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Journal of Lie Theory 18 (2008), No. 2, 253--271
Copyright Heldermann Verlag 2008

A Paley-Wiener Theorem for the Bessel Laplace Transform,
I: the case SU(n,n)/SL(n,C) x R*+

Salem Ben Saïd
Institut Élie Cartan, Dép. de Mathématiques, Université Henri Poincaré, B.P. 239, 54506 Vandoeuvres-Les-Nancy, France


\def\C{{\Bbb C}} \def\R{{\Bbb R}} \def\q{{\frak q}} Let $\q$ be the tangent space to the noncompact causal symmetric space $$SU(n,n)/SL(n,\C)\times \R^*_+$$ at the origin. In this paper we give an explicit formula for the Bessel functions on $\q$. We use this result to prove a Paley-Wiener theorem for the Bessel Laplace transform on $\q$. Further, a flat analogue of the Abel transform is defined and inverted.

Keywords: Non-compactly causal symmetric spaces, multivariable Bessel function, Paley-Wiener theorem, Abel transform.

MSC: 43A85, 43A32, 33C80

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