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

Kashiwara, M.; Vergne, M.

Invent. Math. 44, 1-47 (1978).

MSC:

22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
22E30	Analysis on real and complex Lie groups
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
43A32	Other transforms and operators of Fourier type
43A85	Harmonic analysis on homogeneous spaces

References:

[1]	Anderson, R.L., Fischer, J., Raczka, R.: Coupling problems forU(p,q) ladder representations. Proc. Roy. Soc., Series A, vol.302, 491-500 (1968) · Zbl 0159.29103
[2]	Gelbart, S.: Harmonics on Stiefel manifolds and generalized Hankel transforms. B.A.M.S.78, 451-455 (1972) · Zbl 0235.43012
[3]	Gelbart, S.: Holomorphic discrete series for the real symplectic group. Inventiones math.19, 49-58 (1973) · Zbl 0236.22013
[4]	Gindikin, S.: Invariant generalized functions on homogeneous domains. Functional analysis and its applications9 (1), 56-58 (1975) · Zbl 0332.32022
[5]	Gross, K., Kunze, R.: Fourier Bessel transforms and holomorphic discrete series, Proceedings of the Maryland conference on Harmonic Analysis. Lecture Notes in Mathematics 266. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0361.22007
[6]	Gross, K., Kunze, R.: Bessel functions and representation theory II. To appear in J. Funct. Anal. · Zbl 0322.43014
[7]	Harish-Chandra: Representations of Semi-simple Lie groups. IV. Am. J. Math.77, 743-777 (1955), V. Am. J. Math.78, 1-41 (1956), VI. Am. J. Math.78, 564-628 (1956) · Zbl 0066.35603
[8]	Jacobsen, H., Vergne, M.: Wave and Dirac operators and representations of the conformal group. To appear in J. Funct. Anal.
[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) · Zbl 0346.17008
[10]	Levine, D.A.: Systems of singular integral operators on spheres. Trans. Am. Math. Soc.144, 493-522 (1969) · Zbl 0196.15803
[11]	Mack, G.: All unitary Ray representations of the conformal groupsSU(2,2) with positive energy. Preprint · Zbl 0352.22012
[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
[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) · Zbl 0085.21806
[14]	Shale, D.: Linear symmetrics of free boson fields. Trans. Am. Math. Soc.103, 149-167 (1962) · Zbl 0171.46901
[15]	Sternberg, S., Wolf, J.: Hermitian Lie algebras. Preprint · Zbl 0386.22010
[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
[17]	Strichartz, R.S.: Bochner identities for Fourier transforms. Trans. A.M.S.228, 307-327 (1977) · Zbl 0349.43007
[18]	Tuong-Ton-That: Lie group representations and harmonic polynomials of a matrix variable. Trans. Am. Math. Soc.216, 1-46 (1976)
[19]	Wallach, N.: Analytic continuation of the holomorphic discrete series II. To appear · Zbl 0419.22018
[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 · Zbl 0311.22008
[21]	Weil, A.: Sur certains groupes d’operateurs unitaires. Acta Math.111, 193-211 (1964) · Zbl 0203.03305
[22]	Duflo, M.: Representations unitaires irreductibles des groupes simples complexes de rank 2. Preprint
[23]	Howe, R.: Remark on classical invariant theory. Preprint · JFM 19.1236.02
[24]	Saito, M.: Representations Unitaires des groupes symplectiques. Journ. Math. Soc. of Japan24, 232-251 (1972) · Zbl 0232.22025