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

我们继续了关于一般辛群和酉群的自守L-函数的研究。更具体地说，其中一个主要项目是证明推广的齐格弗里德Boecherer的猜想有关的中心临界值的程度四L-功能的西格尔本征尖形式的程度二。我们的方法是建立某些相对迹公式，这些公式可以看作是Jacquet相对迹公式的自然推广，Jacquet相对迹公式给出了著名的Waldspurger定理关于GL（2）的环面周期与GL（2）的自守L-函数的中心临界值之间关系的另一个证明。我们已经证明了Hecke代数的单位元的基本原理，并将结果发表在AMS Memoirs的第782号中。在此期间的资助，我们致力于延长基本引理从单位元到整个Hecke代数。我们发现，通过将Macdonald多项式理论应用于Bessel模型的显式公式，Hecke代数中一般元的Kloosterman轨道积分的计算可归结为Hecke代数中一般Kostka数和单位元的退化Kloosterman轨道积分的计算.我们已经评估了所有这些。现在，我们剩下的任务是比较对应于迹公式两侧的这些线性组合，并确保它们匹配。

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.