Endoscopic Transfer of Orbital Integrals in Large Residual Characteristic Julia Gordon and Thomas Hales

This article constructs Shalika germs in the context of motivic integration, both for ordinary orbital integrals and κ-orbital integrals. Based on transfer principles in motivic integration and on Waldspurger’s endoscopic transfer of smooth functions in characteristic zero, we deduce the endoscopic transfer of smooth functions in sufficiently large residual characteristic. We dedicate this article to the memory of Jun-Ichi Igusa. The second author wishes to acknowledge the deep and lasting influence that Igusa’s research has had on his work, starting with his work as a graduate student that used Igusa theory to study the Shalika germs of orbital integrals, and continuing today with themes in motivic integration that have been inspired by the Igusa zeta function. This article establishes the endoscopic matching of smooth functions in sufficiently large residual characteristic. The main conclusions are based on four fundamental results: LanglandsShelstad descent for transfer factors [26], Ngô’s proof of the fundamental lemma [28], Waldspurger’s proof that the fundamental lemma implies endoscopic matching of smooth functions in characteristic zero [34], and the Cluckers-Loeser version of motivic integration [12], including transfer principles for deducing results for one nonarchimedean field from another nonarchimedean field with the same residue field [13]. We use recent extensions of the transfer principle to transfer linear dependencies from one field to another [9]. We note that the term transfer is used with two separate meanings in this article. Endoscopic matching1 refers to the matching of κ-orbital integrals on a reductive group with stable orbital integrals on an endoscopic group, in a form made precise by the Langlands-Shelstad transfer factor. On the other hand, transfer principles refer to the transfer of first-order statements or properties of constructible functions from one nonarchimedean field to another nonarchimedean field with the same residual characteristic. In this article, we will refer to endoscopic matching, transfer factors, and transfer principles. In our main results, the constraints on the size of the residual characteristic are not effective. This means that our results are not known to apply to any particular nonarchimedean field of positive characteristic. This seems to be a serious limitation of our methods. Nonetheless, we hope that our results about the constructibility of Shalika germs can serve as a further illustration of the close connection between harmonic analysis of p-adic groups and motivic integration. It is called endoscopic transfer in the literature, title, and abstract, but we prefer to call it endoscopic matching in the body of the article because of the other uses of the word transfer. We avoid the awkward but apt phrase “transfer of transfer.”