Transfer principles and uniform estimates for orbital integrals

: In this talk, I hope to describe the state-of-the-art of the program of applying the theory of motivic integration (which is based on model theory, and will be defined in the talk) to harmonic analysis on p-adic groups. This program was started by T.C. Hales around 1999, and in particular, led to one of the proofs of transfer of the Fundamental Lemma to local fields of characteristic zero, by Cluckers, Hales and Loeser. I will describe their proof, and also talk about the transfer principles of a more analytic nature that were recently proved by Cluckers, Halupczok and myself. These new transfer principles allow one to transfer statements about integrability and boundedness of functions between characteristic zero and (large) positive  characteristic, and in particular, give a proof of local integrability of Harish-Chandra characters in the large positive characteristic case.

Finally, the same model theory methods give some uniform bounds for orbital integrals, normalized by the square root of the discriminant,  on the groups G(K_v),  where G is a connected reductive group defined over a global field K, and K_v varies over almost all completions of K.

This talk is based on the work of several people, and especially on the joint project with R. Cluckers and I. Halupczok.

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