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

We show that the projectivized complex reflection group Γ of the unique (1+i)-modular Hermitian [i]-module of signature (9,1) is a new arithmetic reflection group in PU(9,1) and we construct an automorphic form on H9 with singularities along the mirrors of Γ using Borcherds’ singular theta lift. We find 32 complex reflections of order four generating Γ. 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 𝔽24. The group of automorphisms of D is 24 : (23 : L3(2)) : 2. This group transitively permutes the 32 mirrors of generating reflections and fixes a unique point τ in H9 . These 32 mirrors are precisely the mirrors closest to τ. These results are strikingly similar to the results satisfied by the complex hyperbolic reflection group at the center of Allcock’s monstrous proposal.

complex reflection group, hyperbolic reflection group, arithmetic lattices, Barnes–Wall lattice
Mathematical Subject Classification 2010
Primary: 11H56, 20F55
Secondary: 20F05, 51M10
Received: 17 September 2018
Revised: 21 August 2019
Accepted: 30 January 2020
Published: 8 August 2020
Tathagata Basak
Department of Mathematics
Iowa State University
Ames, IA
United States