Eichler-Selberg type relations for negative weight vector-valued mock modular forms

By comparing two different evaluations of a modified (a la Borcherds) higher Siegel theta lift on even lattices of signature $(r,s)$, we prove Eichler-Selberg type relations for a wide class of negative weight vector-valued mock modular forms. In doing so, we detail several properties of the lift, as well as showing that it produces an infinite family of vector-valued local (and locally harmonic) Maas forms.