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Orbital integrals on \(GL(N,F)\) where \(F\) is a non-archimedean local field. (Intégrales orbitales sur \(GL(N,F)\) où \(F\) est un corps local non archimédien.) (French) Zbl 0914.22020

In this paper proofs are given for the basic results concerning orbital integrals on \({GL}(N,F)\) for a non-archimedean local field \(F\) of arbitrary characteristic. The treatment is independent of the characteristic.
A new normalization of the orbital integrals is introduced, which (I translate) “seems particularly adapted, when the characteristic divides \(N\), to problems connected with the infinite number of conjugacy classes of Cartan subgroups of \({GL}(N,F)\)”. This normalization uses the constructions of Bushnell and Kutzko. The proof of linear independence of the germs in the neighbourhood of an inseparable semi-simple element is particularly difficult. A characterization of orbital integrals is given and their density in the space of invariant distributions is proved.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
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