The fundamental lemma of Jacquet and Ye in positive characteristic. (Le lemme fondamental de Jacquet et Ye en caractéristique positive.) (French) Zbl 1047.11517

Ngô, Báo Châu

Duke Math. J. 96, No. 3, 473-520 (1999).

In the trace formula approach to the lifting problems of automorphic representations of reductive groups, the matching of local orbital integrals has been reduced to that for the unit element of the Hecke algebra related to relevant groups, which is usually called the fundamental lemma. In general, the fundamental lemma is one of the most difficult open problems in the trace formula approach to the theory of automorphic representations.
The fundamental lemma of Jacquet and Ye is the one in the relative trace formula approach to the quadratic base change problem for \(\text{GL}(n)\) [H. M. Jacquet and Y. Ye, Bull. Soc. Math. Fr. 120, 263–295 (1992; Zbl 0785.11032)]. An equivalent formulation of the fundamental lemma of Jacquet and Ye is an identity of two distributions, called Jacquet and Ye’s conjecture. One of the distributions is of generalized Kloosterman integral type and is defined over \(\text{GL}(n,E)\), and the other distribution is of relative Kloosterman integral type and is defined over \(\text{GL}(n,F)\), where \(E/F\) is a quadratic extension. Jacquet and Ye verified their conjecture for \(\text{GL}(2)\) and \(\text{GL}(3)\) over a local non-Archimedean field of characteristic zero.
The aim of the paper under review is to prove the above conjecture of Jacquet and Ye for \(\text{GL}(n)\) over local fields of positive characteristic. The main idea is to interpret the function-field analogies of the distributions of Kloosterman type defined by Jacquet and Ye as the trace of the Frobenius endomorphism over the \(l\)-adic cohomology of a certain algebraic variety, in the sense of Grothendieck. The conjectural identity of Jacquet and Ye in the function-field case becomes a certain quasi-invariant property of the trace with respect to a simply defined involution. It is a beautiful geometric argument.
The original conjecture of Jacquet and Ye over local fields of characteristic zero is still open for \(\text{GL}(n)\), \(n\geq 4\).

Reviewer: Dihua Jiang (Minneapolis)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
14F20	Étale and other Grothendieck topologies and (co)homologies

Citations:

Zbl 0785.11032

References:

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