VECTOR BUNDLES ON MANIFOLDS

在这个项目中，我们从以下不同的角度研究了流形上的向量束。1)代数几何方法。2)研究无穷维李代数和量子群的代数解析方法。， 3)研究经典群上模形式和l函数的数论方法。4)用费曼路径积分研究无穷维李群的量子化的微分几何方法。在1)中，我们通过研究P^n上与2秩束相关的行列式的代数几何、微分几何和拓扑性质，得到了射影空间P^n （n <大于等于bbbb4）上2秩束分裂为线束的充要条件，从而给出了关于P^n上向量束分裂的重要猜想的新途径。在2)中，我们证明了仿射李代数上不可约最高权模的Kazdan-Lusztig型的一个特征公式，并构造了P^1束的Radon变换理论，将这一结果推广到有理最高权情况。并将Grassmann变异上的一般超几何方程推广到Hermite对称空间上的微分方程，研究了上述Radon变换与其完整性的一个条件之间的关系。3)详细研究了正交群上具有半整数权值的尖形升格为模形的l -函数，给出了l -函数的傅里叶系数的积分表达式，并证明了在某些技术条件下的亚纯延拓和泛函数方程，这可以看作是二次型Siegel尖形上Kohnen-Skoruppa结果的推广。4)利用Feynman路径积分构造了无限维李群（如kac - moody李群）的表示模块，并从几何上研究了无限维流形上的积分算子。少

In this project, we studied vector bundles on manifolds from the following various points of view. 1) algebraic geometric method., 2) algebraic analytic method to study infinite dimensional Lie algebras and quantum groups., 3) number theoretic method to study modular forms and L-functions on classical groups., 4) differential geometric method to study quantization of infinite dimensional Lie groups by Feynman path integrals. In 1), we obtained a necessary and sufficient condition for a rank 2 bundle on projective space P^n (n <greater than or equal> 4) to split into line bundles by researching algebro-geometric, differential geometric and topological properties of determinantal varieties associated to rank 2 bundles on P^n which gives us a new approach to important conjectures concerning splitting of vector bundles on P^n. In 2), we proved a character formula of Kazdan-Lusztig type for irreducible highest weight modules over affine Lie algebras with integral highest weight case and als … More o extended this result to the rational highest weight case by constructing a theory on Radon transformations of P^1 bundles. Moreover we generalized generic hypergeometric equations on Grassmann varieties to differential equations on Hermite symmetric spaces and researched relations between the above Radon transformations and a condition concerning their holonomicity. In 3), the L-functions which are liftings of cusp forms with half integer weights to modular forms on orthogonal groups were studied in detail and we gave the L-functions an expression by integrals in terms of their Fourier coefficients and proved the meromorphic continuation and the functional equation under some technical conditions which can be viewed as a generalization of the Kohnen-Skoruppa's result on quadratic Siegel cusp forms. In 4), we constructed representation modules of infinite dimensional Lie groups, e.g.Kac-Moody Lie groups by Feynman path integrals and studied integral operators on infinite dimensional manifolds geometrically. Less