On a hidden property of Borcherds products

Borcherds products are special automorphic forms for the orthogonal
group $O_{2,n}(\R)$ of signature $(2,n)$, lifts from  weakly modular
forms. To state explicit examples we identify $O_{2,2}$ with $SL_2
\times SL_2$ and $O_{2,3}$ with the Siegel modular group of degree
$2$.

Let $j$ be the modular invariant with Fourier coefficients $c(n)$,
normalized with $c(0)=0$.  Then one of the most famous examples is
given by

$$j(z)-j(w) = p^{-1} \prod_{m=1, n\geq -1}^{\infty} \left( 1 - p^m q^n\right)^{c(nm)} 
\quad \left(p:=e^{2 \pi i z}, q:=e^{2 \pi i w}\right), $$

related to the denominator formula of the monster Lie algebra. This
formula had been known before by Koike, Zagier. New was the link to
the Moonshine conjecture and a systematic
construction of automorphic products called Borcherds lifts due to
Borcherds.
The case
of Siegel modular forms has been first studied by Gritsenko and
Nikulin. They found a deep connection between the Igusa function
$\Delta_5$ and a certain Kac-Moody algebra with implications in
representation theory. Borcherds proved that the lifts have Heegner
divisors, and conversely Bruinier proved that in principle every
modular form with Heegner divisors is a Borcherds lift. 
Since modular
forms are usually given by Fourier expansion (for example Eisenstein
series, Theta series, Maass lifts) it is not easy to decide if a
concrete given form is a Borcherds lift. 

Recently we discovered a
hidden property of Borcherds lifts giving a complete
characterization. In this talk we describe the property, sketch the
proof and give applications. This is a joint project with Atsushi
Murase.