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Intégrales orbitales sur \(GL(N)\) et corps locaux proches. (Orbital integrals on \(GL(N)\) and close local fields.). (French) Zbl 0853.22012

Summary: Let \(F\) be a local non-archimedean field, \(N\) an integer \(\geq 2\), \(\underline{G}=GL(N)\), \(n\) a positive integer and \({\mathcal H}(\underline{G}(F),K_{F}^{n})\) the Hecke algebra of \(\underline{G}(F)\) with respect to the congruence subgroup modulo \({\mathcal P}_{F}^{ n}\) of \(\underline{G}({\mathcal O}_{F})\). We prove an explicit formula for the elliptic orbital integrals of functions in \({\mathcal H}(\underline{G}(F),K_{F}^{n})\). Thanks to this formula, for \(\gamma \in \underline{G}(F)\) semi-simple regular, we produce an integer \(r = r(\gamma,n)\geq n\) such that for any local non-archimedean field \(F'\) \(r\)-close to \(F\) (i.e. such that there exists an isomorphism of rings \({\mathcal O}_{F}/{\mathcal P}_{F}^{ r}\simeq {\mathcal O}_{F'}/ {\mathcal P}_{F'}^{ r}\)), there exists \(\gamma' \in \underline{G}(F')\) semi-simple regular such that the orbital integrals at \(\gamma\) of all functions in \({\mathcal H}(\underline{G}(F),K_{F}^{n})\) match, via a given isomorphism of algebras \({\mathcal H}(\underline{G}(F),K_{F}^{n})\simeq {\mathcal H}(\underline{G}(F'),K_{F'}^{n})\), those of functions in \({\mathcal H}(\underline{G}(F'),K_{F'}^{n})\) at \(\gamma'\).

MSC:

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