带有李群作用CR流形上的Szegő核

在CR几何中，Szegő核是至关重要的研究对象，它在多复变、复几何、辛几何、几何量子化和半经典分析等领域有许多应用。在本项目中我们将研究带有紧致李群G作用的CR流形上的Szegő核。主要内容是以下三点：第一，我们将利用微局部分析和CR流形与李群的Fourier分析等方法，尝试把G-等变Szegő核表示成Fourier积分算子的形式，在同时带有圆作用的情况下，将得到它的渐进展开及其前三项系数；第二，我们将定义G-等变Berezin-Toeplitz量子化算子，利用稳定相方法研究它的渐进展开；第三，我们将发展Szegő核函数的一致估计的方法，建立G-等变CR函数的零点等分布定理。

The study of Szegő kernels is very important in CR geometry, which has many applications in several complex variables, complex geometry, symplectic geometry, geometric quantization, semiclassical analysis, etc. In this program, we will study Szegő kernels on a CR manifold with a compact Lie group G action. We mainly consider the following three topics: Firstly, we will try to express G-equivariant Szegő kernels as Fourier integral operators by using microlocal analysis and Fourier analysis on CR manifolds and Lie groups. When the CR manifold admits also a circle action, we will obtain asymptotic expansions of G-equivariant Szegő kernels and compute coefficients of first three lower order terms. Secondly, we will define G-equivariant Berezin-Toeplitz quantization and study its asymptotic expansion by using stationary phase formula. Thirdly, we will establish equidistribution theorems for zeros of G-equivariant CR functions by developing the method of uniform estimates on Szegő kernel functions.