Lefschetz decompositions of Kudla–Millson theta functions

Abstract In the 1980s Kudla and Millson introduced a theta function in two variables. It behaves as a Siegel modular form with respect to the first variable, and is a closed differential form on an orthogonal Shimura variety with respect to the other variable. We prove that the Lefschetz decomposition of the cohomology class of that theta function is also its modular decomposition in Eisenstein, Klingen, and cuspidal parts. We also show that the image of the Kudla–Millson lift is contained in the primitive cohomology of . As an application, we deduce dimension formulas for the cohomology of in low degrees. Our results cover the cases of moduli spaces of compact hyperkähler manifolds.