Separation of unitary representations of certain Cartan motion groups (Q2831502)

Let \(G\) be a Lie group, \((\pi,\mathcal{H}_\pi)\) an irreducible unitary representation of \(G\) and \(\mathcal{H}_\pi^{+\infty}\) the space of smooth vectors of \(\pi\). In [Invent. Math. 98, No. 2, 281--292 (1989; Zbl 0684.22005)], \textit{N. J. Wildberger} defined the moment map \(\psi_\pi\) of \(\pi\). For \(\xi\in \mathcal{H}_\pi^{+\infty}\setminus\{0\}\) and \(x\in \mathfrak{g}\) one has: \(\psi_\pi(\xi)(x)=\frac{1}{i}\frac{\langle d\pi(x)\xi,\xi\rangle}{\|\xi\|^2}\), where \(d\pi\) denotes the derivative of the representation \(\pi\). Let \(\widetilde{\psi}_\pi\) be the generalized moment map defined on the enveloping algebra \(U(\mathfrak{g}^{\mathbb{C}})\) of \(\mathfrak{g}\) by \(\widetilde{\psi}_\pi(\xi)(A)=\mathrm{Re}\{\frac{1}{i}\frac{\langle d\pi(A)\xi,\xi \rangle}{\|\xi\|}^2\}\) and \(J_\pi\) the connex hull of the image of \(\widetilde{\psi}_\pi\). Let \(G\) be an exponential solvable Lie group, the author of this review with \textit{D. Arnal} et al. [J. Lie Theory 10, No. 2, 399--410 (2000; Zbl 0962.22005)] proved that for \(\pi\) and \(\rho\in \widehat{G}\), one has \(\pi\sim \rho\Leftrightarrow J_{\pi}=J_{\rho}\). In the present paper, this question is considered in the context of Cartan motion groups associated to a pair \((G,K)\), where \(G\) is a semisimple connected Lie group of finite center and \(K\) a maximal compact connected subgroup of \(G\). The main result consist in showing that for \(\pi,\,\rho\in \widehat{G}\), \(\pi\simeq \rho\Leftrightarrow \widetilde{\psi}_\pi(\xi_\pi)=\widetilde{\psi}_\rho(\xi_\rho)\), where \(\xi_{\pi}\) and \(\xi_{\rho}\) denote some special vectors in \(\mathcal{H}_\pi^\infty\setminus \{0\}\).