## Tensor products of Minimal Holomorphic Representations

HTML articles powered by AMS MathViewer

- by Genkai Zhang PDF
- Represent. Theory
**5**(2001), 164-190 Request permission

## Abstract:

Let $D=G/K$ be an irreducible bounded symmetric domain with genus $p$ and $H^{\nu }(D)$ the weighted Bergman spaces of holomorphic functions for $\nu >p-1$. The spaces $H^\nu (D)$ form unitary (projective) representations of the group $G$ and have analytic continuation in $\nu$; they give also unitary representations when $\nu$ in the Wallach set, which consists of a continuous part and a discrete part of $r$ points. The first non-trivial discrete point $\nu =\frac a2$ gives the minimal highest weight representation of $G$. We give the irreducible decomposition of tensor product $H^{\frac a2}\otimes \overline {H^{\frac a2}}$. As a consequence we discover some new spherical unitary representations of $G$ and find the expansion of the corresponding spherical functions in terms of the $K$-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.## References

- Richard Askey and James Wilson,
*Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*, Mem. Amer. Math. Soc.**54**(1985), no. 319, iv+55. MR**783216**, DOI 10.1090/memo/0319 - 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 - Alexander Dvorsky and Siddhartha Sahi,
*Tensor products of singular representations and an extension of the $\theta$-correspondence*, Selecta Math. (N.S.)**4**(1998), no. 1, 11–29. MR**1623698**, DOI 10.1007/s000290050023
ehprz:comser M. Engliš, S. C. Hille, J. Peetre, H. Rosengren, and G. Zhang, - 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 - George Gasper and Mizan Rahman,
*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153** - 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**
Hille:thesis S. C. Hille, - Bob Hoogenboom,
*Spherical functions and invariant differential operators on complex Grassmann manifolds*, Ark. Mat.**20**(1982), no. 1, 69–85. MR**660126**, DOI 10.1007/BF02390499 - Hiro-o Yamamoto,
*Pinching deformations of Fuchsian groups*, Tohoku Math. J. (2)**33**(1981), no. 4, 443–452. MR**643228**, DOI 10.2748/tmj/1178229348 - 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 - A. A. Kirillov,
*Merits and demerits of the orbit method*, Bull. Amer. Math. Soc. (N.S.)**36**(1999), no. 4, 433–488. MR**1701415**, DOI 10.1090/S0273-0979-99-00849-6 - A. W. Knapp and B. Speh,
*The role of basic cases in classification: theorems about unitary representations applicable to $\textrm {SU}(N,\,2)$*, Noncommutative harmonic analysis and Lie groups (Marseille, 1982) Lecture Notes in Math., vol. 1020, Springer, Berlin, 1983, pp. 119–160. MR**733464**, DOI 10.1007/BFb0071500
Askey:Scheme R. Koekoek and R. F. Swarttouw, - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - Edward Nelson and W. Forrest Stinespring,
*Representation of elliptic operators in an enveloping algebra*, Amer. J. Math.**81**(1959), 547–560. MR**110024**, DOI 10.2307/2372913
Neretin:plan:beredef Yu. Neretin, - Gestur Ólafsson and Bent Ørsted,
*Generalizations of the Bargmann transform*, Lie theory and its applications in physics (Clausthal, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 3–14. MR**1634480** - Bent Ørsted and Gen Kai Zhang,
*Weyl quantization and tensor products of Fock and Bergman spaces*, Indiana Univ. Math. J.**43**(1994), no. 2, 551–583. MR**1291529**, DOI 10.1512/iumj.1994.43.43023 - Bent Ørsted and Genkai Zhang,
*$L^2$-versions of the Howe correspondence. I*, Math. Scand.**80**(1997), no. 1, 125–160. MR**1466908**, DOI 10.7146/math.scand.a-12615 - Bent Ørsted and Genkai Zhang,
*Tensor products of analytic continuations of holomorphic discrete series*, Canad. J. Math.**49**(1997), no. 6, 1224–1241. MR**1611656**, DOI 10.4153/CJM-1997-060-5 - Jaak Peetre and Gen Kai Zhang,
*A weighted Plancherel formula. III. The case of the hyperbolic matrix ball*, Collect. Math.**43**(1992), no. 3, 273–301 (1993). MR**1252736** - Joe Repka,
*Tensor products of unitary representations of $\textrm {SL}_{2}(\textbf {R})$*, Amer. J. Math.**100**(1978), no. 4, 747–774. MR**509073**, DOI 10.2307/2373909 - Joe Repka,
*Tensor products of holomorphic discrete series representations*, Canadian J. Math.**31**(1979), no. 4, 836–844. MR**540911**, DOI 10.4153/CJM-1979-079-9 - 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 - Wilfried Schmid,
*Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen*, Invent. Math.**9**(1969/70), 61–80 (German). MR**259164**, DOI 10.1007/BF01389889 - Goro Shimura,
*Invariant differential operators on Hermitian symmetric spaces*, Ann. of Math. (2)**132**(1990), no. 2, 237–272. MR**1070598**, DOI 10.2307/1971523 - Goro Shimura,
*Differential operators, holomorphic projection, and singular forms*, Duke Math. J.**76**(1994), no. 1, 141–173. MR**1301189**, DOI 10.1215/S0012-7094-94-07606-0 - Harald Upmeier,
*Toeplitz operators on bounded symmetric domains*, Trans. Amer. Math. Soc.**280**(1983), no. 1, 221–237. MR**712257**, DOI 10.1090/S0002-9947-1983-0712257-2 - 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 - James A. Wilson,
*Some hypergeometric orthogonal polynomials*, SIAM J. Math. Anal.**11**(1980), no. 4, 690–701. MR**579561**, DOI 10.1137/0511064 - Zhi Min Yan,
*Differential operators and function spaces*, Several complex variables in China, Contemp. Math., vol. 142, Amer. Math. Soc., Providence, RI, 1993, pp. 121–142. MR**1208787**, DOI 10.1090/conm/142/1208787 - Gen Kai Zhang,
*Ha-plitz operators between Moebius invariant subspaces*, Math. Scand.**71**(1992), no. 1, 69–84. MR**1216104**, DOI 10.7146/math.scand.a-12411 - Gen Kai Zhang,
*A weighted Plancherel formula. II. The case of the ball*, Studia Math.**102**(1992), no. 2, 103–120. MR**1169281**, DOI 10.4064/sm-102-2-103-120
gz:invdiff —, - Genkai Zhang,
*Berezin transform on line bundles over bounded symmetric domains*, J. Lie Theory**10**(2000), no. 1, 111–126. MR**1748086**

*A new kind of Hankel type operators connected with the complementary series*, Arabic J. Math. Sci.

**6**(2000), 49–80.

*Canonical representations*, Ph.D. thesis, Leiden University, 1999.

*The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue*, Math. report, Delft Univ. of Technology 98-17, 1998.

*Plancherel formula for Berezin deformation of ${L^2}$ on Riemannian symmetric space*, (1999), preprint, Math.RT/9911020.

*Invariant differential operators on hermitian symmetric spaces and their eigenvalues*, Israel J. Math.

**119**(2000), 157–185. gz:shimura —,

*Shimura invariant differential operators and their eigenvalues*, Math. Ann.,

**319**(2001), 235-265.

## Additional Information

**Genkai Zhang**- Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
- Email: genkai@math.chalmers.se
- Received by editor(s): May 23, 2000
- Received by editor(s) in revised form: April 10, 2001
- Published electronically: June 15, 2001
- Additional Notes: Research supported by the Swedish Natural Science Research Council (NFR)
- © Copyright 2001 American Mathematical Society
- Journal: Represent. Theory
**5**(2001), 164-190 - MSC (2000): Primary 22E46, 47A70, 32M15, 33C52
- DOI: https://doi.org/10.1090/S1088-4165-01-00103-0
- MathSciNet review: 1835004