Green functions and relative trace formulas

给出了与任意实数秩的不定酉群相关的对称空间和实数秩为1的正交群的一个相对迹公式。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公式是由酉群的等差商上自同构形式的周期积分的平方范数的平均值所引起的离散测度序列所近似的。(2)给出了l2归一化拉普拉斯特征函数在秩一正交群的等差商上的周期积分的某些平均值的渐近公式；该公式应视为拉普拉斯特征值多重性谱计数函数的Weyl定律的类似物。

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.