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On the Segal-Shale-Weil representations and harmonic polynomials. (English) Zbl 0375.22009

MSC:
22E45Analytic representations of Lie and linear algebraic groups over real fields
22E30Analysis on real and complex Lie groups
32M15Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (analytic spaces)
43A32Other transforms and operators of Fourier type
43A85Analysis on homogeneous spaces
References:
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[2]Gelbart, S.: Harmonics on Stiefel manifolds and generalized Hankel transforms. B.A.M.S.78, 451-455 (1972) · Zbl 0235.43012 · doi:10.1090/S0002-9904-1972-12941-0
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[9]Joseph, A.: The minimal orbit in a simple Lie algebra and its associated maximal Ideal. Annales Scientifiques de l’Ecole Normale Superieure9, 1-30 (1976)
[10]Levine, D.A.: Systems of singular integral operators on spheres. Trans. Am. Math. Soc.144, 493-522 (1969) · doi:10.1090/S0002-9947-1969-0412743-1
[11]Mack, G.: All unitary Ray representations of the conformal groupsSU(2,2) with positive energy. Preprint
[12]Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semi-simple group. Acta Mathematica136, 1-59 (1976) · Zbl 0356.32020 · doi:10.1007/BF02392042
[13]Segal, E.: Foundations of the theory of dynamical systems of infinitely many degrees of freedom. Mat. fys. Medd. Danske Vid. Selsk,31 (12), 1-39 (1959)
[14]Shale, D.: Linear symmetrics of free boson fields. Trans. Am. Math. Soc.103, 149-167 (1962) · doi:10.1090/S0002-9947-1962-0137504-6
[15]Sternberg, S., Wolf, J.: Hermitian Lie algebras. Preprint
[16]Strichartz, R.S.: The explicit Fourier decomposition ofL 2 (SO(n)/SO(n?m)). Can. J. Math.27, 294-310 (1975) · Zbl 0295.43011 · doi:10.4153/CJM-1975-036-x
[17]Strichartz, R.S.: Bochner identities for Fourier transforms. Trans. A.M.S.228, 307-327 (1977) · doi:10.1090/S0002-9947-1977-0433147-6
[18]Tuong-Ton-That: Lie group representations and harmonic polynomials of a matrix variable. Trans. Am. Math. Soc.216, 1-46 (1976) · doi:10.1090/S0002-9947-1976-0399366-1
[19]Wallach, N.: Analytic continuation of the holomorphic discrete series II. To appear
[20]Wallach, N.: On the unitarizability of Representations with Highest weights, non commutative Harmonic analysis. Lecture Notes, in Math.466. Berlin-Heidelberg-New York: Springer 1975
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[22]Duflo, M.: Representations unitaires irreductibles des groupes simples complexes de rank 2. Preprint
[23]Howe, R.: Remark on classical invariant theory. Preprint
[24]Saito, M.: Representations Unitaires des groupes symplectiques. Journ. Math. Soc. of Japan24, 232-251 (1972) · Zbl 0232.22025 · doi:10.2969/jmsj/02420232