Orbital Integrals on p-Adic Lie Algebras

Abstract

Let $G$ be a connected reductive $p$-adic group and let $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{O}$ be any $G$-orbit in $\mathfrak{g}$. Then the orbital integral ${{\mu }_{\mathcal{O}}}$ corresponding to $\mathcal{O}$ is an invariant distribution on $\mathfrak{g}$, and Harish-Chandra proved that its Fourier transform ${{\hat{\mu }}_{\mathcal{O}}}$ is a locally constant function on the set ${\mathfrak{g}}'$ of regular semisimple elements of $\mathfrak{g}$. If $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $\omega $ is a compact subset of $\mathfrak{h}\,\cap \,{\mathfrak{g}}'$, we give a formula for ${{\hat{\mu }}_{\mathcal{O}}}\left( tH \right)$ for $H\,\in \,\omega $ and $t\,\in \,{{F}^{\times }}$ sufficiently large. In the case that $\mathcal{O}$ is a regular semisimple orbit, the formula is already known by work of Waldspurger. In the case that $\mathcal{O}$ is a nilpotent orbit, the behavior of ${{\hat{\mu }}_{\mathcal{O}}}$ at infinity is already known because of its homogeneity properties. The general case combines aspects of these two extreme cases. The formula for ${{\hat{\mu }}_{\mathcal{O}}}$ 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.”