Abstract

We show that the projectivized complex reflection group $\Gamma$ of the unique $\left(1+i\right)$-modular Hermitian $\mathbb{Z}\left[i\right]$-module of signature $\left(9,1\right)$ is a new arithmetic reflection group in $\mathit{PU}\left(9,1\right)$ and we construct an automorphic form on $ℂH^9$ with singularities along the mirrors of $\Gamma$ using Borcherds’ singular theta lift. We find $32$ complex reflections of order four generating $\Gamma$. The mirrors of these $32$ reflections form the vertices of a sort of Coxeter–Dynkin diagram $D$ whose edges are determined by the finite geometry of 16 points and 16 affine hyperplanes in ${\mathbb{𝔽}}_2^4$. The group of automorphisms of $D$ is $2^4:\left(2^3:L_3\left(2\right)\right):2$. This group transitively permutes the $32$ mirrors of generating reflections and fixes a unique point $\tau$ in $ℂH^9$. These $32$ mirrors are precisely the mirrors closest to $\tau$. These results are strikingly similar to the results satisfied by the complex hyperbolic reflection group at the center of Allcock’s monstrous proposal.

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