A Study on Some Spectral Invariants on Cauchy-Riemann Manifolds with Group Action(1/2)

In this project the PI plan to study the asymptotic expansion of the analytic torsion on a compact CR manifold of high codimension with a compact Lie group locally free CR action. Recently S. Finski generalized the Bismut-Vasserot asymptotic formula to setting of compact complex orbifolds. This project should be closely related to the results of Finski on Bismut-Vasserot's asymptotic formula on compact complex orbifolds. One other possible application will be a generalization of the Bismut-Vasserot asymptotic formula for symmetric powers of a positive vector bundle over a compact complex manifold.Another project is jointly with Chin-Yu Hsiao (Academia Sinica) and Guokuan Shao (Sun Yat-Sen University (Zhuhai), China), we study G-equivariant Szego kernels on a compact CR manifold with compact Lie group action G associated with all equivalent classes of irreducible unitary representations of G. We shall establish G-equivariant Boutet de Monvel-Sjostrand type theorems. When the CR manifold is strongly pseudoconvex with S^1 action, we shall compute coefficients of the first few terms of the asymptotic expansions of G-equivariant Szego kernel functions. The main tool of our appraoch is Hormander's stationary phase formula. We expect that the coefficients of the first few lower order terms of the G-equivalent Szego kernel function's expansion for functions will contain some geometric quantities.