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

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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.