## $$*$$-representations of semisimple compact Lie groups. (Représentation $$*$$ des groupes de Lie compacts semi simples.)(French)Zbl 0997.22009

For a Hermitian space $$E$$, on the space $$M(E,\mathbb C)$$ identified with the complexification of the Lie algebra $$\mathfrak u(n)$$, the author considers the covariant Moyal product and a representation of the unitary group $$U(E)$$ on $$L^2(M(E,\mathbb C))$$. All arbitrary compact semisimple Lie groups $$G$$ can be identified under some faithful unitary representation with the unitary groups $$U(V)$$ in some Hermitian space $$V$$. Chosen $$E = \wedge V$$, one considers the representation $$L^G$$ of $$G$$ in $$L^2(E\otimes E) = L^2(\mathfrak u(E)^\mathbb C)$$. The main result (Proposition 5): All the irreducible unitary representations of $$G$$ can be found in $$L^G$$.

### MSC:

 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 53D50 Geometric quantization

### Keywords:

representation; geometric quantization; Moyal product
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### References:

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