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.


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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