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