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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:
 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 2.2e+26 Nilpotent and solvable Lie groups 2.2e+47 Semisimple Lie groups and their representations
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##### References:
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