ON AUTOMORPHIC L-FUNCTIONS OF GENERAL SYMPLECTIC AND UNITARY GROUPS OF RANK TWO

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We have continued the projects concerning the automorphic L-functions of gereral symplectic and unitary groups of rank two. More specifically one of the main projects is to prove the generalization of Siegfried Boecherer's conjecture concerning the central critical values of the degree four L-functions for the Siegel eigen cusp forms of degree two. Our method is to establish certain relative trace formulas, which may be regarded as natural generalizations of Jacquet's relative trace formulas which have given another proof of celebrated Waldspurger's theorem on the relation between the torus period for GL(2) and the central critical values of automorphic L-functions for GL(2).In order to establish a relative trace formula, proving the fundamental lemma is the first and crucial step. We have proved the fundamental for the unit element of the Hecke algebra already and published the result as No. 782 of the Memoirs of the AMS. During the period supported by this grant, we worked on extending the fundamental lemma from the unit element to the entire Hecke algebra. We have discovered that, by applying the theory of Macdonald polynomials to the explicit formulas for the Bessel model, the evaluation of the Kloosterman orbital integral for the general element in the Hecke algebra is reduced to the computation of general Kostka numbers and that of degenerate Kloosterman orbital integrals for the unit element of the Hecke algebra. We have evaluated all of them. Now our remaining task is to compare the linear combinations of these corresponding to the both sides of the trace formula and to make sure they match.