Bounds for Fourier transforms of regular orbital integrals on \(p\)-adic Lie algebras (Q2761190)

Let \(G\) be a connected reductive algebraic group defined over a non-archimedean local field \(F\) of characteristic 0 and let \({\mathfrak g}\) be its Lie algebra. Let \({\mathcal O}\) be a \(G(F)\)-orbit in \({\mathfrak g}\) and let \(\mu\) be the corresponding invariant orbital integral (defined up to a multiple). Harish-Chandra showed that the Fourier transform of \(\mu\) has a very specific asymptotic behaviour at infinity. In this paper the author shows that there is a natural universal polynomial function \(\eta_{\mathfrak g}\) on \({\mathfrak g}\) so that \(|\eta_{\mathfrak g} |^{1/2}\widehat\mu\) is bounded on the set of regular elements if \({\mathcal O}\) is regular, and in a suitable sense, this bound is locally uniform on the set of regular orbits. This result is based on a sharper version of Harish-Chandra's germ expansion at infinity which permits an inductive argument. The proof of this is a delicate local analysis of the functions involved.