\) has a solution for all \(n\) and all \(a \in G\). \(G\) is complete if the solution is unique. It is known that nilpotent groups are complete if and only if they are torsion- free and divisible. The results are first shown for complete groups and then for divisible groups by using the fact that every divisible nilpotent group \(G\) is a quotient of a complete nilpotent group \(H\) by a central group \(K\). The main results are as follows. There is a Kirillov theory for \(G\) in the sense that \(\text{Prim }G\), the primitive ideal space of the \(C^*\)-algebra of \(G\), is homeomorphic to the space of \(K\)-integral \(G\) quasi-orbits in the dual \({\mathfrak h}^*\) of the suitably defined Lie algebra \(\mathfrak h\) of \(H\). Plancherel measure for \(G\) is canonically identified with Haar measure \(\mu\) on \((Z(H)/K)^\wedge\). In fact, with respect to \(\mu\) almost all characters of \(G\) are zero off \(Z(H)/K\) and are faithful circle-valued homomorphisms on \(Z(H)/K\).
scientific article; zbMATH DE number 734582
scientific article