Construction of a singular theta-lifting for unitary groups U(p,q)

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The aim of the present project is the construction of a singular theta-lifting for unitary groups U(p,q) with a Millson-type kernel function.A lifting of this type is well-studied for orthogonal groups and has found a number of arithmetic and geometric applications. It originates in work of Kudla and Millson and was examined further by Bruinier and Funke. It takes weak Maass forms and lifts them to differential forms on the symmetric space of the orthogonal group. Also, it is adjoint to another lifting, the Kudla Millson lift. In the hermitian case, O(p,2) this property is also shared by the geometric Borcherds lift, not however in the general case O(p,q). For unitary groups, the existence of a lifting of this type also follows from the work of Kudla and Millson. However this has not been studied further as yet. Contrastingly, the Borcherds lift is fairly well-studied for the case U(p,1) or U(q,1), respectively. Constructed by Hofmann, its geometric properties were more closely examined by Bruinier, Howard and Yang. Also, a construction for U(p,q) has been given by Hufler.In this general case, however, a lifting with a Millson-type kernel would be of considerable interest for geometric applications. This is the starting point for the project: The kernel function, which is defined by a differential equation, is to be determined explicitly. The singular theta-lifting is to be constructed and attached Green's currents to be calculated. Further, it is projected to study the behavior of the lift on boundary components of the symmetric domain. This is expected to yield identities between generating series with applications to the Kudla program. A further planned objective is to calculate the Fourier-Jacobi expansion of the lift explicitly.