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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