Orbital integrals on p-adic Lie algebras

Let G be a connected reductive p-adic group and let g be its Lie algebra. Let O be any G-orbit in g. Then the orbital integral mu (o) corresponding to O is an invariant distribution on g, and Harish-Chandra proved that its Fourier transform <(<mu>)over cap>o is a locally constant function on the set g' of regular semisimple elements of g. If b is a Cartan subalgebra of g, and omega is a compact subset of b boolean AND g', we give a formula for <(<mu>)over cap>(o) (tH) for H is an element of omega and t is an element of F-X sufficiently large. In the case that O is a regular semisimple orbit. the Formula is already known by work of Waldspurger. In the case that O is a nilpotent orbit, the behavior of <(<mu>)over cap>(o) at infinity is already known because of its homogeneity properties. The general case combines aspects of these two extreme cases. The formula for <(<mu>)over cap>(o) at infinity can be used to formulate a "theory of the constant term" for the space of distributions spanned by the Fourier transforms of orbital integrals. It can also be used to show that the Fourier transforms of orbital integrals are "linearly independent at infinity".