The $c$-function expansion of a basic hypergeometric function associated to root systems

Abstract

We derive an explicit $c$-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as an explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald $q$-difference operators. The $c$-function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of $q$-analogs of the Harish-Chandra series. We express the expansion coefficients in terms of a $q$-analog of the Harish-Chandra $c$-function, which is explicitly given as product of $q$-Gamma functions. The $c$-function expansion shows that the basic hypergeometric function formally is a $q$-analog of the Heckman-Opdam hypergeometric function, which in turn specializes to elementary spherical functions on noncompact Riemannian symmetric spaces for special values of the parameters.