On the Jacobson Element and Generators of the Lie Algebra grt in Nonzero Characteristic

We state a conjecture (due to M. Duflo) analogous to the Kashiwara-Vergne conjecture in the case of a characteristic p > 2 , where the role of the Campbell-Hausdorff series is played by the Jacobson element.We prove a simpler version of this conjecture using Vergne's explicit rational solution of the Kashiwara-Vergne problem.Our result is related to the structure of the Grothendieck-Teichmller Lie algebra grt in characteristic p : we conjecture existence of a generator of grt in degree p -1 , and we provide this generator for p = 3 and p = 5.