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Preprint Number 382

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382. Raf Cluckers, Julia Gordon, and Immanuel Halupczok
Definability results for invariant distributions on a reductive unramified p-adic group

Submission date: 1 December 2011.


Let G be a connected reductive algebraic group over a non-Archimedean local field K, and let g be its Lie algebra. By a theorem of Harish-Chandra, if K has characteristic zero, the Fourier transforms of nilpotent orbital integrals are represented on the set of regular elements in g(K) by locally constant functions, which, extended by zero to all of g(K), are locally integrable. In this paper, we prove that if the group G is unramified, these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability of [R. Cluckers, J. Gordon, I. Halupczok, “Transfer principles for integrability and boundedness conditions for motivic exponential functions”, preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when K is a non-Archimedean local field of sufficiently large positive characteristic. Under some mild hypotheses, this also implies local integrability in a neighbourhood of the identity element of Harish-Chandra characters of admissible representations of G(K), with G an unramified connected reductive algebraic group, and K an equicharacteristic field of sufficiently large (depending on the root datum of G) characteristic.

Mathematics Subject Classification: 22E50 (Primary), 14E18 (Secondary)

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Full text arXiv 1111.7057: pdf, ps.

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