A general Kirillov theory for locally compact nilpotent groups (Q2905182)

The classical Kirillov theory proposes a correspondence between coadjoint orbits of a (connected, simply connected) nilpotent Lie group and its irreducible unitary representations. It has been extended to various other contexts, such as solvable Lie groups [\textit{L. Auslander} and \textit{B. Kostant}, Bull. Am. Math. Soc. 73, 692--695 (1967; Zbl 0203.03302)] and is considered as a fundamental principle in representation theory [\textit{D. Vogan}, Bull. Am. Math. Soc., New Ser. 42, No. 4, 535--544 (2012)].NEWLINENEWLINEThe paper under review discusses a general version of this method for a class of nilpotent locally compact groups, covering a variety of known examples.NEWLINENEWLINEFollowing ideas of \textit{R. E. Howe} [Pac. J. Math. 73, 307--327 (1977; Zbl 0396.43013)], the authors generalize the notion of Lie groups by generalizing the one of Lie algebras. To do so, they introduce the notion of \(k\)-Lie pairs \((G,\mathfrak{g})\) for \(k\in\mathbb{N}\cup\{\infty\}\) in which \(G\) is a locally compact nilpotent group of nilpotence length \(\leq k\) and \(\mathfrak{g}\) is a Lie algebra over the ring \(\Lambda_k=\mathbb{Z}\left[\frac{1}{k!}\right]\) (resp. \(\Lambda_k=\mathbb{Q}\)) if \(k\in\mathbb{N}\) (resp. if \(k=\infty\)), such that \(G\) and \(\mathfrak{g}\) can be identified \textit{via} a bijective homeomorphism \(\exp:\mathfrak{g}\rightarrow G\) satisfying the Campbell-Hausdorff formula.NEWLINENEWLINEIf, moreover, there exists a locally compact \(\Lambda_k\)-module \(\mathfrak{m}\) together with a basic character \(\epsilon:\mathfrak{m}\rightarrow \mathbb{T}\) such that the dual \(\mathfrak{g}^*=\Hom_{\Lambda_k}(\mathfrak{g},\mathfrak{m})\) is isomorphic to the Pontrjagin dual \(\hat{\mathfrak{g}}\) of \(\mathfrak{g}\) \textit{via} \(f\mapsto \epsilon\circ f\), then \((G,\mathfrak{g})\) is said to be \((\mathfrak{m},\epsilon)\)-dualizable.NEWLINENEWLINEUnder these hypotheses, the authors introduce the concept of a standard polarizing subalgebra \(\mathfrak{r}\) of \(\mathfrak{g}\) for \(f\in\mathfrak{g}^*\), such that \(f\) defines a character \(\varphi_f\) of \(R=\exp(\mathfrak{r})\). They prove that the kernel \(P_f\) of the induced representation \(\pi_f=\mathrm{ind}_R^G\,\varphi_f\) is independent of the choice of \(\mathfrak{r}\), thus defining a map \(\kappa\) from \(\mathfrak{g}^*\) to the primitive ideal space \(\mathrm{Prim}(G)\) of the group \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G)\).NEWLINENEWLINEDenoting by \(\mathrm{Ad}^*\) the canonical coadjoint action \(G\rightarrow\mathrm{GL}(\mathfrak{g}^*)\), two elements \(f,f'\) in \(\mathfrak{g}^*\) are said to lie in the same \(\mathrm{Ad}^*(G)\)-quasi-orbit if \(f\in\overline{\mathrm{Ad}^*(G)f'}\) and \(f'\in\overline{\mathrm{Ad}^*(G)f}\), and one then writes \(f\sim f'\). It is proved that \(\kappa\) is constant on coadjoint quasi-orbits, which allows to define the Kirillov-orbit map by NEWLINE\[NEWLINE\tilde{\kappa}:\mathfrak{g}^*/\sim\longrightarrow\mathrm{Prim}(G);\quad [f] \longmapsto P_f=\ker\left(\pi_f\right).NEWLINE\]NEWLINE One of the main results in the article is that \(\tilde{\kappa}\) is continuous and surjective if \((G,\mathfrak{g})\) is an \((\mathfrak{m},\epsilon)\)-dualizable nilpotent \(k\)-Lie pair. Using ideas of \textit{K. I. Joy} [Pac. J. Math. 112, 135--139 (1984; Zbl 0535.22008)], the authors also prove that \(\tilde{\kappa}\) is a homeomorphism under the additional assumption that \((G,\mathfrak{g})\) is regular.NEWLINENEWLINEA further question consists in determining under which conditions the Kirillov orbit method not only computes the primitive ideals but also the irreducible representations of \(G\). The classical theory of (group) \(\mathrm{C}^*\)-algebras [\textit{J. Dixmier}, \(C^*\)-algebras. North-Holland Mathematical Library. Vol. 15. Amsterdam - New York - Oxford: North-Holland Publishing Company (1977; Zbl 0372.46058)] shows that a unitary irreducible representation \(\pi\) of \(G\) is completely determined by the kernel of the corresponding representation of \(\mathrm{C}^*(G)\) if and only if it contains the algebra \(\mathcal{K}(H_\pi)\) of compact operators on the carrying Hilbert space \(H_\pi\). Such representations are called GCR (or postliminal). If the converse inclusion holds, so that \(\pi\left(\mathrm{C}^*(G)\right)=\mathcal{K}(H_\pi)\), then \(\pi\) is called CCR (or liminal). Equivalently, \(\pi\) is GCR (resp. CCR) if and only if the singleton \(\{\pi\}\) is locally closed (resp. closed) in the dual space of \(\mathrm{C}^*(G)\), equipped with the Fell-Jacobson topology.NEWLINENEWLINEIn the present context, the authors prove that if \((G,\mathfrak{g})\) is a regular \((\mathfrak{m},\epsilon)\)-dualizable nilpotent \(k\)-Lie pair and \(f\in\mathfrak{g}^*\) is such that \(\mathrm{Ad}^*(G)f\) is locally closed (resp. closed) in \(\mathfrak{g}^*\), then \(\pi_f\) is GCR (resp. CCR). In particular, if all the coadjoint orbits are locally closed in \(\mathfrak{g}^*\), the authors obtain a well-defined homeomorphism NEWLINE\[NEWLINE\widehat{\kappa}:\mathfrak{g}^*/\mathrm{Ad}^*(G)\longrightarrow\widehat{G};\quad \mathrm{Ad}^*(G)f \longmapsto \pi_f.NEWLINE\]NEWLINENEWLINENEWLINEThe version of Kirillov's theory developed in this article is general enough to cover a wide class of nilpotent locally compact groups, containing in particular:NEWLINENEWLINE- connected and simply connected real nilpotent Lie groups, i.e. the classical situation studied by \textit{A. A. Kirillov} [Russ. Math. Surv. 17, No. 4, 53--104 (1962); translation from Usp. Mat. Nauk 17, No. 4(106), 57--110 (1962; Zbl 0106.25001)];NEWLINENEWLINE- unipotent groups over \(\mathbb{Q}_p\), studied by \textit{C. Moore} in [Ann. Math. (2) 82, 146--182 (1965; Zbl 0139.30702)] and by \textit{M. Boyarchenko} and \textit{M. Sabitova} in [Isr. J. Math. 165, 67--91 (2008; Zbl 1165.20022)];NEWLINENEWLINE- quasi-\(p\) groups with small nilpotence length, studied by \textit{R. E. Howe} [Pac. J. Math. 73, 307--327 (1977; Zbl 0396.43013)];NEWLINENEWLINE- countable torsion-free divisible groups, studied by \textit{A. Carey, W. Moran} and \textit{C. Pearce} in [Math. Proc. Camb. Phil. Soc. 96, 123--137 (1984; Zbl 0549.43004)] and [Math. Ann. 301, No. 1, 119--133 (1995; Zbl 0832.20054)].NEWLINENEWLINEAnother extension of Kirillov's method, also adopting the point of view of primitive ideals in \(\mathrm{C}^*\)-algebras and methods may be found in [\textit{D. L. Moore}, Kirillov theory for \(\mathrm{C}^*(G,\Omega)\). PhD Thesis. University of Colorado (1995)] in the case of group transformation \(\mathrm{C}^*\)-algebras associated to nilpotent Lie groups, following the ideas of \textit{D. Williams} [Trans. Am. Math. Soc. 266, 335--359 (1981; Zbl 0474.46057)] and \textit{S. Echterhoff} [Math. Ann. 292, No. 1, 59--84 (1992; Zbl 0739.22006)].