Coinvariants of metaplectic representations on moduli of abelian varieties

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.