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On the Benson-Ratcliff invariant of coadjoint orbits on nilpotent Lie groups. (English) Zbl 1143.22007
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 \(\wedge^{2q+1}({\mathfrak g}^*)\). C. Benson and 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, C. Benson and 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 \(i(\pi)\).

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22D10 Unitary representations of locally compact groups