Unipotent Representations and Unipotent Classes SL(N)

Abstract

To achieve the stabilization of the trace formula for a reductive group [18], there are two related problems that must first be solved. One is the matching of the sorbital integrals of smooth functions on a reductive group G (over a p-adic field of characteristic zero) with stable orbital integrals on its endoscopic groups H. Igusa theory clarifies the nature of this first problem [11], [12], [13], [19], [22], [23]. The second problem is the fundamental lemma, which asserts the compatibility of the matching of smooth functions with the Hecke algebras on G and H. Buildings have been used to study this problem [25]. The statements of the problems are closely related, and the main results of this paper give a method of reducing one problem to the other. If the first problem (matching smooth functions) is solved for SL(n), then the second problem (the fundamental lemma for the Hecke algebras) follows as a consequence, at least for the identity of the algebra. The solutions of the two problems are linked by making a detailed study of the representations and the classes of SL(n). Representations with Iwahori fixed vectors on SL(n, F) are called unipotent in this paper to emphasize the close relationship between conjugacy classes and these representations. Not long after this paper was written, Waldspurger refined, reformulated and greatly extended the results of this paper. This paper made a modest contribution to this subsequent work of his. He eliminated the hypothesis that functions on SL(n) have matching smooth functions, by making clever use of a strengthened version of Kazhdan's lemma. The opinion of the experts is still divided, will the solution of the fundamental lemma for other reductive groups come from the matching of smooth functions, or will it come from even stronger versions of Kazhdan's lemma. This much is clear: the fundamental lemma no longer needs to be viewed as an isolated combinatorial problem in buildings. Howe's conjecture, uniform germ expansions, homogeneity of germs, compact traces, orbits, and Kazhdan's density theorem give us a framework through which the fundamental lemma may be understood. In the final portion of the introduction, I would like to state a theorem that has considerable interest beyond its usefulness for the fundamental lemma. This theorem will be used in Section 1 to produce a uniform germ expansion. The