On the Benson-Ratcliff invariant of coadjoint orbits on nilpotent Lie groups (Q2471361)

Let \(G\) be a connected Lie group with Lie algebra \({\mathfrak g}\) and \({\mathfrak g}^*\) the dual vector space of \({\mathfrak g}\). The cohomology of the complex \(\wedge({\mathfrak g}^*)\) is denoted by \(H^*({\mathfrak g})\). Let \({\mathcal O}\subset{\mathfrak g}^*\) be a coadjoint orbit of \(G\) with dimension \(2q\). For any \(\ell\in{\mathcal O}\), regarded as an element of \(\wedge^1({\mathfrak g}^*)\), the differential form \(\ell\wedge(d\ell)^q\) is a closed form belonging to \(\wedge^{2q+1}({\mathfrak g}^*)\). \textit{C. Benson} and \textit{G. Ratcliff} [Mich. Math. J. 34, 23--30 (1987; Zbl 0618.22005)] proved that the cohomology class \([\ell\wedge(d\ell)^q]\in H^{2q+1}({\mathfrak g})\) is independent of the choice of \(\ell\in{\mathcal O}\). When \(G\) is an exponential solvable Lie group, every irreducible unitary representation \(\pi\) of \(G\) is uniquely associated with a coadjoint orbit \({\mathcal O}_\pi\) via the Kirillov-Bernat mapping. Let us define \[ i(\pi)= i({\mathcal O}_\pi)= [\ell\wedge (d\ell)^q]\in H^{2q+1}({\mathfrak g}),\quad \ell\in{\mathcal O}_\pi. \] In the paper cited above, \textit{C. Benson} and \textit{G. Ratcliff} presented the following conjecture. Let \(G\) be a connected and simply connected nilpotent Lie group with one-dimensional center. Let \(\ell\in{\mathfrak g}^*\) be a linear form dual to a basis element of the center and \(\pi_\ell\) the irreducible unitary representation of \(G\) corresponding to the coadjoint orbit \(G\cdot\ell\). Then \(i(\pi_\ell)\neq 0\). In this paper the authors first give a counterexample to this conjecture, then they study some cases where the conjecture holds. They also try to separate irreducible unitary representations of \(G\) by means of slightly modified \(i(\pi)\).