On the coefficients of an asymptotic expansion of spherical functions on symmetric spaces

Abstract:

The asymptotic expansion of a spherical function on a symmetric space of noncompact type, obtained by Harish-Chandra, is a finite linear combination of expansions of the form ${\Phi _\theta } = {\sum _\mu }’{\Gamma _\mu }(\theta ){e^{\theta - \mu }}$. In this paper it is proved that ${\lim _{t \to \infty }}’{\Gamma _\mu }(te - \bar \rho )$ is finite and rational for any e, where $\bar \rho$ is the restriction of half the sum of the positive roots.