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Functions whose orbital integrals and those of their Fourier transforms are at topologically nilpotent support. (Fonctions dont les intégrales orbitales et celles de leurs transformées de Fourier sont à support topologiquement nilpotent.) (English) Zbl 1517.20075

Summary: Let \(F\) be a \(p\)-adic field and let \(G\) be a connected reductive group defined over \(F\). We assume \(p\) is large. Denote \(\mathfrak{g}\) the Lie algebra of \(G\). To each vertex \(s\) of the reduced Bruhat-Tits’ building of \(G\), we associate as usual a reductive Lie algebra \({\mathfrak{g}_s}\) defined over the residual field \({\mathbb{F}_q} \). We normalize suitably a Fourier-transform \(f\mapsto \hat{f}\) on \({C_c^{\infty }}(\mathfrak{g}(F))\). We study the subspace of functions \(f\in{C_c^{\infty }}(\mathfrak{g}(F))\) such that the orbital integrals of \(f\) and of \(\hat{f}\) are 0 for each element of \(\mathfrak{g}(F)\) which is not topologically nilpotent. This space is related to the characteristic functions of the character-sheaves on the spaces \({\mathfrak{g}_s}({\mathbb{F}_q})\), for each vertex \(s\), which are cuspidal and with nilpotent support. We prove that our subspace behave well under endoscopy.

MSC:

20G25 Linear algebraic groups over local fields and their integers
17B08 Coadjoint orbits; nilpotent varieties
22E50 Representations of Lie and linear algebraic groups over local fields
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References:

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