One of the major problems in the representation theory of reductive groups over \(p\)-adic fields is the local Langlands conjecture, which classifies irreducible representations in terms of Langlands parameters. The problem has been settled for \(GL_n\), and this article deals with the case of more general groups. Given a representation \(\pi\) of a \(p\)-adic group \(G\) in a vector space \(V_\pi\) and a locally constant compactly supported function \(f\) on \(G\), one can consider an operator \(\pi (f)\) on \(V_\pi\) defined by \(\pi (f) (v) = \int_G f(g) \pi (g) (v) dg\) for \(v \in V_\pi\). When \(\pi\) is irreducible, the operator \(\pi (f)\) has finite rank and therefore has a trace. Then the character \(\Theta_\pi\) of \(\pi\) is the distribution on \(G\) defined by \(\Theta_\pi (f) = \text{tr}\, \pi (f)\).
In this paper, after discussing detailed background materials, the author proves the local trace formula on the Lie algebra \(\mathfrak g\) of \(G\) as well as the linear independence of Shalika germs and the density of regular semisimple orbital integrals. He also considers an invariant distribution \(I\) whose support is bounded modulo conjugation, and shows that the Fourier transform \(\widehat{I}\) of \(I\) is represented by a function. Finally, he proves the Lie algebra analog of Harish-Chandra’s local character expansion for \(\Theta_\pi\).
For the entire collection see [Zbl 1083.11002].