Parabolic Transfer for Real Groups

This paper is the second of two articles in real harmonic analysis. In the first paper [A14], we established asymptotic formulas for some natural distributions on a real group. In this paper we shall establish important relationships among the distributions as the group varies. The group is the set of real points of a connected reductive group G over R. The distributions are weighted orbital integrals JM (γ, f) on G(R), and their invariant counterparts IM (γ, f). Here, M ⊂ G is a Levi subgroup of G, while γ ⊂ MG-reg(R) is a strongly G-regular conjugacy class in M(R). The relationships are defined by the invariant transfer of functions on G(R) to functions attached to endoscopic groups of G. This necessitates our working with the invariant distributions IM (γ, f). We refer the reader to the introduction of [A14] for some general remarks on these objects. The distributions IM (γ, f) are the generic archimedean terms in the invariant trace formula. We cannot review the trace formula here. The reader might consult the introductions to [A13] and its two predecessors for a brief summary. The purpose of the paper [A13] was to stabilize the invariant trace formula, subject to a condition on the fundamental lemma that has been established in some special cases. The stable trace formula is a milestone of sorts. It is expected to lead to reciprocity laws, which relate fundamental arithmetic data attached to automorphic representations on different groups. The stable trace formula of [A13] relies upon the results of this paper (as well as a paper [A16] in preparation). This has been our guiding motivation. The relevant identities among the nonarchimedean forms of the distributions IM (γ, f) were actually established in [A13]. They were a part of the global argument that culminated in the stable trace formula. As such, they are subject to the condition on the fundamental lemma mentioned above. Our goal here is to establish the outstanding archimedean identities. We shall do so by purely local means, which are independent of the fundamental lemma. To simplify the Introducton, we assume that the derived group of G is simply connected. The identities then relate the invariant distributions onG(R) with stable distributions on endoscopic groups G′(R). We recall that a stable distribution on G′(R) depends only on the average values assumed by a test function over strongly regular stable conjugacy classes in Greg(R), which is to say, intersections of G ′ reg(R)