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

Sahi教授将应用代数和分析技巧研究厄米特对称空间上某些不变微分算子的谱。其中预期的应用是(I)不变量理论：建立推广经典Capelli恒等式的恒等式，(Ii)表示理论：研究某些可约化和单一化问题，(Iii)分析：描述形式实Jordan代数上的齐次分布，并将Maxwell的球谐极点理论推广到此设置，(Iv)数论：推广了Shimura关于某些非全纯向量值Eisenstein级数算术性的结果。上述工作涉及群的表示理论。群论基本上是对称论。举一个简单的例子，当所讨论的系统在空间原点的位置改变时是不变的，自然就产生了翻译群。虽然群是抽象的对象，但特殊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.