Section 0. Introduction Let F be p-adic field of characteristic zero, and G a connected reductive algebraic group defined over F. A main objective for harmonic analysis on G(F) is to understand the invariant distributions on G(F), e.g. the characters of the irreducible admissible representations of G(F). By analogy with the case of a finite group and the experience from real groups, the orbital integrals ought to be the simpler invariant distributions to study. The germ expansion of Shalika, as well as the local admissible distribution expansion of Harish Chandra, and the recent work of Waldspurger (cf. [33]), suggest that unipotent (nilpotent) orbital integrals have a special role to play among general orbital integrals. As was first observed by Langlands, the invariant local (and global) harmonic analysis is complicated by the difference between conjugacy and stable conjugacy. To facilitate such a study, the notion of endoscopic datum was introduced. The main objective of the theory of endoscopy in local harmonic analysis is to understand the G(F)-invariant distributions by comparing them to stable (in the sense of Langlands, see [17]) H(F)-invariant distributions for the various endoscopic groups H attached to G. This comparison is dual to a, still conjectured, map: / »-• fH called smooth matching, between smooth functions on G(F) and H(F) satisfying precisely defined identities between ^-regular semi-simple orbital integrals of / and stable regular semi-simple orbital integrals of fH. In light of these considerations, two of the main problems in the study of unipotent orbital integrals are the following: Problem 1. Given a stable unipotent class in G(F), organize the Q{F)-classes within it into "packets" such that an appropi- ate linear combination of the integrals over the G(F)-orbits within a given packet (with respect to related measures) is a stable distribution. Problem 2. Given a packet as above, find the explicit form for the endoscopic transfer of the stable distribution associated with that packet. Stated differently, the problem is to find the 'transfer factors" for unipotent orbital integrals. It is worth pointing out two important and concrete reasons for seeking a solution to the above stated problems. The first reason is related to the problems of matching semi-simple orbital integrals. A precise answer (or compelling evidence for a suggested answer) will provide a good understanding of the precise mechanism according to which the stable Shalika germs in an endoscopic group do transfer to /c-germs. This knowledge (or even insight) may very well be needed in approaching the fundamental lemma. A second reason is related to the problem of finding working criteria for deciding when the character of a virtual admissible representation is a stable distribution. In view of Waldspurger's result (cf. [35]) stating that the Fourier transform of a stable distribution is a stable distribution, an answer to Problem 1 will provide us with a necessary condition for the stability of a virtual admissible character in terms of the constants appearing in the local character expansion of Harish-Chandra. Since the endoscopic transfer of a stable unipotent distribution ought to be a linear combination of unipotent orbital integrals, an answer to Problem 2 requires a map between stable unipotent orbits in H(F) and stable unipotent conjugacy classes in G(F). A good candidate for such a map seems to be endoscopic induction which we attribute to Lusztig, since it was first introduced by him (cf. [20] page 345 see also [30]) on the set of stable special orbits. 1
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