Some recurrence formulas for spherical polynomials on tube domains

Zhang, Gen Kai

doi:10.1090/S0002-9947-1995-1249896-7

Some recurrence formulas for spherical polynomials on tube domains
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Trans. Amer. Math. Soc. 347 (1995), 1725-1734 Request permission

Abstract:

For a tube domain $G/K$ we study the tensor products of two spherical representations of the maximal compact group $K$ and the product of the corresponding spherical polynomials. When one of these is a fundamental representation, we prove that the spherical representations appear with multiplicity at most one and we then find all the coefficients in the recurrence formula for the product of the spherical polynomials. This generalizes the previous result of L. Vretare and proves for certain cases a conjecture of R. Stanley on Jack symmetric polynomials.

References

N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Hermann, Paris, 1975 (French). Actualités Sci. Indust., No. 1364. MR 0453824

Groupes et algebres de Lie

Thomas J. Enright and Anthony Joseph, An intrinsic analysis of unitarizable highest weight modules, Math. Ann. 288 (1990), no. 4, 571–594. MR 1081264, DOI 10.1007/BF01444551
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
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
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
Bertram Kostant and Siddhartha Sahi, The Capelli identity, tube domains, and the generalized Laplace transform, Adv. Math. 87 (1991), no. 1, 71–92. MR 1102965, DOI 10.1016/0001-8708(91)90062-C

Bounded symmetric domains and Jordan pairs

I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
I. G. Macdonald, Commuting differential operators and zonal spherical functions, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 189–200. MR 911140, DOI 10.1007/BFb0079238
Bent Ørsted and Gen Kai Zhang, Reproducing kernels and composition series for spaces of vector-valued holomorphic functions on tube domains, J. Funct. Anal. 124 (1994), no. 1, 181–204. MR 1284609, DOI 10.1006/jfan.1994.1104
Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/70), 61–80 (German). MR 259164, DOI 10.1007/BF01389889
Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073, DOI 10.1016/0001-8708(89)90015-7
Harald Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), no. 1, 1–25 (1986). MR 821311, DOI 10.2307/2374466
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
Lars Vretare, Elementary spherical functions on symmetric spaces, Math. Scand. 39 (1976), no. 2, 343–358 (1977). MR 447979, DOI 10.7146/math.scand.a-11667

Compact Lie groups and their representations