Matching theorems for twisted orbital integrals (Q1918406)

Let \(F\) be a \(p\)-adic field and \(E\) a cyclic extension of \(F\) of degree \(d\) corresponding to the character \(\kappa\) of \(F^\times\). For any positive integer \(m\), we can consider \(H = GL (m,E)\) as a subgroup of \(G = GL (md,F)\). In this paper we discuss matching of orbital integrals between \(H\) and \(G\). Specifically, ordinary orbital integrals corresponding to regular semisimple elements of \(H\) are matched with orbital integrals on \(G\) which are twisted by the character \(\kappa\). For the general situation we only match functions which are smooth and compactly supported on the regular set. If the extension \(E/F\) is unramified, we are able to match arbitrary smooth, compactly supported functions.