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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 \(i(\pi)\).

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