非黎曼局部对称空间的Poincaré级数
Poincaré series for non-Riemannian locally symmetric spaces
摘要：
拉普拉斯算子的离散谱在约化对称空间和黎曼局部对称空间上得到了广泛的研究。在这里，我们研究它的第一次在一般设置的非黎曼，还原，局部对称空间。对于任何非黎曼的约化对称空间X，其上拉普拉斯算子的离散谱是非空的，并且对于任何等距的离散群Gamma，其在X上的作用是充分适当的，我们构造了X_{Gamma}上拉普拉斯算子的L^2-本征函数：=Gamma \X对于一个无限集的本征值。这些本征函数作为广义Poincaré级数得到，即作为X上拉普拉斯算子的本征函数在伽玛轨道上的和到X_{Gamma}的投影。证明了我们构造的Poincaré级数在Gamma的任何小变形之后，对于一大类群Gamma仍然收敛，并且定义了非零的L^2-函数。换句话说，我们构造的特征值的无限集在小变形下是稳定的。这与非零离散谱在紧致黎曼曲面的Teichmüller空间上变化的经典设置形成对比。我们实际上构造了X_{Gamma}上不变微分算子的整个交换代数的联合L^2-本征函数。
英文摘要：
The discrete spectrum of the Laplacian has been extensively studied on reductive symmetric spaces and on Riemannian locally symmetric spaces. Here we examine it for the first time in the general setting of non-Riemannian, reductive, locally symmetric spaces. For any non-Riemannian, reductive symmetric space X on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Gamma whose action on X is sufficiently proper, we construct L^2-eigenfunctions of the Laplacian on X_{Gamma}:=Gamma\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to X_{Gamma} of sums, over the Gamma-orbits, of eigenfunctions of the Laplacian on X. We prove that the Poincaré series we construct still converge, and define nonzero L^2-functions, after any small deformation of Gamma, for a large class of groups Gamma. In other words, the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface. We actually construct joint L^2-eigenfunctions for the whole commutative algebra of invariant differential operators on X_{Gamma}.
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