Unitary Schwartz Forms & the Weil Representation

Abstract

Stephen Kudla has conjectured a relationship between the Fourier coefficients of Eisenstein series, and the arithmetic heights of certain special cycles. Luis Garcia and Siddarth Sankaran confirmed the conjecture for certain Shimura varieties of type U(p, q), arising as the quotient of a symmetric space by a group action, when q=1. An essential step in their argument relies on establishing that a specific form is a highest weight vector of a particular weight, for the Weil representation. In an effort to extend the results of Garcia and Sankaran, we show that the aforementioned forms are highest weight vectors of the expected weight, under the action of the Weil representation, in various cases when q>1. In particular, we show that this result holds for all cases when q=2. We prove this result by using an inductive argument, which depends on a technical result about immersed submanifolds, and various results about splitting the action of the Weil representation on tensor products. The base cases are intractable to carry out by hand, and thus the final section of the thesis contains Sage code which was written to carry out the computations of the base cases.