Publishing house / Banach Center Publications / All volumes

Abstract

We study the regularity properties of functions that can be represented on a positive Weyl chamber $A^+$ by generalized Harish-Chandra expansions. We adopt the approach of Heckman and Opdam by allowing arbitrary Weyl-group-invariant complex multiplicities. The generalized Harish-Chandra expansions that we consider are associated with arbitrary parabolic system $\Theta$ of roots and a root multiplicity function. They are given as sum over the Weyl group of $\Theta$ of generalized hypergeometric functions. They are analytic on $A^+$ and meromorphic in the spectral parameter $\lambda$. We prove that they extend as $W_\Theta$-invariant holomorphic functions on a tubular neighborhood of $(W_\Theta\cdot \overline{A^+})^0$. The possible location of the $\lambda$-singularities is shown to be the polar set of an explicit function naturally constructed from the fixed data. For reduced root systems with even multiplicities we refine our result and show that the $ł$-singularities lie on a specific finite family of affine hyperplanes. Finally, when all multiplicities are equal to $2$, we generalize the classical explicit formula for spherical functions on Riemannian symmetric spaces with a complex structure.