Local Maa{\ss} forms and Eichler--Selberg type relations for negative weight vector-valued mock modular forms

TL;DR

This paper establishes Eichler--Selberg type relations for negative weight vector-valued mock modular forms by analyzing two evaluations of a modified higher Siegel theta lift on even lattices, revealing new properties and local Maa{f} forms.

Contribution

It introduces a novel comparison method of theta lift evaluations to derive relations for negative weight mock modular forms and explores their properties and local Maa{f} forms.

Findings

Proves Eichler--Selberg type relations for a broad class of mock modular forms.

Identifies properties of the modified Siegel theta lift.

Constructs infinite families of local Maa{f} forms on Grassmanians.

Abstract

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 local (and locally harmonic) Maa{\ss} forms on Grassmanians in certain signatures.