Endoscopy and the Fourier transform of minimal unipotent orbital integrals for spherical functions on p-adic SO(2n+1)

Let G be a split orthogonal group in an odd number of variables, defined over a p-adic field F. Let H denote an elliptic endoscopic group of G. Let O-H denote the (stable) unipotent orbit in G(F) obtained, via endoscopic induction, from the trivial orbit in H. We prove that the restriction to the spherical Hecke algebra of the Fourier transform of the orbital integral over O-H is given (up to an explicit constant) by integrating the Satake transform against the spherical Plancherel measure of H(F). This settles in the affirmative, for the groups mentioned above, the hypothesis formulated in [2] We also explicitly calculate the integrals, of a basis for the spherical Hecke algebra of any split orthogonal group in an even number of variables, over the minimal unipotent orbits.