Orbital integrals on reductive Lie algebras. (Intégrales orbitales sur les algèbres de Lie réductives.) (French) Zbl 0814.22005

Bouaziz, Abderrazak

Invent. Math. 115, No. 1, 163-207 (1994).

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.

Reviewer: F.Rouvière (Nice)

MSC:

22E60	Lie algebras of Lie groups
43A80	Analysis on other specific Lie groups
17B20	Simple, semisimple, reductive (super)algebras

Keywords:

reductive Lie group; Lie algebra; adjoint action; compactly supported smooth functions; orbital integrals; regular elements; \(G\)-invariant smooth functions; regular semisimple elements; Schwartz space; dual spaces; \(G\)-invariant distributions

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