\(*\)-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 \(L^G\).


22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
53D50 Geometric quantization
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[1] Arnal, D.), Cahen, M.) et Gutt, S.). - Representation of compact Lie groups and quantisation by deformation. Académie Royale de Belgique. Bulletin de la classe des sciences. 5 série - Tome LXXIV (1988), pp. 123-141. · Zbl 0681.58016
[2] Arnal, D.), Cortet, J.C.). - *-product in the method of orbits for nilpotent groups. J. Geom. and Phys.2,2, p. 85-116 (1985). · Zbl 0599.22012
[3] Bayen, F.), Flato, M.), Fronsdal, C.), Lichnérowicz, A.), Sternheimer, D.). - Deformation Theory and quantization. Ann. of physics.111 (1978), pp. 61-110 et 111-151. · Zbl 0377.53025
[4] Ben Ammar, M.) et Tlili, M.H.). - Produit * sur le fibré tangent du groupe U(n). Travaux Mathématiques. Luxembourg. Fascicule X (1998), pp. 1-13. · Zbl 0930.22010
[5] Gutt, S.). - An explicit *-product on the cotangent bundle of a Lie group. Letters in Mathematical physics, 7 (1983), pp. 249-258. · Zbl 0522.58019
[6] Helgason, S.). - Differentiel Geometry, Lie groups, and Symmetric spaces. Academic Press, New York (1978). · Zbl 0451.53038
[7] Pukanszky, L.). - Leçons sur les représentations des groupes. Monographies de la Société Mathématique de France. DunodParis (1967). · Zbl 0152.01201
[8] Pontryagin, L.S.). - Topological groups. Edition Gordon and Breach, New York (1966).
[9] Theodor, B.) and Tammo, D.). - Representation of compact Lie groups. Springer Verlag, Berlin (1998). · Zbl 0581.22009
[10] Zahir, H.). - Représentation *-régulière des groupes de Lie compacts. Thèse de l’université de Metz (1992). · Zbl 0790.22009
[11] Varadarajan, V.S.). - Lie groups, Lie algebras, and their representations. Springer Verlag, Berlin (1984). · Zbl 0955.22500
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