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

Tathagata Basak

Vol. 307 (2020), No. 1, 13–51

DOI: 10.2140/pjm.2020.307.13

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.

Keywords

complex reflection group, hyperbolic reflection group, arithmetic lattices, Barnes–Wall lattice

Mathematical Subject Classification 2010

Primary: 11H56, 20F55

Secondary: 20F05, 51M10

Milestones

Received: 17 September 2018

Revised: 21 August 2019

Accepted: 30 January 2020

Published: 8 August 2020