Spherical functions on symmetric cones

Sawyer, P.

doi:10.1090/S0002-9947-97-01505-5

Spherical functions on symmetric cones
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Trans. Amer. Math. Soc. 349 (1997), 3569-3584 Request permission

Abstract:

In this note, we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real Jordan algebra. We use this formula as an inspiration for a similar recursive formula involving the Jack polynomials.

References

Hel Braun and Max Koecher, Jordan-Algebren, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin-New York, 1966 (German). MR 0204470
Jacques Faraut, Algèbres de Jordan et cônes symétriques, Ecole CIMPA - Université de Poitiers, 22 Août - 16 Septembre, 1988.
Jacques Faraut, Fonctions de Legendre sur une algèbre de Jordan, CWI Quarterly 5 (1992), no. 4, 309–320 (French, with French summary). MR 1213745
Kenneth I. Gross and Donald St. P. Richards, Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions, Trans. Amer. Math. Soc. 301 (1987), no. 2, 781–811. MR 882715, DOI 10.1090/S0002-9947-1987-0882715-2
Sigurđur Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239–299. MR 117681, DOI 10.1007/BF02564248
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
Ulrich Hirzebruch, Über Jordan-Algebren und kompakte Riemannsche symmetrische Räume vom Rang $1$, Math. Z. 90 (1965), 339–354 (German). MR 188944, DOI 10.1007/BF01112353
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
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
Robb J. Muirhead, Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982. MR 652932
Patrice Sawyer, The heat equation on the symmetric space associated to $\mathbf {SL}(n, \mathbf {R})$, Thesis, McGill University, 1989.
P. Sawyer, The heat equation on the spaces of positive definite matrices, Canad. J. Math. 44 (1992), no. 3, 624–651. MR 1176373, DOI 10.4153/CJM-1992-038-7
P. Sawyer, On an upper bound for the heat kernel on $\textrm {SU}^\ast (2n)/\textrm {Sp}(n)$, Canad. Math. Bull. 37 (1994), no. 3, 408–418. MR 1289778, DOI 10.4153/CMB-1994-059-x
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
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Zhi Min Yan, Generalized hypergeometric functions and Laguerre polynomials in two variables, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 239–259. MR 1199131, DOI 10.1090/conm/138/1199131