Coinvariants of metaplectic representations on moduli of abelian varieties

TL;DR

This paper constructs and analyzes coinvariant spaces related to metaplectic representations on moduli spaces of abelian varieties, revealing their geometric and algebraic structures and their relation to line bundles and Atiyah algebras.

Contribution

It introduces a new construction of coinvariant spaces for infinite-dimensional Lie algebra actions on abelian varieties and explores their geometric properties and associated differential operators.

Findings

Coinvariant spaces are constructed for abelian varieties.

These spaces form twisted D-modules on moduli spaces.

The Atiyah algebra of relevant line bundles is explicitly determined.

Abstract

We construct spaces of coinvariants at principally polarized abelian varieties with respect to the action of an infinite-dimensional Lie algebra. We show how these spaces globalize to twisted $\mathcal{D}$-modules on moduli of principally polarized abelian varieties, and we determine the Atiyah algebra of a line bundle acting on them. We prove analogous results on the universal abelian variety. An essential aspect of our arguments involves analyzing the Atiyah algebra of the Hodge and canonical line bundles on moduli of abelian varieties and the universal abelian variety.