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Some results on unipotent orbital integrals. (English) Zbl 0737.22005
This paper deals with some questions concerning orbital integrals important in applications of the Selberg trace formula. Let \(F\) be a local field and \(G\) be a reductive algebraic group defined over \(F\). Let \(H\) be an endoscopic group of \(G\). Then one should be able to transfer functions on \(G(F)\) to functions on \(H(F)\) so that certain corresponding orbital integrals are equal. Moreover this transfer should extend the natural map between Hecke algebras. At present this is not known in general for non-archimedean fields and the purpose of this paper is to produce evidence for it. One approach to the problem requires the analysis of certain unipotent orbital integrals for functions in the Hecke algebra of \(G\) and of corresponding functions for \(H\) with the intention of proving linear relations between them. The author does this by direct and difficult computations in the cases \(G=SO(2k+1)\), \(H=SO(2k- 1)\times PGL(2)\) and \(G=Sp(2k)\), \(H=SO(2k)\).

MSC:
22E35 Analysis on \(p\)-adic Lie groups
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E50 Representations of Lie and linear algebraic groups over local fields
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