A special Case of the Fundamental Lemma I

Abstract

In [36] the existence of matching functions (f, f^M) for endoscopic orbital integrals is shown for the group GSp(4, F) and its unique endoscopic group M. However, the proof in [36] only gives the existence of matching functions. It does not explicitly describe the correspondence between the functions that have matching orbital integrals. However, it is this additional information which is relevant for many applications. In particular, it is important to know this correspondence for all functions f in the K-spherical Hecke algebra of all K-bi-invariant functions with compact support on G(F). The fundamental lemma asserts that there exists a specific ring homomorphism b between the spherical Hecke algebras of the groups G(F) and M(F) for which the pairs (f, fM) = (f, b(f)) define matching functions (if the transfer factors and measures are suitably normalized). Using the trace formula, one can reduce this assertion to the case of one particular function in the spherical Hecke algebra, the unit element f = 1K. In this special case, of course, f^M = b(1K) = 1K^M. The reduction of the fundamental lemma to this special case can be found in [35] and also in Chap. 9. So it is enough to prove that f = 1^K and f^M = 1K^M are functions with matching orbital integrals.