Jolanta Marzec

Construction of Poincaré-type series by generating kernels

Let Gamma be a discrete subgroup of PSL_2(R) of the first kind whose fundamental domain is of finite volume, and let GGamma be its cover in SL_2(R). Consider the space of twice continuously differentiable, square-integrable functions on H, which transform in a suitable way with respect to a multiplier system of real weight k under the action of GGamma. The space of such functions admits action of the hyperbolic Laplacian Delta_k of weight k. Following an approach of Jorgenson, von Pippich and Smajlović (where k=0), we use spectral expansion associated to Delta_k to construct wave distribution and then identify conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincare-type series. As we will show, one of advantages of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of Delta_k. This is joint work with Y. Kara, M. Kumari, K. Maurischat, A. Mocanu and L. Smajlović.