Abstract
This chapter explains the asymptotic behavior of spherical functions on semisimple symmetric spaces. A K-finite simultaneous eigenfunction of the invariant differential operators on G/H is called a spherical function, where K is a maximal compact subgroup of G modulo center. The chapter presents the construction of linearly independent eigenfunctions for Weyl group invariant differential operators with constant coefficients defined on a root space. The eigenfunctions holomorphically depends on the eigenvalue. The chapter reviews the boundary value maps for the eigenfunctions of invariant differential operators on riemannian symmetric spaces of the noncompact type, which are first introduced to prove Helgason's conjecture in [K-]. It also discusses intertwining operators between locally defined sections of principal series for G/K.
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