Orbital integrals on reductive Lie algebras
Let \(G\) be a reductive Lie group of Harish-Chandra class, with Lie algebra \(\mathfrak g\). Let \(U\) be a completely \(G\)-invariant open subset of \(\mathfrak g\) (under the adjoint action), and let \({\mathcal D}(U)\) be the space of compactly supported smooth functions on \(U\). To each \(f\) in \({\mathcal D}(U)\) correspond orbital integrals \(Jf(X)\), obtained when integrating \(f\) on the orbits \(G\cdot X\) of regular elements \(X\) in \(U\), by means of the Liouville measure on these orbits. The main result of the paper is a complete description of the image of the map \(J\), as a space of \(G\)-invariant smooth functions on the subset of regular semisimple elements in \(U\), which satisfy certain jump conditions. Among the tools used in its proof are \(G\)-invariant partitions of unity and Harish-Chandra's method of descent: the latter reduces the problem for \(\mathfrak g\) to a similar problem for the centralizer of a semisimple element, thus allowing a proof by induction on the dimension. An analogous result is proved for the Schwartz space \({\mathcal S}({\mathfrak g})\) replacing \({\mathcal D}(U)\) above. Transposing the map \(J\), the author also describes the dual spaces \({\mathcal D}'(U)^ G\) and \({\mathcal S}'({\mathfrak g})^ G\) of \(G\)-invariant distributions. For orbital integrals on the group \(G\) itself, see [the author, Ann. Sci. Ec. Norm. Supér., IV. Sér. 27, 573-609 (1994)], which relies on the results of the present paper.
scientific article; zbMATH DE number 549842