Whittaker Vectors and the Helgason Conjecture on Riemannian Manifolds

Helgason定理是黎曼对称空间调和分析中最基本的定理之一，它指出任何调和函数都可以表示为Furstenberg边界上的泊松积分。这里，“调和”意味着函数被所有没有常数项的不变微分算子所湮灭。对于厄米对称空间，有一些定理的推广，将Furstenberg边界替换为Shilov边界，将不变微分算子替换为更大的系统，如Hua-Johnson-Koranyi算子或Berline-Vergne算子或Lassalle算子。在这个建议中，我们提出推广Helgason理论，以及Hua-Johnson-Koranyi理论，到齐次Kaehler流形的背景下。谐波函数在科学和工程中反复出现，在热流、电学、波传播和粒子理论等领域，仅举几例。许多试图理解自然基本属性的尝试都是基于更加复杂的空间。在这样的空间中，许多重要的物理和工程问题的解决都涉及到调和函数。虽然目前的技术在黎曼对称空间中工作得相当好，但很明显，如果空间变得稍微复杂一点，当前技术必须取得根本性的进步。此外，所需要的新技术也必将产生黎曼对称空间谐波理论的新信息和新技术。我们建议继续我们以前旨在开发这项技术的工作。

AbstractPenneyOne of the most fundamental theorems in harmonic analysis on Riemannian symmetric spaces is the Helgason theorem which states that any harmonic function may be represented as a Poisson integral over the Furstenberg boundary. Here, "harmonic" means that the function is annihilated by all of the invariant differential operators which have no constant term. For a Hermitian symmetric space, there are generalizations of this theorem which replace the Furstenberg boundary with the Shilov boundary and the invariant differential operators with larger systems such as the Hua-Johnson-Koranyi operators or the Berline-Vergne operators or the Lassalle operators. In this proposal we propose generalizing the Helgason theory, as well as the Hua-Johnson-Koranyi theory, to the context of homogeneous Kaehler manifolds.Harmonic functions occur repeatedly in science and engineering, in contexts such as heat flow, electricity, wave propagation and particle theory, just to name a few. Many attempts to understand the fundamental properties of nature are based on ever more complicated spaces. The solutions to many important physical and engineering problems on such spaces involve harmonic functions. While the current techniques work reasonably well for Riemanniansymmetric spaces, it it is clear that fundamental advances in the current technology must be made if the spaces become even slightly more complicated. Furthermore, the required new technology will certainly yield new information and new techniques in harmonic theory on Riemannian symmetric spaces as well. We propose continuing our previous work which is aimed at the development of this technology.