Towards stable-stable transfer involving symplectic groups

This thesis investigates the transfer formulas for orbital integrals, in the context of the modern Langlands' program for reductive algebraic groups. In the modern theory, there are two very different transfer theorems to be accomplished. First, there is endoscopic transfer, which relates, via an appropriate embedding of their L-groups, a given group to a particular family of groups, its endoscopic groups. Here, deep theorems are known in great generality. Such a theory, however, is preliminary to the second transfer, which is much less understood. At the same time, this second transfer is generally viewed as the more fundamental of the two, involving any connected reductive group related to the given group by an L-homomorphism. Prompted by the results for endoscopic transfer, our study focuses first on groups defined over an archimedean field. To do so, we study the geometric objects, orbital integrals, on real or complex reductive Lie groups, for which there is a basic theory due to Harish-Chandra on which to build, focusing on the split and hyperbolic symplectic groups to develop details. Concrete expressions of the final transfer formulas are notably different from those for endoscopic transfer, and the algebraicity condition on the ambient group is critical in their development. Specifically, our main focus is on a refined version of the structure of the lattice of maximal tori and on the role this plays in developing the concrete expressions for transfer. Our structural results apply to symplectic groups of all sizes and their inner forms, and we develop an explicit transfer formula in the rank one case.Ph.D.Includes bibliographical reference