1 J an 1 99 5 THE FUNDAMENTAL LEMMA FOR

The authors establish the fundamental lemma for a relative trace formula. This trace formula compares generic automorphic representations of GSp(4) with auto-morphic representations of GL(4) which are distinguished with respect to a character of the Shalika subgroup, the subgroup of matrices of 2 × 2 block form g 0 X g. The fundamental lemma, giving the equality of two orbital integrals, amounts to a comparison of certain exponential sums arising from these two different groups. On the GSp(4) side one has the Kloosterman sums for this group, and on the GL(4) side certain, new, relative Kloosterman sums. To show that these are equal for each relevant Weyl group element, the authors compute the Mellin transforms of the sums and match them in all cases.