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Coadjoint orbits of Sternberg type and their geometric quantization
Coadjoint orbits of Sternberg type and their geometric quantization
教师介绍

本讲教师:孟国武
所属学科:理科
人  气:276

课程介绍
摘要:Let $kge 1$ be an integer and $mu$ be the half of a {it nonzero} integer. The following statements hold for the elliptic co-adjoint orbit of the real Lie algebra $mathfrak{so}(2, 2k+2)$ that corresponds to the dominant weight $(underbrace{|mu|, ldots, |mu|}_{k+1}, mu)$. 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 $2k+1$.

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