Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces

Ding, Hongming

doi:10.1090/S0002-9947-99-02461-7

Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces
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Trans. Amer. Math. Soc. 351 (1999), 4205-4243 Request permission

Abstract:

This paper defines $\pi$-weighted Laplace transforms on the spaces of $\pi$-covariant functions. By the inverse Laplace transform we define operator-valued Bessel functions. We also study the holomorphic discrete series of the automorphism group of a Siegel domain of type II.

References

Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Salomon Bochner, Lectures on Fourier integrals. With an author’s supplement on monotonic functions, Stieltjes integrals, and harmonic analysis, Annals of Mathematics Studies, No. 42, Princeton University Press, Princeton, N.J., 1959. Translated by Morris Tenenbaum and Harry Pollard. MR 0107124
Hacen Dib, Fonctions de Bessel sur une algèbre de Jordan, J. Math. Pures Appl. (9) 69 (1990), no. 4, 403–448 (French). MR 1076516
Hongming Ding, Operator-valued Bessel functions on Schrödinger-Fock spaces and Siegel domains of type $\textrm {II}$, J. Funct. Anal. 106 (1992), no. 1, 189–225. MR 1163469, DOI 10.1016/0022-1236(92)90068-T
Hongming Ding, Operator-valued Bessel functions, holomorphic discrete series, and harmonic representations of $\textrm {U}(p,q)$, J. Funct. Anal. 108 (1992), no. 1, 187–230. MR 1174162, DOI 10.1016/0022-1236(92)90150-H
Hongming Ding, Bessel functions on quaternionic Siegel domains, J. Funct. Anal. 120 (1994), no. 1, 1–47. MR 1262244, DOI 10.1006/jfan.1994.1021
Hongming Ding, Hilbert spaces of vector-valued holomorphic functions and irreducibility of multiplier representations, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 119–122. MR 1199123, DOI 10.1090/conm/138/1199123
Hongming Ding and Kenneth I. Gross, Operator-valued Bessel functions on Jordan algebras, J. Reine Angew. Math. 435 (1993), 157–196. MR 1203914
—, Operator-valued Bessel functions on Jordan algebras, II: Analytic continuation, in preparation.
J. Dorfmeister, Quasisymmetric Siegel domains and the automorphisms of homogeneous Siegel domains, Amer. J. Math. 102 (1980), no. 3, 537–563. MR 573102, DOI 10.2307/2374115
Josef Dorfmeister, Homogeneous Siegel domains, Nagoya Math. J. 86 (1982), 39–83. MR 661219
Thomas Enright, Roger Howe, and Nolan Wallach, A classification of unitary highest weight modules, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97–143. MR 733809
J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), no. 1, 64–89. MR 1033914, DOI 10.1016/0022-1236(90)90119-6
Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489
S. G. Gindikin, Analysis on homogeneous domains, Russian Math. Surveys 19 (1964), 1–90.
Kenneth I. Gross and Donald St. P. Richards, Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions, Trans. Amer. Math. Soc. 301 (1987), no. 2, 781–811. MR 882715, DOI 10.1090/S0002-9947-1987-0882715-2
Kenneth I. Gross and Ray A. Kunze, Bessel functions and representation theory. I, J. Functional Analysis 22 (1976), no. 2, 73–105. MR 0415214, DOI 10.1016/0022-1236(76)90015-x
Kenneth I. Gross and Ray A. Kunze, Bessel functions and representation theory. II. Holomorphic discrete series and metaplectic representations, J. Functional Analysis 25 (1977), no. 1, 1–49. MR 0453928, DOI 10.1016/0022-1236(77)90030-1
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
Roger Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535–552. MR 985172, DOI 10.1090/S0894-0347-1989-0985172-6
Alan T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964), 475–501. MR 181057, DOI 10.1214/aoms/1177703550
Soji Kaneyuki, Homogeneous bounded domains and Siegel domains, Lecture Notes in Mathematics, Vol. 241, Springer-Verlag, Berlin-New York, 1971. MR 0338467
M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. MR 463359, DOI 10.1007/BF01389900
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
M. Koecher, Jordan algebras and their applications, Lecture notes, Univ. of Minn., 1962.
R. A. Kunze, On the irreducibility of certain multiplier representations, Bull. Amer. Math. Soc. 68 (1962), 93–94. MR 133405, DOI 10.1090/S0002-9904-1962-10735-6
O. Loos, Bounded symmetric domains and Jordan pairs, Lecture notes, Univ. of California, Irvine, 1977.
Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721
Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936
Shingo Murakami, On automorphisms of Siegel domains, Lecture Notes in Mathematics, Vol. 286, Springer-Verlag, Berlin-New York, 1972. MR 0364690
I. I. Pyatetskii–Shapiro, Automorphic functions and the geometry of classical domains, Fizmatgiz, New York, 1961; English transl., New York, 1969.
Hugo Rossi and Michèle Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Functional Analysis 13 (1973), 324–389. MR 0407206, DOI 10.1016/0022-1236(73)90056-6
M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1976), no. 1-2, 1–59. MR 480883, DOI 10.1007/BF02392042
I. E. Segal, Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand. 13 (1963), 31–43. MR 163183, DOI 10.7146/math.scand.a-10686
David Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149–167. MR 137504, DOI 10.1090/S0002-9947-1962-0137504-6
Nolan R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. MR 531967, DOI 10.1090/S0002-9947-1979-0531967-2
Nolan R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. MR 531967, DOI 10.1090/S0002-9947-1979-0531967-2
George H. Weiss, A note on a generalized elliptic integral, J. Res. Nat. Bur. Standards Sect. B 68B (1964), 1–2. MR 165150