On the values of some singular currents on Shimura varieties of orthogonal type

In a previous paper (arxiv:1409.7353), we introduced a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we proved that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In this paper, we will evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts. Using the Siegel-Weil formula and ideas of Piatetskii-Shapiro and Rallis, we show that the result involves near central special values of standard $L$-functions for $Sp_{4,F}$. An example concerning products of Shimura curves will be given at the end of the paper.