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

An orthogonal family of polynomials on the generalized unit disk and ladder representations of $U(p,q)$
HTML articles powered by AMS MathViewer

by John D. Lorch PDF

Proc. Amer. Math. Soc. 126 (1998), 3755-3762 Request permission

Abstract:

Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk $\textbf {D}_{p,q} =U(p,q)/K$. This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on $\textbf {D}_{p,q}$. 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.

References

Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 463358, DOI 10.1007/BF01389783
V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 157250, DOI 10.1002/cpa.3160140303
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
Mark G. Davidson, The harmonic representation of $\textrm {U}(p,q)$ and its connection with the generalized unit disk, Pacific J. Math. 129 (1987), no. 1, 33–55. MR 901255, DOI 10.2140/pjm.1987.129.33
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, DOI 10.1090/memo/0455
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
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
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
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR 0171936, DOI 10.1090/mmono/006
J. Lorch, Unitary structures for ladder representations of $U(p,q)$, Ph.D. thesis, Oklahoma State University, 1995.
John D. Lorch and Lisa A. Mantini, Inversion of an integral transform and ladder representations of $\textrm {U}(1,q)$, Representation theory and harmonic analysis (Cincinnati, OH, 1994) Contemp. Math., vol. 191, Amer. Math. Soc., Providence, RI, 1995, pp. 117–138. MR 1365539, DOI 10.1090/conm/191/02332
J. Lorch, An integral transform and ladder representations of $U(p,q)$, Pacific J. Math., in press.
Lisa A. Mantini, An integral transform in $L^2$-cohomology for the ladder representations of $\textrm {U}(p,q)$, J. Funct. Anal. 60 (1985), no. 2, 211–242. MR 777237, DOI 10.1016/0022-1236(85)90051-5
Lisa A. Mantini, An $L^2$-cohomology construction of negative spin mass zero equations for $\textrm {U}(p,q)$, J. Math. Anal. Appl. 136 (1988), no. 2, 419–449. MR 972147, DOI 10.1016/0022-247X(88)90095-9
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 274657, DOI 10.2307/1970635
R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR 318398, DOI 10.2307/1970892
I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Mathematics and its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Translated from the Russian. MR 0252690
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058