The Kudla-Millson form via the Mathai-Quillen formalism

TL;DR

This paper demonstrates how the Kudla-Millson form, important in theta lifting, can be derived using the Mathai-Quillen formalism, extending previous results to arbitrary signature symmetric spaces.

Contribution

It shows how to recover the Kudla-Millson form through the Mathai-Quillen construction for general signatures, broadening the scope of prior work.

Findings

Kudla-Millson form can be obtained via Mathai-Quillen formalism.

Extension of results to arbitrary signature symmetric spaces.

Provides a unified approach to Thom forms in different signatures.

Abstract

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.