Two examples of denominator formulas of generalized Kac-Moody algebras are the denominator formula for the monster Lie algebra $$j(\sigma)-j(\tau)=p^{-1}\prod_{m>0,n\in {\bf Z}}(1-p^mq^n)^{c(mn)}$$ (where $p=e^{2\pi i\sigma}$, $q=e^{2\pi i\tau}$, and $j(\tau)=744+\sum_nc(n)q^n=q^{-1}+744+196884q+\cdots$ is the elliptic modular function) and the denominator formula for the fake monster Lie algebra $$\Phi(v)=\sum_{w\in W}\sum_{n>0}\det(w)\tau(n)e^{-2\pi in(w(\rho),v)}=e^{-2\pi i(\rho,v)}\prod_{r>0}(1-e^{-2\pi i(r,v)})^{p_{24}(1-r^2/2)}$$ (where $v\in {\bf C}^{26}$, $p_{24}(n)$ is the number of partitions of $n$ into parts of 24 colors, $\tau(n)$ is Ramanujan's $\tau$ function, and $W$ is a certain reflection group in 26 dimensional Lorentzian space with Weyl vector $\rho$). The function in these two identities are automorphic forms of weights 0 and 12 for the groups $O_{2,2}({\bf R})$ and $O_{26,2}({\bf R})$. These are two of the simplest members of an infinite family of automorphic forms for the groups $O_{n,2}9{\bf R})$ that can be written as infinite products. This talk will be about these automorphic forms and their relationship with generalized Kac-Moody algebras, hyperbolic reflection groups, positive definite lattices, and modular forms.
By specializing these automorphic forms we can often write classical modular forms as infinite products. For example, if we put $$F(\tau)=\sum_{n>0,n{\rm odd}}\sigma_1(n)q^n=q+4q^3+bq^5+\cdots$$ $$\theta(\rau)=\sum_{n\in Z}q^{n^2}=1+2q+2q^4+\cdots$$ $$\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=1-24q+252q^2-\cdots$$ $$E_6(\tau)=1-504\sum_{n>0}\sigma_5(n)q^n=1-504q-16632q^2-\cdots$$ (where $\sigma_m(n)=\sum_{d|n}d^m)$ and define $a(n)$ by $$\displaylines{\sum_na(n)q^n=3F(\tau)\theta(\tau)(\theta(\tau)^4-2F(\tau))(\theta(\tau))(\theta(\tau)^4-16F(\tau))E_6(4\tau)/\Delta(4\tau)+168\theta(\tau)\hfill\cr\hfill{} =3q^{-3}-744q+80256q^4-257985q^5+5121792q^8-12288744q^9+\cdots\cr}$$ then we get the following product formula for the elliptic modular function $j(\tau)$. $$\aligned j(\tau)=&q^{-1}+744+196884q+21493760q^2+\cdots\\ =&q^{-1}\prod_{n>0}(1-q^n)^{a(n^2)}\\ =&q^{-1}(1-q)^{-744}(1-q^2)^{80256}(1-q^3)^{-12288744}\cdots\endaligned$$