Decomposition of splitting invariants in split real groups

For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad construct in [LS1] a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group. In applications of harmonic analysis and representation theory of reductive groups over local fields to questions in number theory, a central role is played by the theory of endoscopy. This theory associates to a given connected reductive group G over a local field F a collection of connected reductive groups over F, often denoted by H, which have smaller dimension (except when H = G), but are usually not subgroups of G. The geometric side of the theory is then concerned with transferring functions on G(F) to functions on H(F) in such a way that suitable linear combinations of their orbital integrals are comparable, while the spectral side is concerned with transferring “packets” of representations on H(F) to “packets” of representations on G(F) in such a way that suitable linear combinations of their characters are comparable. In both cases, the comparison involves certain normalizing factors, called geometric or spectral transfer factors. Over the real numbers, the theory of endoscopy was developed by Diana Shelstad in a series of profound papers including [S1], [S2], [S3], [S4] (but see also [S6], [S7], [S8] for a more modern point of view and additional results), in which she defines geometric and spectral transfer factors and proves that these factors indeed give a comparison of orbital integrals and character formulas between G and H. A very subtle and complicated feature of the transfer factors was the need to assign a ±-sign to each maximal torus in G in a coherent manner, and Shelstad was able to prove that this is possible. A uniform and explicit definition of geometric transfer factors for all local fields was given in [LS1]. An explicit construction of spectral transfer factors over the real numbers was given in [S7], while over the p-adic numbers their existence is still conjectural (see however [Kal] for a proof of the spectral transfer in a special case). The structure of transfer factors is quite complex – both the geometric and the real spectral ones are a product of multiple terms of group-theoretic or Galois-cohomological nature. There are numerous choices involved in the construction of each individual term, but the product is independent of most choices. One term that is common to both the geometric and the real spectral transfer factors is called ∆I . It is regarded as the most subtle and is the one that makes explicit the choice of coherent collection of signs in Shelstad’s earlier work. At its heart is a Galois-cohomological object, called the splitting invariant. The splitting invariant is an element of H1(F,T ) associated to any maximal torus T of a quasi-split semi-simple simply-connected group G, whose construction occupies the first half of [LS1, Sec.2]. It depends on the choice of a splitting (T0, B0, {Xα}α∈∆) of G as well as a-data {aβ}β∈R(T,G). AMS subject classification: Primary 11F70, 22E47; Secondary 11S37, 11F72, 17B22