# 01-08【孟国武】五教5206 吴文俊数学重点实验室数学物理系列报告之2019-19

1. This orbit is diffeomorphic to $\mathrm{SO}_0(2, 2k+2)/\mathrm{U}(1, k+1)$. As a result, it is pre-quantizable.

2. This orbit is the total space of a fiber bundle with base space being the total cotangent space of the punctured euclidean space of dimension $2k+1$ and the fiber being diffeomorphic to $\mathrm{SO}(2n)/\mathrm{U}(n)$. As a result, it admits a canonical polarization.

3. The geometric quantization of this orbit with its canonical polarization yields the Hilbert of square integrable sections of a Hermitian vector bundle over the punctured Euclidean space in dimension $2k+1$; moreover, this Hilbert space provides a geometric realization for the unitary highest weight $\frak{so}(2, 2k+2)$-module with highest weight $(-k-|\mu|, \underbrace{ |\mu|, \ldots, |\mu|}_k, \mu).$

The above results in Lie theory is obtained from the study of magnetized Kepler models in dimension $2k+1$.