Développement fin de la contribution unipotente à la formule des traces sur un corps global de caractéristique p>0, I
Bertrand Lemaire
TL;DR
The paper develops a comprehensive rational theory of Kempf-Rousseau-Hesselink unipotent F-strata for connected reductive groups G over a field F, including the definition of F-lames and F-strata and the Kirwan-Ness framework in both geometric and rational settings. It then applies this theory to the unipotent contribution in the trace formula over global function fields of characteristic p>0, outlining a finite stratification and a PolExp-type expansion for the unipotent kernel, while noting the need for future work to realize a full local-global product decomposition in step II.2 and to handle infinite orbit counts in positive characteristic. The analysis systematically treats induction by parabolic subgroups, descent across separable extensions, and the interaction between unipotent varieties and their Lie algebras, setting a robust foundation for open problems in positive characteristic trace formulas. The work also clarifies how F-strata generalize Arthur-style geometric orbits and how rationality aspects influence the stratification and stability criteria, with implications for future arithmetic applications.
Abstract
For a field $F$ and a connected reductive group $G$ defined over $F$, we develop a theory of Kempf-Rousseau-Hesselink unipotent $F$-strata in $G(F)$ that should allow us to attack open problems in positive characteristic. As an application, we use this theory to establish the fine expansion of the unipotent contribution to the (non-twisted) trace formula over a global field of characteristic $p>0$. The unipotent $F$-strata play here the role of the unipotent geometric orbits in Arthur's work over a number field. The expansion in terms of products of local distributions is not discussed here; it will be the subject of further work.