The Kudla–Millson form via the Mathai–Quillen formalism

Abstract

Abstract A crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$ -form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature.