Differential-difference operators and radial part formulas for non-invariant elements

الأصل · EN

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 Crad whose object is a pair of a (gC,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 Crad. We construct natural functors which send an H-module to a (gC,K)-module and have some universal properties intimately related to Crad.