Vol. 307, No. 1, 2020

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

Tathagata Basak

Vol. 307 (2020), No. 1, 13–51
Abstract

We show that the projectivized complex reflection group $\Gamma$ of the unique $\left(1+i\right)$-modular Hermitian $ℤ\left[i\right]$-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 $ℂ{H}^{9}$ 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 $ℂ{H}^{9}$. 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.

Keywords
complex reflection group, hyperbolic reflection group, arithmetic lattices, Barnes–Wall lattice
Mathematical Subject Classification 2010
Primary: 11H56, 20F55
Secondary: 20F05, 51M10