A new complex reflection group in $PU(9,1)$ and the Barnes-Wall lattice

Abstract

We show that the projectivized complex reflection group $\Gamma$ of the unique $(1+i)$-modular Hermitian $\mathbb{Z}[i]$-module of signature $(9,1)$ is a new arithmetic reflection group in $PU(9,1)$. 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 $D$ for $\Gamma$ that encode Coxeter-type generators and relations for $\Gamma$. The vertices of $D$ can be indexed by sixteen points and sixteen affine hyperplanes in $\mathbb{F}_2^4$. The edges of $D$ are determined by the finite geometry of these points and hyperplanes. The group of automorphisms of the diagram $D$ is $2^4 \colon (2^3 \colon L_3(2)) \colon 2$. This group transitively permutes the $32$ mirrors of generating reflections and fixes an unique point $\tau$ in $\mathbb{C} H^9$. 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.