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\(*\)-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
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References:

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