Canonical Kernels on Hermitian Symmetric Spaces

By Mads Aunskjær Bech
PhD Dissertations
October 2017

This thesis studies symmetric spaces $G/H$ with $G$ a semisimple Lie group and where the isotropy subgroup $H$ has a non-discrete center; we will consider the cases when $G/H$ is either a Hermitian, pseudohermitian, or parahermitian symmetric space. For non-compact $G$ with has finite center and $H=K$ is a maximal compact subgroup, $G/K$ is a Hermitian symmetric space of the non-compact type and the Harish-Chandra embedding realizes $G/K$ as a bounded symmetric domain $D$. Clerc and Ørsted expressed the symplectic area of a geodesic triangle in terms of the Bergman kernel $k_D$ of $D$. We prove a similar formula for the compact dual $U/K$ using a slightly different kernel $k_c$. We give a geometric characterization of the zeroes of this kernel.

Semisimple parahermitian symmetric spaces are also studied using a generalized Borel embedding due to Kaneyuki. We introduce a suitable kernel function and relate it to the symplectic area of geodesic triangles. We also treat complex parahermitian symmetric spaces $G_{\mathbb{C}}/H_{\mathbb{C}}$ separately. Here $G_{\mathbb{C}}$ and $H_{\mathbb{C}}$ are complex Lie groups with $G_{\mathbb{C}}$ simple. In this case, we introduce a holomorphic kernel function $k_{\mathbb{C}}$ and calculate the (complex) symplectic area of geodesic triangles. Finally we show how the other kernels $k_D$ and $k_c$ may be recovered from the complex kernel $k_{\mathbb{C}}$ as suitable restrictions.

Thesis advisor: Bent Ørsted
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