VECTOR BUNDLES ON MANIFOLDS

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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