Eisenstein classes and generating series of modular symbols in ${\mathrm{SL}}_N$

Abstract

We define a theta lift between the homology in degree $N-1$ of a locally symmetric space associated to ${\mathrm{SL}}_N(\mathbb{R})$ and the space of modular forms of weight $N$, similar to the Kudla-Millson lift in the orthogonal setting. We show that the Fourier coefficients of this lift are Poincaré duals of modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. When $N=2$, we show that the lift surjects on the space of weight 2 modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing L-function.

v3-->v4 Corrected some typos and minor mistakes, removed some standard results about theta series