A generalized Harish-Chandra method of descent for reductive Lie algebras

Let G be a connected real reductive group and M a connected reductive subgroup of G with Lie algebras g and m, respectively. We assume that g and m have the same rank. We define a map from the space of orbital integrals of m into the space of orbital integrals of g which we call a transfer. We then consider the transpose of the transfer. This can be viewed as a map from the space of G-invariant distributions of g to the space of M-invariant distributions of m and can be considered as a restriction map from g to m. We prove that this map extends Harish-Chandra method of descent and we obtain a generalization of the radial component theorem. We give an application.