Green functions and relative trace formulas

对任意真实的秩的不定酉群所对应的对称空间和真实的秩为1的正交群，给出了一个相对迹公式。作为应用，我们证明了两类渐近公式：（1）证明了酉群对称空间的Plancherel公式可由酉群算术商上自守形式的周期积分的平方模的平均所产生的离散测度序列逼近。(2)本文给出了一阶正交群等差商上的L ~ 2正规化拉普拉斯本征函数周期积分的一个渐近公式，该公式可看作是谱计数函数的Weyl定律在拉普拉斯本征值重数下的一个类比.

We develop a version of relative trace formula for symmetric spaces associated with indefinite unitary groups of arbitrary real rank and for orthogonal groups of real rank one. As an application, we prove two kinds of asymptotic formulas ; (1) It is proved that the Plancherel formula of the symmetric space for unitary group is approximated by certain sequence of discrete measures arising from the average of square norm of period integrals of automorphic forms on an arithmetic quotient of the unitary group. (2) We showed an asymptotic formula for certain average of period integrals of L2-normalized Laplace eigenfunctions on an arithmetic quotient of rank one orthogonal group ; the formula should be regarded as an analogue of Weyl's law for spectral counting function for multiplicity of Laplace eigenvalues.