An analogue to the Duistermaat–Kolk–Varadarajan estimate for the spherical functions associated with the root systems of type A

References

[1] Delorme P., Espace de Schwartz pour la transformation de Fourier hypergéométrique, J. Funct. Anal. 168 (1999), no. 1, 239–312. 10.1006/jfan.1999.3449Search in Google Scholar

[2] Duistermaat J. J., Kolk J. A. C. and Varadajan V. S., Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compos. Math. 49 (1983), 309–398. Search in Google Scholar

[3] Graczyk P. and Sawyer P., On the kernel of the product formula on symmetric spaces, J. Geom. Anal. 14 (2004), no. 4, 653–672. 10.1007/BF02922174Search in Google Scholar

[4] Gradshteyn I. S. and Ryzhik I. M., Table of Integrals, Series, and Products, 7th ed., Elsevier, Amsterdam, 2007. Search in Google Scholar

[5] Heckman G. J., Root systems and hypergeometric functions II, Compos. Math. 64 (1987), 353–374. Search in Google Scholar

[6] Helgason S., Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, Math. Surveys Monogr. 83, American Mathematical Society, Providence, 2000. 10.1090/surv/083/03Search in Google Scholar

[7] Helgason S., Differential Geometry, Lie Groups and Symmetric spaces, Grad. Stud. Math. 34, American Mathematical Society, Providence, 2001. 10.1090/gsm/034Search in Google Scholar

[8] Macdonald I. G., Commuting differential operators and zonal spherical functions, Algebraic Groups (Utrecht 1986), Lecture Notes in Math. 1271, Springer, Berlin (1987), 189–200. 10.1007/BFb0079238Search in Google Scholar

[9] Narayanana E. K., Pasquale A. and Pusti S., Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications, Adv. Math. 252 (2014), 227–259. 10.1016/j.aim.2013.10.027Search in Google Scholar

[10] Opdam E. M., Root systems and hypergeometric functions IV, Compos. Math. 67 (1988), no. 2, 191–209. Search in Google Scholar

[11] Opdam E. M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121. 10.1007/BF02392487Search in Google Scholar

[12] Opdam E. M., Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups, MSJ Mem. 8, Mathematical Society of Japan, Tokyo, 2000. Search in Google Scholar

[13] Pasquale A., Asymptotic analysis of Θ-hypergeometric functions, Invent. Math. 157 (2004), no. 1, 71–122. 10.1007/s00222-003-0349-9Search in Google Scholar

[14] Sawyer P., Estimates for the heat kernel on SL(n,R)/SO(n), Canad. J. Math. 49 (1997), no. 2, 359–372. 10.4153/CJM-1997-018-1Search in Google Scholar

[15] Sawyer P., Spherical functions on symmetric cones, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3569–3584. 10.1090/S0002-9947-97-01505-5Search in Google Scholar

[16] Sawyer P., The heat kernel on the symmetric space SL(n,F)/SU(n,F), The Ubiquitous Heat Kernel (Boulder 2003), Contemp. Math. 398, American Mathematical Society, Providence (2006), 369–391. 10.1090/conm/398/07497Search in Google Scholar

[17] Schapira B., Contributions to the hypergeometric function theory of Heckman and Opdam: Sharp estimates, Schwartz space, heat kernel, Geom. Funct. Anal. 18 (2008), no. 1, 222–250. 10.1007/s00039-008-0658-7Search in Google Scholar