If G is a reductive group
quasi-split over a number field F and K the kernel of the trace formula, one can
integrate K in the two variables against a generic character of a maximal
unipotent subgroup N to obtain the Kuznietsov trace formula. If H is the
fixator of an involution of G, one can also integrate K in one variable over
H and in the other variable against a generic character of N: one obtains
then a “relative” version of the Kuznietsov trace formula. We propose as a
conjecture that the relative Kuznietsov trace formula can be “matched” with
the Kuznietsov trace formula for another group G′. A consequence of this
formula would be the characterization of the automorphic representations of G
which admit an element whose integral over H is non-zero: they should
be functorial image of representations of G′. In this article, we study the
case where H is the symplectic group inside the linear group; we prove the
“fundamental lemma” for the situation at hand and outline the identity of the trace
formulas. This case is elementary and should serve as a model for the general
case.
