The Kudla-Millson form via the Mathai-Quillen formalism

In \cite{km2}, Kudla and Millson constructed a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space using Howe's differential operators. It is a crucial ingredient in their theory of theta lifting. This form can be seen as a Thom form of a real oriented vector bundle. In \cite{mq} Mathai and Quillen constructed a {\em canonical} Thom form and we show how to recover the Kudla-Millson form via their construction. A similar result was obtained by \cite{garcia} for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to an arbitrary signature.