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Journal of Lie Theory 24 (2014), No. 2, 421--438
Copyright Heldermann Verlag 2014

Integral Formula and Upper Estimate of I and J-Bessel Functions on Jordan Algebras

Ryosuke Nakahama
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan

We give a new integral expression of I and J-Bessel functions on simple Euclidean Jordan algebras, integrating on a bounded symmetric domain. From this we easily get the upper estimate of Bessel functions. As an application we give an upper estimate of the integral kernel function of the holomorphic 1-dimensional semi-group acting on the space of square integrable functions on symmetric cones.

Keywords: Euclidean Jordan algebras, Bessel functions, holomorphic discrete series representations, holomorphic semigroups.

MSC: 33C10, 33C67, 17C30, 22E45, 47D06

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