  The classical radial part formula for the invariant differential operators
and the K-invariant functions on a Riemannian symmetric space G/K is
generalized to some non-invariant cases by use of Cherednik operators and a
graded Hecke algebra H naturally attached to G/K. We introduce a category
C_{rad} whose object is a pair of a (g_C,K)-module and an H-module satisfying
some axioms which are formally the same as the generalized Chevalley
restriction theorem and the generalized radial part formula. Various pairs of
analogous notions in the representation theories for G and H, such as the
Helgason-Fourier transform and the Opdam-Cherednik transform, are unified in
terms of C_{rad}. We construct natural functors which send an H-module to a
(g_C,K)-module and have some universal properties intimately related to
C_{rad}.
