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The convexity of the moment map of a Lie group. (La convexité de l’application moment d’un groupe de Lie.) (French) Zbl 0763.22006
Let \(G\) be a real Lie group with Lie algebra \(\mathfrak g\). Let \(\pi\) be a unitary representation of \(G\) in a Hilbert space \(\mathcal H\). Wildberger defined the moment map \(\Psi_ \pi: {\mathcal H}^ \infty\backslash\{0\}\to{\mathfrak g}^*\); \(\Psi_ \pi(\xi)(X)=-i\langle X\cdot\xi, \xi\rangle/\langle\xi,\xi\rangle\) \((X\in{\mathfrak g})\), where \({\mathcal H}^ \infty\) denotes the space of \(C^ \infty\)-vectors of \(\pi\). If \(G=\exp{\mathfrak g}\) is nilpotent and if \(\pi\) is irreducible, he showed that the moment set \(I_ \pi\) of \(\pi\), namely the closure in \({\mathfrak g}^*\) of the image of \(\Psi_ \pi\), is just the convex hull of the coadjoint \(G\)-orbit associated to \(\pi\) by the orbit method [N. J. Wildberger, Invent. Math. 98, 281-292 (1989; Zbl 0684.22005)].
In this paper, the authors study the moment set \(I_ \pi\) for the solvable and compact cases. Their main results are as follows. If \(G\) is connected solvable, \(I_ \pi\) is always convex and equal, if \(\pi\) is irreducible, to the closure of the convex hull of the Kirillov-Pukanszky orbit associated to \(\pi\) [cf. L. Pukanszky, Ann. Sci. Éc. Norm. Supér., IV. Sér. 4, 457-608 (1971; Zbl 0238.22010)]. A unitary representation \(\rho\) is said to be convex if \(I_ \rho\) is convex. Then there are two essential steps in their proof: the induction procedure preserves the convexity and the holomorphically induced representation is convex [cf. P. Bernat, et al., Représentations des groupes de Lie résolubles, Dunod, Paris (1972; Zbl 0248.22012)].
If \(G\) is connected compact semisimple and \(\pi\) is irreducible, then \(I_ \pi\) is the convex hull of the highest weight \(\Lambda\) of \(\pi\) if and only if \(\prod^ n_{j=1}\langle 2\Lambda-\alpha_ j, \alpha_ j\rangle\neq 0\), \(\alpha_ j\) \((1\leq j\leq n)\) denoting the simple roots of \(\mathfrak g\).

MSC:
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E25 Nilpotent and solvable Lie groups
22E46 Semisimple Lie groups and their representations
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