BOUNDS FOR FOURIER TRANSFORMS OF REGULAR ORBITAL INTEGRALS ON p-ADIC LIE ALGEBRAS

Let G be a connected reductive p-adic group and let g be its Lie algebra. LetO be a G-orbit in g. Then the orbital integral μO corresponding to O is an invariant distribution on g, and Harish-Chandra proved that its Fourier transform μ̂O is a locally constant function on the set g′ of regular semisimple elements of g. Furthermore, he showed that a normalized version of the Fourier transform is locally bounded on g. Suppose that O is a regular semisimple orbit. Let γ be any semisimple element of g, and let m be the centralizer of γ. We give a formula for μ̂O(tH) (in terms of Fourier transforms of orbital integrals on m), for regular semisimple elements H in a small neighborhood of γ in m and t ∈ F× sufficiently large. We use this result to prove that HarishChandra’s normalized Fourier transform is globally bounded on g in the case that O is a regular semisimple orbit.