An orthogonal family of polynomials on the generalized unit disk and ladder representations of 𝑈(𝑝,𝑞)

Lorch, John

doi:10.1090/S0002-9939-98-04506-7

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ISSN 1088-6826(online) ISSN 0002-9939(print)

An orthogonal family of polynomials
on the generalized unit disk
and ladder representations of

Author: John D. Lorch
Journal: Proc. Amer. Math. Soc. 126 (1998), 3755-3762
MSC (1991): Primary 22E45, 22E70; Secondary 32L25, 32M15, 58G05, 81R05, 81R25
MathSciNet review: 1458255
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Abstract: Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk ${\bf D}_{p,q} =U(p,q)/K$ . This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on ${\bf D}_{p,q}$ . These polynomials, which play a crucial role in the present work, are shown to be orthogonal with respect to Lebesgue measure, and their norms are computed. The orthogonal family is then used to invert a certain integral transform, giving the desired inner product structures.

1. Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 0463358 (57 #3310)
2. V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 0157250 (28 #486)
3. T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, 1978. Mathematics and its Applications, Vol. 13. MR 0481884 (58 #1979)
4. Mark G. Davidson, The harmonic representation of 𝑈(𝑝,𝑞) and its connection with the generalized unit disk, Pacific J. Math. 129 (1987), no. 1, 33–55. MR 901255 (89c:22021)
5. Mark G. Davidson, Thomas J. Enright, and Ronald J. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. 94 (1991), no. 455, iv+102. MR 1081660 (92c:22034), http://dx.doi.org/10.1090/memo/0455
6. 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 (90m:32049), http://dx.doi.org/10.1016/0022-1236(90)90119-6
7. Harish-Chandra, Representations of semisimple Lie groups. V, Amer. J. Math. 78 (1956), 1–41. MR 0082055 (18,490c)
8. Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767 (86c:22017)
9. L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I., 1963. MR 0171936 (30 #2162)
10. J. Lorch, Unitary structures for ladder representations of , Ph.D. thesis, Oklahoma State University, 1995.
11. John D. Lorch and Lisa A. Mantini, Inversion of an integral transform and ladder representations of 𝑈(1,𝑞), Representation theory and harmonic analysis (Cincinnati, OH, 1994), Contemp. Math., vol. 191, Amer. Math. Soc., Providence, RI, 1995, pp. 117–138. MR 1365539 (96m:22022), http://dx.doi.org/10.1090/conm/191/02332
12. J. Lorch, An integral transform and ladder representations of , Pacific J. Math., in press.
13. Lisa A. Mantini, An integral transform in 𝐿²-cohomology for the ladder representations of 𝑈(𝑝,𝑞), J. Funct. Anal. 60 (1985), no. 2, 211–242. MR 777237 (87a:22029), http://dx.doi.org/10.1016/0022-1236(85)90051-5
14. Lisa A. Mantini, An 𝐿²-cohomology construction of negative spin mass zero equations for 𝑈(𝑝,𝑞), J. Math. Anal. Appl. 136 (1988), no. 2, 419–449. MR 972147 (90c:22040), http://dx.doi.org/10.1016/0022-247X(88)90095-9
15. M. S. Narasimhan and K. Okamoto, An analogue of the Borel-Weil-Bott theorem for hermitian symmetric pairs of non-compact type, Ann. of Math. (2) 91 (1970), 486–511. MR 0274657 (43 #419)
16. R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR 0318398 (47 #6945)
17. I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Translated from the Russian. Mathematics and Its Applications, Vol. 8, Gordon and Breach Science Publishers, New York, 1969. MR 0252690 (40 #5908)
18. Walter Rudin, Function theory in the unit ball of 𝐶ⁿ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York, 1980. MR 601594 (82i:32002)
19. Gabor Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939. American Mathematical Society Colloquium Publications, v. 23. MR 0000077 (1,14b)