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

Abstract

By comparing two different evaluations of a modified (à 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 local (and locally harmonic) Maaß forms on Grassmanians in certain signatures.

21 pages, no figures, corrigendum added