Tensor products of Minimal Holomorphic Representations

Zhang, Genkai

doi:10.1090/S1088-4165-01-00103-0

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

A new kind of Hankel type operators connected with the complementary series

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, 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

Canonical representations

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

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

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

Plancherel formula for Berezin deformation of ${L^2}$ on Riemannian symmetric space

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

Invariant differential operators on hermitian symmetric spaces and their eigenvalues

119

Shimura invariant differential operators and their eigenvalues

319

Genkai Zhang, Berezin transform on line bundles over bounded symmetric domains, J. Lie Theory 10 (2000), no. 1, 111–126. MR 1748086