The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4)

We announce, in the case of the group GSp(4), an equality of two local inte grals. One is a Kloosterman integral on the Bessel subgroups of GSp(4) and the other is a Kloosterman integral on the Novodvorsky subgroups of GSp(4). We conjecture that Jacquet's relative trace formula for GL(2) in [7], wher e Jacquet has given another proof of Waldspurger's result [9], generalizes to GSp(4). We believe that this approach will lead us to a proof and also a precise formulation of a conjecture of Bocherer [1]. Support for this conj ecture may be found in the important paper of Bocherer and Schulze-Pillot [ 2]. Our result serves as the fundamental lemma for our conjectural relative trace formula for the main relevant double cosets. (C) Academie des Scienc es/Elsevier, Paris.