Mathematical Sciences: Spectral Analysis of Differential Operators on Hermitian Symmetric Spaces

Sahi教授将运用代数和分析技术 研究某些不变微分算子的谱 关于Hermitian对称空间 在预期的应用中， 是（一）不变理论：建立恒等式推广 经典的Capelli恒等式，（ii）表示理论： 研究某些可约性和可统一性问题，（iii） 分析：描述一个 形式上的真实的Jordan代数并推广了麦克斯韦的理论 极点的球谐函数，以这种设置，（四）数论： 推广了Shimura关于某些非- 全纯向量值Eisenstein级数 上述工作涉及的理论 代表团体。 群论基本上就是 对称性。 举一个简单的例子，当系统 问题是不变的下的位置的变化， 空间的起源，翻译组自然产生。 虽然组是抽象对象，但特定情况需要 对称群的具体实现或“表示”。

Professor Sahi will apply algebraic and analytic techniques to study the spectrum of certain invariant differential operators on Hermitian symmetric spaces. Among the intended applications are (i) invariant theory: to establish identities generalizing the classical Capelli identity, (ii) representation theory: to study certain reducibility and unitarizability questions, (iii) analysis: to describe the homogeneous distributions on a formally real Jordan algebra and to extend Maxwell's theory of poles of spherical harmonics to this setting, (iv) number theory: to extend results of Shimura on the arithmeticity of certain non- holomorphic vector-valued Eisenstein series. The work described above involves the theory of representations of groups. Group theory is basically the theory of symmetry. To take a simple example, when the system in question is invariant under a change in the position of the origin of space, the group of translations naturally arises. While groups are abstract objects, particular situations demand concrete realizations or "representations" of the symmetry group.