Definability results for invariant distributions on a reductive unramified p-adic group

Details Definability results for invariant distributions on a reductive unramified p-adic group Definability results for invariant distributions on a reductive unramified p-adic group

2011-11-30

arxiv.org/abs/1111.7057v1

Mathematics

Representation Theory

Scientific paper

Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $\mathfrak g$ 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 ${\mathfrak g}(K)$ by locally constant functions, which, extended by zero to all of ${\mathfrak g}(K)$, 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.

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