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Unipotent representations and unipotent classes in \(SL(N)\). (English) Zbl 0810.22008

There are (at least) two basic problems encountered in attempts to stabilize the trace formula of a reductive group \(G\). The first is the matching of the unstable orbital integrals of smooth functions on \(G\) (over a \(p\)-adic field of characteristic zero) with stable orbital integrals on its endoscopic groups \(H\); the second is the fundamental lemma asserting the compatibility of this matching with the Hecke algebras of \(G\) and \(H\). In this paper, the solutions of these problems for \(SL(n)\) are linked by making a detailed study of the unipotent representations and the unipotent classes of \(SL(n)\). In particular, it is shown that if the first problem is solved (the matching of smooth functions), then the second problem follows as a consequence, at least for the identity element of the algebras. One of the main tools used is a uniform germ expansion for the Iwahori Hecke algebra \({\mathcal H}(G, B)\) (the algebra of Hecke functions on \(G\) bi-invariant for an Iwahori subgroup \(B\) in the maximal compact \(K\)). Not long after this paper was written, J.-L. Waldspurger refined, reformulated and greatly extended its results [Can. J. Math. 43, 852-896 (1991; Zbl 0760.22026)]. Although the proof of the fundamental lemma is not yet complete for arbitrary \(G\), it should no longer be viewed as an isolated, combinatorial problem in the theory of buildings; but rather as a consequence of the general theory of germ expansions, Howe’s conjecture, etc.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 0760.22026
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