%
% dja.tex
% Daniel Allcock (allcock@math.utah.edu)
% 17 December 1996
%
% A simple set of macros for plain tex
% documentation is in dja.doc, a text file
%
% suppresses duplicate inputs of this file:
\ifx\djatexLoaded\relax\endinput\else\let\djatexLoaded=\relax\fi
% FONTS
%
% calligraphic font
\newfam\calfont
\font\tencal=eusm10 \font\sevencal=eusm7 \font\fivecal=eusm5
\textfont\calfont=\tencal
\scriptfont\calfont=\sevencal
\scriptscriptfont\calfont=\fivecal
\def\cal{\fam=\calfont}
% fraktur
\newfam\frakturfont
\font\tenfrak=eufm10 \font\sevenfrak=eufm7 \font\fivefrak=eufm5
\textfont\frakturfont=\tenfrak
\scriptfont\frakturfont=\sevenfrak
\scriptscriptfont\frakturfont=\fivefrak
\def\frak{\fam=\frakturfont}
% blackboard bold (includes many math symbols)
\newfam\bbbfont
\font\tenbbb=msbm10 \font\sevenbbb=msbm7 \font\fivebbb=msbm5
\textfont\bbbfont=\tenbbb
\scriptfont\bbbfont=\sevenbbb
\scriptscriptfont\bbbfont=\fivebbb
\def\bbb{\fam=\bbbfont}
% script and script-script fonts for typewriter font (all in
% 10pt for portability)
\scriptfont\ttfam=\tentt
\scriptscriptfont\ttfam=\tentt
% PROGRAMMING STUFF
%
% `if not defined' :
\long\def\ifndef#1{\expandafter\ifx\csname#1\endcsname\relax}
% sometimes more convenient, but you can't say
% ``\ifdef...\else...\fi''
\long\def\ifdef#1{\ifndef{#1}\else}
% My warning messages
\def\mywarning#1{\ifnum\warningsoff=0
\immediate\write16{l.\the\inputlineno: #1}\fi}
\ifndef{warningsoff}\def\warningsoff{0}\fi
\def\givenowarnings{\def\warningsoff{1}}
\def\givewarnings{\def\warningsoff{0}}
% FRONT MATTER
%
% note: \author, \title and \note get redefined when
% \bibliography is called.
\long\def\author#1{\def\Zauthor{#1}}
\long\def\title#1{\def\Ztitle{#1}}
\long\def\address#1{\def\Zaddress{#1}}
\long\def\email#1{\def\Zemail{#1}}
\long\def\homepage#1{\def\Zhomepage{#1}}
\long\def\date#1{\def\Zdate{#1}}
\long\def\subject#1{\def\Zsubject{#1}}
\long\def\note#1{\def\Znote{#1}}
\def\plaintitlepage{{\parindent=0pt
\ifdef{Ztitle}\par{\bf\Ztitle}\medskip\fi
\ifdef{Zauthor}\par{\Zauthor}\fi
\ifdef{Zdate}\par{\Zdate}\fi
\smallskip
\ifdef{Zemail}\par{\it \Zemail}\fi
\ifdef{Zhomepage}\par{web page: \it\Zhomepage}\fi
\ifdef{Zaddress}\par{\Zaddress}\fi
\ifdef{Zsubject}\smallskip\par{1991 mathematics subject
classification: \Zsubject}\fi
\ifndef{Znote}\else\smallskip\par{\Znote}\fi
\vskip 0pt plus 10pt}}
\def\abstract{\bigbreak\noindent{\bf Abstract.}\par\noindent}
% SECTIONS, THEOREMS, ETC.
%
%\let\section=\beginsection
\outer\def\section#1\par{\bigbreak\noindent{\bf #1}\nobreak\medskip\noindent}
% Proclaimed Things (theorems, lemmas, etc.)
% (modeled on plain tex's ``\proclaim'', but theorems can be more
% than one paragraph long.)
\outer\def\beginproclaim#1. {\medbreak
\noindent{\bf #1.\enspace}\begingroup\sl}
\def\endproclaim{\endgroup\par\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
% Proofs (n.b. \drawbox is defined in `OTHER')
\def\beginproof#1{\smallskip{\it#1\/}}
\def\endproof{\leavevmode\vrule height0pt width0pt
depth0pt\nobreak\hfill\proofbox\smallskip}
\def\QED{\endproof}
\def\proofbox{\drawbox{1.2ex}{1.2ex}{.1ex}}
% Remarks (n.b. remarks are only one par long)
\def\remark#1#2\par{\ifdim\lastskip<\smallskipamount
\removelastskip\penalty55\smallskip\fi
{\it #1\/}#2\smallbreak}
% CROSS-REFERENCES AND BIBLIOGRAPHY
%
% If the tag has not been defined (say by \deftag), then
% \tag warns and inserts question
% marks. If the tag is defined, it
% just produces the tag (symbolic if \userawtags=0,
% or symbolic:raw, otherwise.) Default is \userawtags=1.
\def\rawtags{\def\userawtags{1}}
\def\symbolictags{\def\userawtags{0}}
\ifndef{userawtags}\def\userawtags{1}\fi
\def\tag#1{\ifndef{#1}\mywarning{tag `#1'
undefined.}{\hbox{\bf?`}\tt #1\hbox{\bf ?}}\else
\ifnum\userawtags=0 \csname #1\endcsname
\else\csname #1\endcsname{ \tt [#1]}\fi\fi}
\def\Tag#1{\tag{#1}}
\def\eqtag#1{(\tag{#1})}
\def\eqTag#1{\ifnum\userawtags=0
\eqtag{#1}\else
\lower12pt\hbox{\eqtag{#1}}\fi}
\def\cite#1{[\tag{#1}]}
\def\ecite#1#2{[\tag{#1}, #2]}
\def\nocite#1{}
% syntax: \deftag{label to insert in text}{label in file}
\def\deftag#1#2{\ifndef{#2}\else\mywarning{tag `#2' defined
more than once.}\fi
\expandafter\def\csname #2\endcsname{#1}}
\def\defcite#1#2{\deftag{#1}{#2}}
% bibliography entries
\def\bibitem#1{\ifnum\userawtags=0
\item{[{\tag{#1}}]}\else
\noindent\hangindent=\parindent\hangafter=1
\cite{#1}\enskip\fi}
% FONT SHORTCUTS
%
\def\rom#1{({\it\romannumeral#1\/})}
\def\Rom#1{{\rm\uppercase\expandafter{\romannumeral#1}}}
\def\a{\alpha} \def\b{\beta} \def\c{\gamma}
\def\d{\delta} \def\e{\varepsilon}
\def\cala{{\cal A}} \def\calj{{\cal J}} \def\cals{{\cal S}}
\def\calb{{\cal B}} \def\calk{{\cal K}} \def\calt{{\cal T}}
\def\calc{{\cal C}} \def\call{{\cal L}} \def\calu{{\cal U}}
\def\cald{{\cal D}} \def\calm{{\cal M}} \def\calv{{\cal V}}
\def\cale{{\cal E}} \def\caln{{\cal N}} \def\calw{{\cal W}}
\def\calf{{\cal F}} \def\calo{{\cal O}} \def\calx{{\cal X}}
\def\calg{{\cal G}} \def\calp{{\cal P}} \def\caly{{\cal Y}}
\def\calh{{\cal H}} \def\calq{{\cal Q}} \def\calz{{\cal Z}}
\def\cali{{\cal I}} \def\calr{{\cal R}}
% SHORTCUTS AND MNEMONICS FOR STANDARD MATH SYMBOLS
%
\def\Z{{\bbb Z}} % integers
\def\Q{{\bbb Q}} % rational numbers
\def\R{{\bbb R}} % real numbers
\def\C{{\bbb C}} % complex numbers
\def\F{{\bbb F}} % A finite field
\let\sset=\subseteq \let\dimension=\dim%
\let\equivalent=\equiv%
\let\congruent=\equiv%
\def\setminus{\mathop{\bbb \char"72}\nolimits}
\def\semidirect{\mathop{\bbb \char"6E}\nolimits}
\let\tensor=\otimes
\let\del=\partial
\let\isomorphism=\cong \let\iso=\cong
\let\to=\rightarrow
\def\aut{\mathop{\rm Aut}\nolimits}
\def\hom{\mathop{\rm Hom}\nolimits}
\def\re{\mathop{\rm Re}\nolimits}
\def\im{\mathop{\rm Im}\nolimits}
\def\tr{\mathop{\rm Tr}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
% quick parentheses and brackets
\def\({\left(} \def\){\right)}
\def\[{\left[} \def\]{\right]}
% OTHER STUFF
%
\def\hideboxes{\overfullrule=0pt}
\def\frac#1/#2{\leavevmode\kern.1em\raise.5ex\hbox{\the
\scriptfont0 #1}\kern-.1em/\kern-.15em\lower.25ex
\hbox{\the\scriptfont0 #2}}
% 2x2 matrices for use in text:
\def\smallmatrix#1#2#3#4{%
\left({#1\atop #3}\;{#2\atop #4}\right)}
% syntax: ``\drawbox{internal width}{internal height}{line
% thickness}''; arguments should be dimens.
\def\drawbox#1#2#3{\leavevmode\vbox{\hrule height #3%
\hbox{\vrule width #3 height #2\kern #1%
\vrule width #3}\hrule height #3}}
% does an \halign but centers it with a little vertical space
% around it. Uses diplay math mode.
\def\calign#1{$$\vbox{\halign{#1}}$$}
% an \llap that you can use inside formulas without
% tex changing to textstyle inside the \hbox of \llap
\def\mathllap#1{\mathchoice
{\llap{$\displaystyle #1$}}%
{\llap{$\textstyle #1$}}%
{\llap{$\scriptstyle #1$}}%
{\llap{$\scriptscriptstyle #1$}}}
% the set of all #1 such that #2; makes the braces and the
% middle bar big enough to accomodate everything.
\def\set#1#2{\left\{\,#1\mathllap{\phantom{#2}}\mathrel{}\right|\left.#2\mathllap{\phantom{#1}}\,\right\}}
\parskip=0pt
% -----------------end-of-dja.tex-------------------------------
% This is a plain tex file:
% % Daniel Allcock
%
% The Leech Lattice and Complex Hyperbolic Reflections
% load the style file
%
%----------Options-(comment-or-uncomment-as-desired)-------------
%
\magnification=1100
%
% for doublespacing
%\baselinesfooKip=24pt
%
% eliminates the nasty overfullbox bars:
%\hideboxes
%
% uses your symbols rather than tag names for
% cross-referencing:
\symbolictags
%
% suppresses djatex warnings (e.g., undefined tags)
%\givenowarnings
%
%-----------------Cross-References-and-Citations------------------
%
% Do not tamper with the following line.
% ---begin deftags---
\deftag{1}{sec-intro}
\deftag{2}{sec-background}
\deftag{2.1}{eq-formula-for-reflection}
\deftag{3}{sec-null-vector}
\deftag{3.1}{eq-translation-matrix}
\deftag{3.2}{eq-product-of-translations}
\deftag{3.3}{eq-inverse-of-translation}
\deftag{3.4}{eq-commutator-of-translations}
\deftag{3.5}{eq-conjugate-of-translation}
\deftag{3.1}{thm-got-translations}
\deftag{3.6}{eq-got-funny-central-translation}
\deftag{3.2}{thm-got-all-translations}
\deftag{3.3}{thm-nuke-old-conjecture}
\deftag{4}{sec-ch13}
\deftag{4.1}{thm-ch13}
\deftag{5}{sec-ch9-and-ch5}
\deftag{5.1}{fig-figure}
\deftag{5.1}{thm-ch9-and-ch5}
\deftag{5.1}{eq-formula-for-r}
\deftag{5.2}{eq-formula-for-w}
\deftag{5.2}{thm-Eisenstein-E8}
\deftag{6}{sec-quaternions}
\deftag{6.1}{thm-got-translations-quaternionic-case}
\deftag{6.2}{thm-thm-quaternionic-reflection-groups}
\deftag{6.1}{eq-formula-for-r-quaternionic-case}
\deftag{6.2}{eq-foo}
\deftag{7}{sec-aut-forms}
\deftag{7.1}{thm-automorphic-forms-complex-case}
\deftag{7.1}{eq-e4squared-over-delta}
\deftag{7.2}{thm-automorphic-forms-quaternionic-case}
\deftag{7.2}{eq-foo2}
% ---end deftags---
% Do not tamper with the previous line or the following line.
% ---begin defcites---
\defcite{1}{allcock:oh2}
\defcite{2}{dja:complex-reflection-groups}
\defcite{3}{borcherds:aut-forms-singularities-grassmannians}
\defcite{4}{jhc:26dim}
\defcite{5}{jhc:leech-radius}
\defcite{6}{splag}
\defcite{7}{coxeter:finite-groups-generated-by-unitary-reflections}
\defcite{8}{deligne:monodromy-of-hypergeometric-functions}
\defcite{9}{feit:unimodular-Eisenstein}
\defcite{10}{mostow:picard-lattices-from-half-inegral-conditions}
\defcite{11}{mostow:monodromy-groups-on-the-complex-n-ball}
\defcite{12}{thurston:shapes-of-polyhedra}
\defcite{13}{vinberg:nonexistence-crystallographic-reflection-groups}
\defcite{14}{raw:quaternionic-leech}
\defcite{15}{raw:complex-leech}
% ---end defcites---
% Do not tamper with the preceeding line.
%-----------------Beginning-of-Document---------------------------
%
% RINGS, ETC
\def\H{{\bbb H}} % quaternions
\def\i{i}
\def\j{j}
\def\k{k}
\def\Eisenstein{\cale}
\def\w{\omega} % cube root of 1
\def\wbar{{\bar\w}}
\def\thetabar{{\bar\theta}} % square root of -3
\def\Hurwitz{{\cal H}}
\def\p{p} % p=1-i generates the relevant ideal in
% the hurwitz case.
\def\pbar{\bar{\p}}
\def\ideal{{\frak p}} % the ideal generated by p
\def\O{{\bbb O}} % octaves
\def\ocint{{\cal K}} % integer octaves
% CODES
\def\code{\calc_{12}} % ternary golay code
\def\tcode{\calc_{4}} % tetracode
% LATTICES
\def\cell{H} % 2x2 isotropic, Eisenstein, determinant 3
\def\qcell{H} % 2x2 isotropic, Hurwitz, determinant 2.
%
\def\cll{{\Lambda_{12}^\Eisenstein}} % complex leech lattice
\def\clla{\Lambda} % complex leech lattice (abbreviation)
\def\qll{{\Lambda_6^\Hurwitz}} % Hurwitz leech lattice
\def\rll{{\Lambda_{24}}} % standard Leech lattice
%
\def\ee{{\Lambda_4^\Eisenstein}} % Eisenstein E_8
\def\qee{{\Lambda_2^\Hurwitz}} % Hurwitz E_8
\def\oee{{\Lambda_1^\ocint}} % octave E_8
%
\def\II#1,#2{{\rm II}_{#1,#2}} % even unimodular Z-lattices
% SYMMETRIES AND GROUPS
\def\reflec#1{R(#1)}
\def\suz{{\it Suz}} % Suzuki group
\def\csuz{6{\cdot}\suz} % aut group of complex leech
\def\G{\Gamma} % largest of Deligne-Mostow examples
\def\braid{B} % spherical braid group on 12 strands
\def\Sp#1#2{{\rm Sp}_{#1}(#2)} % symplectic group
\def\SL#1#2{{\rm SL}_{#1}#2} % special linear group
% translations; the complexity is so the typesetting will look pretty.
\def\Tinv#1#2{T_{#1;#2}^{-1}}
\newbox\Tbox
\setbox\Tbox=\hbox{$T$}
\newdimen\Theight
\Theight=\ht\Tbox
\def\T#1#2{\setbox\Tbox=\hbox{$T_{#1;\,#2}^{\phantom{-1}}$}\ht\Tbox=\Theight\box\Tbox}
% SPACES
\def\ch#1{\C H^{#1}} % hyperbolic space
\def\hh#1{\H H^{#1}}
\def\oh#1{\O H^{#1}}
\def\chn{\ch{n}}
\def\cp#1{\C P^{#1}} % projective space
% MISC
\def\ip#1#2{\langle#1{|}#2\rangle}
\def\spanof#1{\langle #1\rangle}
\def\bbar{{\bar\b}}
\def\height#1{\mathop{\rm ht}\nolimits #1}
% LATIN
\def\ie{{\it i.e., }}
%
% END of macros
\author {Daniel Allcock}
\title {The Leech Lattice and Complex Hyperbolic Reflections}
\address{Department of Mathematics\par
University of Utah\par
Salt Lake City, UT 84112.}
\email {allcock@math.utah.edu}
\homepage{http://www.math.utah.edu/$\sim$allcock}
\date {19 November 1997}
%\subject{ }
%\note { }
\plaintitlepage
\abstract
We construct a natural sequence of finite-covolume reflection
groups acting on the complex
hyperbolic spaces $\ch{13}$, $\ch9$ and $\ch5$, and show that
the $9$-dimensional example coincides with the largest of the
groups of Mostow \cite{mostow:picard-lattices-from-half-inegral-conditions}. These reflection groups
arise as automorphism groups of certain Lorentzian lattices over
the Eisenstein integers, and we obtain our largest example by
using the complex Leech lattice. We also construct
finite-covolume reflection groups on the quaternionic hyperbolic
spaces $\hh7$, $\hh5$ and $\hh3$, again using the Leech lattice,
and apply results of Borcherds
\cite{borcherds:aut-forms-singularities-grassmannians}
to obtain automorphic forms for our groups.
\section{\Tag{sec-intro} Introduction}
In \cite{dja:complex-reflection-groups} we constructed a large number of reflection groups
acting on
complex and quaternionic hyperbolic spaces. For the most part,
the groups appeared as symmetry groups of selfdual Lorentzian
lattices (\ie those of signature $(1,n)$) over the rings of
Eisenstein, Gaussian and Hurwitz integers. We were
surprised to find examples in even higher dimensions
by using certain non-selfdual
lattices. These are the subject of the paper.
Although similar in
spirit to \cite{dja:complex-reflection-groups}, we do not need
results from there.
The basic idea, as in \cite{dja:complex-reflection-groups}, is that one can build reflection
groups acting on $\ch{n+1}$ or $\hh{n+1}$ from suitable
negative-definite $n$-dimensional lattices; we will use
lattices over the Eisenstein integers
$\Eisenstein=\Z[\w]$, where $\w$ is a primitive cube root of
unity, and over the quaternionic ring $\Hurwitz$ of Hurwitz integers.
If $\Lambda$ is such
a lattice then one constructs a Lorentzian lattice $L$ by taking
the direct sum of $\Lambda$ and a suitable $2$-dimensional
Lorentzian lattice $\cell$. In \cite{dja:complex-reflection-groups} we took $\cell$ to be
given by the inner product matrix $\smallmatrix0110$, whereas
here we use
$$
\pmatrix{0&\thetabar\cr\theta&0\cr}
\hbox{\qquad or\qquad}
\pmatrix{0&1+\i\cr1-\i&0\cr}\;,
$$
where $\theta\in\Eisenstein$ is a square root of $-3$. If the covering radius
of $\Lambda$ is small enough, then $\aut L$ is generated up to
finite index by reflections.
The small change in $\cell$ has surprising effects,
allowing us to use certain lattices $\Lambda$ that did not work
before. In particular, there are Eisenstein and Hurwitz versions
of the Leech lattice, but their covering radii are too large
for the technology of \cite{dja:complex-reflection-groups} to apply. Our main results are
that the automorphism groups of the Eisenstein lattices
$$
\ee\oplus\cell,\quad
\ee\oplus\ee\oplus\cell
\hbox{\quad and\quad}
\cll\oplus\cell
$$
are generated (up to finite index, in the last case) by
reflections. Here, $\ee$ and $\cll$ are the Eisenstein versions
of the $E_8$ and Leech lattices, and
$\cell=\smallmatrix{0}{\thetabar}{\theta}{0}$.
Alternately, these are the unique Lorentzian $\Eisenstein$-lattices $L$ of
dimensions $6$, $10$ and $14$ satisfying $L=\theta L'$, where
$L'$ is the dual of $L$. In particular, $\cll\oplus\cell$ may
also be described as $(\ee)^3\oplus\cell$, so that our three
groups form a natural sequence. Furthermore, we identify our
$10$-dimensional $\Eisenstein$-lattice with a lattice described by Thurston
\cite{thurston:shapes-of-polyhedra} in terms of combinatorial
triangulations of the sphere $S^2$, and we identify the
automorphism group of the lattice with the largest of the groups
described by Mostow
\cite{mostow:picard-lattices-from-half-inegral-conditions}.
(See also Deligne and Mostow
\cite{deligne:monodromy-of-hypergeometric-functions} and
Thurston \cite{thurston:shapes-of-polyhedra}.)
We also carry out a quaternionic version of this program,
proving that the automorphism groups of the lattices
$$
\qee\oplus\qcell,\quad
\qee\oplus\qee\oplus\qcell
\hbox{\quad and\quad}
\qll\oplus\qcell
$$
are virtually generated by reflections. Here, $\qee$ and $\qll$
are the Hurwitz versions of the $E_8$ and Leech lattices, and
$\qcell=\smallmatrix{0}{1+\i}{1-\i}{0}$. The lattices above may
be described as the unique Lorentzian $\Hurwitz$-lattices $L$ in
dimensions $4$, $6$ and $8$ satisfying $L=L'\cdot(1+i)$. By the
isomorphism
$\qll\oplus\qcell\isomorphism(\qee)^3\oplus\qcell$, we see
that these three lattices also form a natural sequence.
These two sequences are obviously similar to each other, and
they are also very similar to a sequence of real hyperbolic
reflection groups, namely those of the lattices
$$
E_8\oplus\cell,\quad
E_8\oplus E_8\oplus\cell
\hbox{\quad and\quad}
\rll\oplus\cell\;,
$$
where $\rll$ is the Leech lattice and
$\cell=\smallmatrix0110$. These lattices are the unique
even selfdual Lorentzian $\Z$-lattices in dimensions $10$, $18$
and $26$. As before, $\rll\oplus\cell\isomorphism
E_8^3\oplus\cell$, so that these lattices form a natural
sequence. The subgroup of $\aut L$ generated by reflections has
index $2$, $4$ or $\infty$ in $\aut L$ for these three lattices $L$. In
the last case, although the reflection group has infinite index,
it is still ``almost all'' of $\aut L$, in the sense that the
quotient by it is virtually $\Z^{24}$. See \cite{jhc:26dim} for details.
Finally, we construct automorphic forms on complex and
quaternionic hyperbolic space that are automorphic with respect
to the symmetry groups of the various lattices listed
above. This construction relies on the work of Borcherds
\cite{borcherds:aut-forms-singularities-grassmannians}, and allows us to explicitly describe the zeros and
singularities of these forms. In the complex case, the forms are
holomorphic with simple zeros along the mirrors of the
reflection groups, and in the quaternionic case the forms are
real analytic except at their singularities, which lie along the
mirrors of the groups. The analogy with the real hyperbolic case
extends to the construction of these automorphic forms.
\section{\Tag{sec-background}. Background on lattices}
We write $\w$ for a primitive cube root of unity
and define the Eisenstein integers
$\Eisenstein$ to be $\Z[\w]$. It will be useful to denote
$\w-\wbar$, a square root of $-3$, by $\theta$. For
$x\in\Eisenstein$ we define $\im x=(x-\bar x)/2$. A lattice over
$\Eisenstein$ is a free (right) module over $\Eisenstein$
equipped with a Hermitian form $\ip{ }{ }$ taking values in
$\Eisenstein$. If $x$ lies in a lattice then the norm $x^2$ of
$x$ is defined to be $\ip{x}{x}$; some authors call this the
squared norm. We will consider only nondegenerate forms---in fact
only those of signature $(0,n)$ or $(1,n)$. The former are
negative-definite and the latter are called Lorentzian.
We use the convention that $\ip{ }{ }$ be linear in its
second argument rather than in its first. This is because it is convenient
to regard lattice elements as column vectors, with linear
transformations acting on the left. This means that in the
quaternionic case (section~\tag{sec-quaternions}), scalars should act on the
right and this makes it natural for $\ip{ }{ }$ to be linear in
its second argument. Until
section~\tag{sec-quaternions} we will allow ourselves to write
the action of scalars on either side.
The dual $L'$ of a lattice $L$ is the set of all $x\in
L\tensor\Q$ satisfying $\ip{x}{\lambda}\in\Eisenstein$ for all
$\lambda\in L$. For the most part we will consider lattices $L$
satsifying $L\sset\theta L'$, so that all inner products of
lattice vectors are divisible by $\theta$. We call $r\in L$ a
root of $L$ if $r^2=-3$ and $r\in\theta L'$. Then the complex
$\xi$-reflection in $r$,
$$
w\mapsto w-r(1-\xi){\ip{r}{w}\over r^2}\;,
\eqno\eqTag{eq-formula-for-reflection}
$$
is an isometry of $L$ if $\xi$ is a cube root of unity. If
$\xi\neq1$ then the reflection has order 3 and is called a
triflection. We define $\reflec{L}$ to be the subgroup of $\aut
L$ generated by the triflections in the roots of $L$.
An element $x$ of a lattice $L$ is called primitive if it is not
of the form $y\a$ with $y\in L$ and $\a$ a non-unit of
$\Eisenstein$. A sublattice $M$ of $L$ is called primitive if
$L\cap (M\tensor\Q)=M$. A deep hole of a positive-definite
lattice $L$ is a point of $L\tensor\R$ at maximal distance from
$L$; this distance is called the covering radius of $L$. It is
awkward to directly adapt this definition for negative-definite lattices,
so we make the following definition instead. The covering norm $N$ of a
negative-definite lattice $L$ is the largest (negative) number
$N$ such that
for each $\ell\in L\tensor\R$ there is a lattice vector $\lambda$
satisfying $(\ell-\lambda)^2\in[N,0]$. The points $\ell$ of
$L\tensor\R$ for which there are no $\lambda\in L$ satisfying
$(\ell-\lambda)^2>N$ are called the deep holes of $L$.
If $L$ is a Lorentzian $\Eisenstein$-lattice then $\aut L$ acts
on the set of positive-definite subspaces of $L\tensor\R$; this
space is called complex hyperbolic space and denoted $\chn$,
where $n=\dimension L-1$.
\section{\Tag{sec-null-vector}. The stabilizer of a
null vector}
We define $\cell$ to be the 2-dimensional lattice over
$\Eisenstein$ with inner product matrix
$H=\smallmatrix{0}{\thetabar}{\theta}{0}$. That is, elements of
$\cell$ are column vectors with entries in $\Eisenstein$, so
that the inner product $\ip{v}{w}$ of lattice vectors $v$ and
$w$ is $v^*\cell w$, where $v^*$ denotes the conjugate-transpose
of $v$.
Let $\Lambda$ be any $\Eisenstein$-lattice and let
$L=\Lambda\oplus\cell$. We write elements of $L$ in the form
$(\lambda;\mu,\nu)$ with $\lambda\in \Lambda$ and
$\mu,\nu\in\Eisenstein$. We give the vector $(0;0,1)$ the name
$\rho$ and define the height $\height v$ of $v\in L\tensor\R$ to
be the inner product $\ip{\rho}{v}$. For $v=(\lambda;\mu,\nu)$,
the height of $v$ is just $\theta\mu$.
If $\Lambda$ is negative-definite then since $\cell$ is Lorentzian,
$L$ is also Lorentzian.
Let $\lambda\in \Lambda\tensor\R$ and suppose
$z\in\im\C$. Then the map
$$
\T{\lambda}{z}:
\eqalign{
(\ell;0,0)
&\mapsto(\ell;0,\theta^{-1}\ip{\lambda}{\ell})\cr
(0;1,0)
&\mapsto(\lambda;1,\thetabar^{-1}(z-\lambda^2/2))\cr
(0;0,1)
&\mapsto(0;0,1)
\cr}
$$
is an isometry of $L\tensor\R$ that preserves $\rho$. Every
isometry preserving $\rho$ and acting trivially on
$\rho^\perp/\spanof{\rho}$ has this form, where $\spanof{\rho}$
denotes the complex span of $\rho$. We call the maps
$\T{\lambda}{z}$ the translations by $\lambda$. Regarding
elements of $L\tensor\R$ as column
vectors, $\T{\lambda}{z}$ acts by left multiplication by the
matrix
$$
\T{\lambda}{z}=
\pmatrix{
I&\lambda&0\cr
0&1&0\cr
\theta^{-1}\lambda^*&\thetabar^{-1}(z-\lambda^2/2)&1\cr
}
\eqno\eqTag{eq-translation-matrix}
$$
where $I$ represents the identity map of $\Lambda$ and
$\lambda^*$ is the linear map $\ell\mapsto\ip{\lambda}{\ell}$
on $\Lambda$ induced by $\lambda$.
One verifies the relations
$$\eqalignno{
\T{\lambda}{z}\T{\lambda'}{z'}
&=\T{\lambda+\lambda'}{z+z'+\im\ip{\lambda'}{\lambda}}
&\eqTag{eq-product-of-translations}
\cr
\Tinv{\lambda}{z}
&=\T{-\lambda}{-z}
&\eqTag{eq-inverse-of-translation}
\cr
\Tinv{\lambda}{z}\Tinv{\lambda'}{z'}\T{\lambda}{z}\T{\lambda'}{z'}
&=\T{0}{2\im\ip{\lambda'}{\lambda}}\;.
&\eqTag{eq-commutator-of-translations}
\cr
}$$
These show that the translations form a group whose
center and commutator subgroup coincide and consist of those
$\T{\lambda}{z}$ with $\lambda=0$. We call these the central translations.
Furthermore, if $U$ is an isometry of $\Lambda$ then we may
regard it as acting on $L$, fixing $\cell$ pointwise. Then we
have
$$
U\T{\lambda}{z}U^{-1}=\T{U \lambda}{z}\;.
\eqno\eqTag{eq-conjugate-of-translation}
$$
In order for a translation to preserve $L$, it is obviously
necessary that $\lambda\in \Lambda$. Furthermore, by considering
the lower-left entry of \eqtag{eq-translation-matrix}, we see
that the inner product $\ip{\lambda}{\ell}$ must be divisible
by $\theta$, for all $\ell\in\Lambda$. Finally, by considering
the bottom middle entry of \eqtag{eq-translation-matrix} we see
that $z\in\im\C$ must be chosen so that
$\thetabar^{-1}(z-\lambda^2/2)\in\Eisenstein$. We can choose
such a $z$ just if $\thetabar^{-1}\lambda^2/2$ is an integer
multiple of $\theta/2$. This condition holds because by our
previous condition we know that
$\lambda^2=\ip{\lambda}{\lambda}$ is real and divisible by
$\theta$, hence divisible by $3=-\theta^2$. We conclude that
$\aut{L}$ contains a translation by $\lambda$ just if
$\lambda\in
\Lambda\cap\theta \Lambda'$.
\beginproclaim Theorem
{\Tag{thm-got-translations}}.
Let $\Lambda$ be an $\Eisenstein$-lattice and
let $L=\Lambda\oplus\cell$. Then
for each $\lambda\in \Lambda\cap\theta\Lambda'$, $\reflec{L}$
contains a translation by $\lambda$.
\endproclaim
\beginproof{Proof:}
We begin by working in $\cell$, so we suppress all but the last
pair of vectors' coordinates. Let $r_1=(1,\wbar)$ and
$r_2=(1,-\w)$, each a root of $L$, and let $R_1$ and $R_2$ be
the $\w$-reflections in
$r_1$ and $r_2$, respectively.
Computation reveals that matrices for
the action of the $R_i$ on $\cell$ are
$$
R_1=\pmatrix{0&\wbar\cr -\wbar&-\wbar\cr}
\qquad\hbox{and}\qquad
R_2=\pmatrix{-\wbar&\wbar\cr -\wbar&0\cr}\;.
$$
Then the action of $R_1R_2$ on $\cell$ is
$$
R_1R_2=\pmatrix{-\w&0\cr 2\w&-\w\cr}=-\w\T{0}{2\theta}\;.
\eqno\eqTag{eq-got-funny-central-translation}
$$
Of course, $R_1R_2$ acts trivially on $\cell^\perp=\Lambda$.
We write the action of $R_1R_2$ on $L$ as $-\w K\T{0}{2\theta}$,
where $-\w$ indicates scalar multiplication by $-\w$ on all of
$L$ and $K$ indicates scalar multiplication by $-\wbar$ on
$\Lambda$.
Suppose $\lambda\in \Lambda\cap\theta \Lambda'$, so that
$\aut{L}$ contains a translation
$\T{\lambda}{z}$ by $\lambda$. We will show that $\reflec{L}$
contains a translation by $-\w \lambda$.
The commutator
$$
\T{\lambda}{z}R_1R_2\Tinv{\lambda}{z}(R_1R_2)^{-1}
$$
is a member of $\reflec{L}$, since $\reflec{L}$ is a normal
subgroup of $\aut{L}$. By using our expression for $R_1R_2$,
together with
\eqtag{eq-product-of-translations}--\eqtag{eq-conjugate-of-translation},
we discover that
$$\eqalignno{
\T{\lambda}{z}R_1R_2\Tinv{\lambda}{z}(R_1R_2)^{-1}
&=\T{\lambda}{z}(-\w K)\T{0}{2\theta}\Tinv{\lambda}{z}
\T{0}{-2\theta}(-K^{-1}\wbar)
\cr
&=\T{\lambda}{z}K\Tinv{\lambda}{z}K^{-1}\cr
&=\T{\lambda}{z}\Tinv{K\lambda}{z}\cr
&=\T{\lambda}{z}\Tinv{-\wbar\lambda}{z}\cr
&=\T{\lambda}{z}\T{\wbar\lambda}{-z}\cr
&=\T{\lambda+\wbar\lambda}{\im\ip{\wbar\lambda}{\lambda}}\cr
&=\T{-\w\lambda}{\im(\w\lambda^2)}\cr
&=\T{-\w\lambda}{\theta\lambda^2/2}\;.&\proofbox\cr
\cr}
$$
\beginproclaim Theorem
{\Tag{thm-got-all-translations}}.
Let $\Lambda$ be an $\Eisenstein$-lattice and
$L=\Lambda\oplus\cell$. If $\Lambda=\theta \Lambda'$ then
$\reflec{L}$ contains all the translations of $L$ and all the
scalars of $\cell$.
\endproclaim
\beginproof{Proof:}
By theorem~\tag{thm-got-translations}, $\reflec{L}$ contains a
translation by $\lambda$, for each $\lambda\in \Lambda$. Thus,
to show that $\reflec{L}$ contains all the translations, it
suffices to show that it contains the central translations. By
the condition $\Lambda=\theta \Lambda'$, we may find
$\lambda,\lambda'\in\Lambda$ such that
$\ip{\lambda'}{\lambda}=-\theta\w$, so that
$\im\ip{\lambda'}{\lambda}=\theta/2$. Choosing translations
$\T{\lambda}{z}$ and $\T{\lambda'}{z'}$ that lie in $\reflec{L}$,
we see by \eqtag{eq-commutator-of-translations} that
$\reflec{L}$ contains
$$
\Tinv{\lambda}{z}\Tinv{\lambda'}{z'}
\T{\lambda}{z}\T{\lambda'}{z'}=\T{0}{\theta}\;.
$$
It remains to show that $\reflec{L}$ contains the scalars of
$\cell$. In the proof of theorem~\tag{thm-got-translations} we
saw that $\reflec{L}$ contains the product
$-\w\T{0}{2\theta}$. Since $\reflec{L}$ contains the central
translations, it must also contain the scalar $-\w$, which
generates the group of all scalars of $\cell$.
\endproof
A consequence of these results is that if $\Lambda$ is
negative-definite then the stabilizer of $\rho$ in $\reflec L$
has finite index in the stabilizer in $\aut L$. This holds even
if there are no reflections of $L$ stabilizing $\rho$---we will
see an example of this in the next section.
There are no analogues of theorems~\tag{thm-got-translations} and
\tag{thm-got-all-translations} for $\Z$-lattices. In
particular, if $L$ is a
Lorentzian $\Z$-lattice, $\rho$ is any null vector of $L$, and
$G$ is a subgroup of $\aut L$ generated by reflections, then the
stabilizer of $\rho$ in $G$ is also generated by
reflections. The most notatble example of this phenomenon occurs
with $L=\rll\oplus\smallmatrix0110$ and $\rho=(0;0,1)$, where
$\rll$ is the Leech lattice. From the fact that $\rll$
admits no reflections one can deduce that no reflections of $L$
stabilize $\rho$, so the stabilizer of $\rho$ in the reflection
group of $L$ is trivial, so the reflection group has infinite
index in $\aut L$. (However, the stabilizer of $\rho$
is essentially the entire difference between the two groups. See
\cite{jhc:26dim} for details.)
The following theorem is not needed elsewhere in the paper but
casts light on a conjecture made in \cite{dja:complex-reflection-groups}.
\beginproclaim Theorem
{\Tag{thm-nuke-old-conjecture}}.
Let $\Lambda$ be a negative definite $\Eisenstein$-lattice and let
$L=\Lambda\oplus\smallmatrix0110$. Then the subgroup of $\aut
L$ generated by reflections in norm $-1$ vectors contains a
finite-index subgroup of
the stabilizer of $\rho=(0;0,1)$.
\endproclaim
\beginproof{Proof sketch:}
Take $R_1$ and $R_2$ to be the $(-\w)$-reflections in the norm
$-1$ vectors $r_1=(1,\wbar)$ and $r_2=(1,\w)$ of the second
summand of $L$. Then argue as in the proofs of
theorems~\tag{thm-got-translations} and
\tag{thm-got-all-translations}. (Note that the matrices for
translations of $L$ are given by formula (4.2) of
\cite{dja:complex-reflection-groups} rather than by
\eqtag{eq-translation-matrix}, because we have replaced $\cell$
by $\smallmatrix0110$.)
\endproof
Conjecture~9.1 of \cite{dja:complex-reflection-groups} asserts
that the lattice $\Eisenstein^{1,n+1}$ is reflective just if
each negative-definite selfdual $\Eisenstein$-lattice of
dimension $n$ is virtually spanned by its elements of norms $-1$
and $-2$. (A lattice $L$ is called reflective if the subgoup of
$\aut L$ generated by reflections has finite index in $\aut L$.)
The purpose of the last condition was to assure that the
reflection group of $L$ contains a finite-index subgroup of the
stabilizer of each null vector. Theorem~\tag{thm-nuke-old-conjecture} shows
that this conclusion holds with no hypotheses at all$\,\!$! In
light of this, the natural reformulation of the conjecture is
simply that $\Eisenstein^{1,n+1}$ is reflective for all
$n$. This sounds too good to be true, considering
Vinberg's result
\cite{vinberg:nonexistence-crystallographic-reflection-groups}
that
there are no reflective Lorentzian $\Z$-lattices in high
dimensions.
\section{\Tag{sec-ch13}. A reflection group on
$\ch{13}$}
The Leech lattice $\rll$ is the unique even selfdual positive-definite
lattice of dimension 24 with minimal norm $4$. Its automorphism
group contains an element of order 3 that fixes no vectors
except the origin. We may regard this transformation as defining
an action of $\Eisenstein$ on $\rll$, making $\rll$ into a
12-dimensional $\Eisenstein$-lattice.
Wilson~\cite{raw:complex-leech} describes this lattice,
the complex Leech lattice, in detail. By $\cll$
(in this section, $\clla$) we denote his lattice with all inner products
multiplied by $-3$. Then $\Lambda$ is a negative-definite
$\Eisenstein$-lattice whose real form is the standard Leech
lattice with inner products
muliplied by $-3/2$. By the main result of \cite{jhc:leech-radius},
the covering norm of $\clla$ is $-3$. The minimal vectors of
$\clla$ (those of norm $-6$) span the lattice, and Wilson's
computation (p.~158) of the inner products of minimal vectors
with each other
shows that $\clla\sset\theta\clla'$. By \ecite{feit:unimodular-Eisenstein}{p.~248} the
determinant of $\clla$ is $\theta^{12}$, so
$\clla=\theta\clla'$. The automorphism group of $\clla$ is the
universal central extension $\csuz$ of Suzuki's sporadic simple
group. For further background on $\clla$ we refer the reader to
\cite{raw:complex-leech}.
% OLD INTRO; GIVES THE CONSTRUCTION OF COMPLEX LEECH LATTICE.
%
%Let $\code$ be the ternary Golay code, which is to say the
%unique $6$-dimensional subspace of $\F_3^{12}$ each of whose
%elements has at least $6$ nonzero coordinates. See \cite{splag} for a
%construction of this code and a discussion of its uniqueness up to
%permutation and sign-changes.
%The complex
%Leech lattice $\cll$ (denoted $\clla$ in this section)
%is defined to be the set of vectors
%$(x_1,\ldots,x_{12})\in\Eisenstein^{12}$ for which there is an
%$m\in\{-1,0,+1\}$ such that
%\item{1.} $x_i\congruent m\pmod{\theta}$ for each $i$,
%\item{2.} $\sum x_i\congruent -3m\pmod{3\theta}$, and
%\item{3.} $(x_1',\ldots,x_{12}')\in\code$, where $x_i'$ is the
%image in $\Eisenstein/\theta\Eisenstein\isomorphism\F_3$ of
%$(x_i-m)/\theta\in\Eisenstein$.
%
%\noindent
%The Hermitian form is defined by
%$$
%\ip{(x_1,\ldots,x_{12})}{(y_1,\ldots,y_{12})}
%=-{1\over3}\sum \bar x_i y_i\;.
%$$
%
%According to \cite{raw:complex-leech},
%$\clla$
%has no roots, its real form
%is a scaled version of the Leech lattice, and any inner product of
%its minimal vectors is divisible by $\theta$.
%(Note that our inner products are Wilson's multiplied by $-3$.
%We have chosen the smallest scale at which $\clla$ is integral as
%an $\Eisenstein$-lattice.)
%Since the minimal vectors of $\clla$ span the lattice, we
%have $\clla\sset\theta\clla'$. By
%\ecite{feit:unimodular-Eisenstein}{p.~248}, the determinant of
%$\clla$ is $\theta^{12}$. This proves $\clla=\theta\clla'$.
%By
%the main result of \cite{jhc:leech-radius}, suitably scaled, we know that
%for each $\ell\in\clla\tensor\R$ there is a lattice
%vector $\lambda$ such that $(\ell-\lambda)^2$ lies in $[-3,0]$.
%The automorphism group of $\clla$ is a central extension
%$\csuz$ of Suzuki's sporadic simple group.
%For further background on $\clla$ we refer to Wilson
%\cite{raw:complex-leech}.
We set $L=\clla\oplus\cell$ and note that because $\clla$ has no
roots, $\rho=(0;0,1)$ is
orthogonal to no roots of $L$.
\beginproclaim Theorem
\Tag{thm-ch13}.
The group $\reflec{L}$ acts transitively on the primitive null
vectors of $L$ that are orthogonal to no roots of $L$.
The obvious subgroup $\csuz$ of $\aut{L}$ maps
onto the quotient $\aut{L}/\reflec{L}$.
In
particular, the index
of $\reflec{L}$ in $\aut{L}$ is finite.
\endproclaim
\beginproof{Proof:}
Suppose that $v\in L$ has norm $0$, is not a multiple of
$\rho$ (so that it has nonzero height), and is orthogonal to no
roots of $L$.
We claim that by applying a reflection of $L$ we may reduce the
height of $v$. By this we mean that we may carry $v$ to a vector
$v'$ with $|\height{v'}|<|\height{v}|$.
To prove this it suffices to reduce the height of any scalar
multiple $w$ of $v$, say the one of the form
$w=(\ell;1,\a-\theta\ell^2/6)$, with $\ell\in\clla\tensor\R$ and
$\a\in\R$. (The imaginary part of the last coordinate is
determined by the condition $w^2=0$.) We may choose
$\lambda\in\clla$ such that $(\ell-\lambda)^2\in[-3,0]$. Because
$\lambda^2\congruent0(3)$ we may choose
$\b\in{1\over2}\Z$ such that $\b+\theta(-3-\lambda^2)/6$ lies in
$\Eisenstein$, and then
$r'=(\lambda;1,\b+\theta(-3-\lambda^2)/6)$ is a root of $L$. For
any $n\in\Z$ the vector
$$
r=r'+(0;0,n)=(\lambda;1,n+\b+\theta(-3-\lambda^2)/6)
$$
is also a root of $L$, and we will show that for suitable $n$,
some triflection in $r$ reduces the height of $w$.
We will need to know $\ip{r}{w}$:
$$\eqalign{
\ip{r}{w}
%&=\ip{\lambda}{\ell}+
% \pmatrix{1& n+\b+\thetabar(-3-\lambda^2)/6\cr}
% \pmatrix{&\thetabar\cr\theta&\cr}
% \pmatrix{1\cr\a-\theta\ell^2/6\cr}\cr
&=\ip{\lambda}{\ell}+\thetabar\a-{\ell^2\over2}
+n\theta+\b\theta+{-3-\lambda^2\over2}\cr
&=-{3\over2}-{1\over2}\(\ell^2-2\ip{\lambda}{\ell}+\lambda^2\)
+ n\theta+\thetabar\a+\b\theta\cr
&=-{3\over2}-{1\over2}(\ell-\lambda)^2
+\im\ip{\lambda}{\ell}+n\theta+\thetabar\a+\b\theta\cr
&=-3\left[{1\over2}+{(\ell-\lambda)^2\over6}
+n\theta^{-1}+\a\thetabar^{-1}+\b\theta^{-1}
-{\im\ip{\lambda}{\ell}\over3}\right]\cr
&=-3[a+b]\;,
\cr}$$
where $a={1\over2}+{1\over6}(\ell-\lambda)^2$ is the real part
of the term in brackets and $b$ is the imaginary part. By
construction of $\lambda$ we have $a\in[0,{1\over2}]$. By choice
of $n$ we may suppose
$b\in[\theta^{-1}/2,\thetabar^{-1}/2]=[\thetabar/6,\theta/6]$.
Let $w'$ be the image of $w$ under $\xi$-reflection in
$r$, where $\xi$ is a cube root of unity that we will choose
later.
%That is,
%$$
%w'=w-r(1-\xi){\ip{r}{w}\over\ip{r}{r}}\;.
%$$
Using \eqtag{eq-formula-for-reflection} and the computation above, we can compute the height of $w'$:
$$\eqalign{
\ip{\rho}{w'}
&=\ip{\rho}{w}-\ip{\rho}{r}(1-\xi){-3(a+b)\over-3}\cr
&=\theta-\theta(1-\xi)(a+b)\;.
\cr}$$
For the absolute value of this to be smaller than that of
$\ip{\rho}{w}=\theta$, we need to choose $\xi$ so that
$1-(1-\xi)(a+b)$ has norm less than $1$. If $b\in[0,\theta/6]$
(resp. $b\in[\thetabar/6,0]$) then taking $\xi=\w$
(resp. $\xi=\wbar$) accomplishes this unless $a=b=0$. We have
assumed that $v$ is orthogonal to no roots, so $w$ cannot be
orthogonal to $r$, which rules out the case $a=b=0$. This
proves the claim.
We have shown that we may reduce the height of $v$ with a
reflection. Repeating this as necessary, we may suppose that $v$
has height $0$, so that it is a multiple of $\rho$.
By theorem~\tag{thm-got-all-translations}, $\reflec{L}$ contains the scalars of $\cell$,
so $\reflec{L}$ acts transitively on the unit scalar multiples
of $\rho$. This proves the first claim of the theorem. Also by
theorem~\tag{thm-got-all-translations}, $\reflec{L}$ contains the
translations. Since these act transitively on the the null
vectors of height $\theta$, we conclude that a complete set of
coset representatives for $\reflec{L}$ in $\aut{L}$ may be taken
from the simultaneous stabilizer of $(0;0,1)$ and
$(0;1,0)$. Since this stabilizer is just $\aut\Lambda=\csuz$,
the proof is complete.
\endproof
\section{\Tag{sec-ch9-and-ch5}. Further examples}
We can construct other reflection groups by following the
arguments of section~\tag{sec-ch13} with other lattices in place
of $\cll$. In this section we construct two further examples,
one acting on $\ch9$ and the other on $\ch5$. The first of these
has already been discovered in a
different guise---it is the largest of the groups discovered by
Mostow
\cite{mostow:picard-lattices-from-half-inegral-conditions}
(see also Deligne and Mostow
\cite{deligne:monodromy-of-hypergeometric-functions}, Mostow
\cite{mostow:monodromy-groups-on-the-complex-n-ball} and
Thurston \cite{thurston:shapes-of-polyhedra}). We conjecture
below that the 5-dimensional group coincides with a certain one
of the other groups found in
\cite{mostow:picard-lattices-from-half-inegral-conditions}.
The $E_8$ root lattice may be regarded as an Eisenstein lattice,
and is most easily described in terms of the tetracode
$\tcode$. This is the 2-dimensional subspace of $\F_3^4$
consisting of the scalar multiples of the images of the vectors
$(0,1,1,1)$ and $(1,0,1,-1)$ under cyclic permuation of the last
three coordinates. We define $\ee$ to be the set of vectors in
$\Eisenstein^4$ whose coordinates are elements of $\tcode$ when
reduced modulo $\theta$. Here we regard $\Eisenstein^4$ as being
equipped with the standard (negative-definite) inner product
$$
\ip{(x_1,\ldots,x_4)}{(y_1,\ldots,y_4)}=-\sum \bar{x}_i y_i\;.
$$
This is the smallest scale at which $\ee$ is an integral
$\Eisenstein$-lattice, and we have $\ee=\theta(\ee)'$.
%The
%group $\aut\ee$ is generated by triflections in roots of $\ee$,
%is isomorphic to $3\times2.U_4(2)$, and acts transitively on the
%lattice vectors of norms $-3$ and on those of norm $-6$.
The largest of Mostow's groups is his
$\def\d{\discretionary{}{}{}}\Gamma({\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3})$,
for which we will simply write $\G $. The only fact we will
need about $\G $ is that it is an infinite image of the spherical
braid
group $\braid$ on 12 strands, with the standard generators
$b_1,\ldots,b_{11}$ mapping to triflections
$S_1,\ldots,S_{11}$. We will first construct an explicit set of
reflections that generate a copy of $\G $.
It is easy to check that the $S_i$ are all distinct---if any two
coincide then all coincide. (This follows by repeated use of the
fact that if $S_i=S_j$ then
any $S_k$ which braids with one of them and commute with the
other coincides with both of them.)
Since the $b_i$ are conjugate in $\braid$, the
$S_i$ are either all $\w$-reflections or all
$\wbar$-reflections. Which of these is the case is irrelevant to
the question of determining $\G $, so we suppose that they are
all $\w$-reflections. (In any case, the two possibilities are
exchanged by complex conjugation, or equally well by the
automorphism of $\braid$ exchanging each $b_i$ with its
inverse.) We may choose vectors $s_i$ of norm $-3$ in
$\C^{1,9}$ such that each $S_i$ is the $\w$-reflection in
$s_i$. In order for $S_i$ and $S_{i+1}$ to braid, matrix
computations show that we must have
$|\ip{s_i}{s_j}|=\sqrt{3}$. If $S_i$ and $S_j$ commute then
(since $S_i\neq S_j$) we must have $s_i\perp s_j$. Therefore we
may successively replace each of $s_2,\ldots,s_{11}$ by a scalar
multiple of itself
norm so that $\ip{s_i}{s_j}=0$ unless $|i-j|=1$, when
$\ip{s_i}{s_j}=\theta$ or $\thetabar$ according to whether or
not $j$
is closer than $i$ to $6$. That is, we may take the
vectors $s_i$ to be those elements of
$\ee\oplus\ee\oplus\cell$ given in figure~\tag{fig-figure}. We chose
our strange convention about $\ip{s_i}{s_j}$ when $|i-j|=1$
so that the ``diagram automorphism'' $s_i\mapsto s_{12-i}$ would
be an
isometry.
\topinsert
% I use \0 to represent coordinates with value zero. To make the
% zeros visible, use \def\0{0} and comment out the explanation
% in the caption of what blank entries mean.
\def\0{ }
\def\-{\phantom{-}}
\def\s#1{$s_{#1}$}
\def\B{$\bullet$}
\def\space{\noalign{\smallskip}}
\setbox0=\vbox{%
\halign{%
% vertex labels and nodes
#&\hfil#\hfil%
% coordinates
&\qquad(\hfil$\,#$,&\hfil$\,#$,&\hfil$\,#$,&\hfil$\,#$;&$\;$\hfil$\,#$,&\hfil$\,#$,&\hfil$\,#$,&\hfil$\,#$;&$\;\;$\hfil$\,#$,&\hfil$\,#$%
% closing parentheses
&#\cr
% end of \halign header
%
% top end of the spine of the diagram
&\smash{\vrule depth 100pt width 1pt height -8pt}\cr
% the 11 vectors
\s {1}&\B& \thetabar&\0&\0&\0& \0&\0&\0&\0& \0&\0 &)\cr
\space
\s {2}&\B& -1&-1&1&\0& \0&\0&\0&\0& \0&\0 &)\cr
\space
\s {3}&\B& \0&\theta&\0&\0& \0&\0&\0&\0& \0&\0 &)\cr
\space
\s {4}&\B& \0&1&\-1&\-1& \0&\0&\0&\0& \0&\0 &)\cr
\space
\s {5}&\B& \0&\0&\0&\thetabar& \0&\0&\0&\0& \0&1 &)\cr
\space
\s {6}&\B& \0&\0&\0&\0& \0&\0&\0&\0& 1&\wbar &)\cr
\space
\s {7}&\B& \0&\0&\0&\0& \0&\0&\0&\thetabar& \0&1 &)\cr
\space
\s {8}&\B& \0&\0&\0&\0& \0&1&\-1&\-1& \0&\0 &)\cr
\space
\s {9}&\B& \0&\0&\0&\0& \0&\theta&\0&\0& \0&\0 &)\cr
\space
\s{10}&\B& \0&\0&\0&\0& -1&-1&1&\0& \0&\0 &)\cr
\space
\s{11}&\B& \0&\0&\0&\0& \thetabar&\0&\0&\0& \0&\0 &)\cr
% bottom end of the spine of the diagram
&\smash{\vrule depth -16pt width 1pt height 100pt}\cr
% about to end the \halign and \vbox
}}
% draw the figure, centered
\leavevmode\hfil\box0\hfil
% give the caption
\narrower\medskip\noindent
{\bf Figure~\Tag{fig-figure}.} For each $i=1,\ldots,11$, $S_i$ is the
$\w$-reflection in $s_i$.
The reflections $S_i$ and $S_j$ braid
(resp. commute) if the corresponding nodes of the diagram are
joined (resp. not joined). All of the $s_i$ lie in
$\ee\oplus\ee\oplus\cell$, with coordinates given in an obvious
notation%
%
; each blank entry represents
the value $0$%
%
.
The diagram automorphism $s_i\mapsto s_{12-i}$
corresponds to the bodily exchange of the first two blocks of
coordinates.
\endinsert
We write $K_1$ and $K_2$ for the first and second summands
of $L=\ee\oplus\ee\oplus\cell$, and set $L_0=K_2\oplus\cell$. We
define $\G_0$ to be the
subgroup of $\G $ generated by $S_6,\ldots,S_{11}$.
\beginproclaim Theorem
{\Tag{thm-ch9-and-ch5}}.
$\G $ {\rm(}resp. $\G_0${\rm)} coincides with $\aut L$ {\rm(}resp. $\aut
L_0${\rm)} and acts transitively on
the primitive null vectors of $L$ {\rm(}resp. $L_0${\rm)}.
\endproclaim
\beginproof{Proof:}
All the $s_i$ are roots of $L$ and
$s_6,\ldots,s_{11}$ lie in $L_0$, so $\G\sset\aut L$ and
$\G_0\sset\aut L_0$.
By theorem~\tag{thm-Eisenstein-E8} below, $S_8,\ldots,S_{11}$
generate $\aut K_2$, so $\G_0$
contains $\aut K_2$. Similarly,
$S_1,\ldots,$ $S_4$ generate $\aut K_1$. Since the automorphism
$b_i\mapsto b_{12-i}$ of $\braid$ is inner, $\G $
contains an element sending each $s_i$ to $s_{12-i}$ (up to a
scalar). Such a transformation exchanges $K_1$ with $K_2$, so
$\G $ contains $\aut(K_1\oplus K_2)$.
By theorem~\tag{thm-Eisenstein-E8}, $\aut K_2$ is transitive on
the roots of $K_2$, so $\G_0$ contains the $\w$-reflection $S'$
in $\lambda=(0,0,0,\thetabar)\in K_2$. Computation reveals that
$S'S_7^{-1}$ is the translation $\T{\lambda\wbar}{\theta/2}$ of
$L_0$. The conjugates of this translation by $\aut K_2$ are the
translations $\T{x}{\theta/2}$ where $x$ varies over the roots
of $K_2$. Since the roots of $K_2$ span $K_2$, $\G_0$ contains a
translation by $\lambda$ for each $\lambda\in K_2$. By taking
commutators of these translations, as in the proof of
theorem~\tag{thm-got-all-translations}, we see that $\G_0$
contains the central translations, hence all the translations of
$L_0$. By applying this result together with its image under the
diagram automorphism, we see that $\G $ contains all the
translations of $L$. Together with the previous paragraph, this implies that
$\G$ (resp. $\G_0$) contains the full stabilizer of $\rho$ in
$\aut L$ (resp. $\aut L_0$).
Observe that the roots $r_1$ and $r_2$ appearing in the proof of
theorem~\tag{thm-got-translations} are the vectors $s_6$ and
$\T{0}{-\theta}(s_6)$. We conclude from
\eqtag{eq-got-funny-central-translation} that $\G_0$ (and hence
$\G $) contains the scalars of $\cell$, and therefore acts
transitively on the unit scalar multiples of $\rho$.
Now we study the orbits under $\G $ of the primitive null vectors
of $L$. Suppose $v\in L$ is such a vector, is not a multiple
of $\rho$, and has minimal height in its $\G $-orbit. Then for
some multiple $w$ of $v$ we have
$w=(\ell_1;\ell_2;1,\a-\theta\ell^2/6)$, where
$\a\in\R$ and $\ell=(\ell_1;\ell_2)$ with
$\ell_1,\ell_2\in\ee\tensor\R$.
The argument for theorem~\tag{thm-ch13} proves the following
statement: if $\lambda=(\lambda_1;\lambda_2)\in K_1\oplus K_2$
satisfies $(\ell-\lambda)^2\in[-3,0]$ then there is a root $r$
of $L$ of the
form
$$
r=\(\lambda_1;\,\lambda_2;\,1,\b+{\theta(-3-\lambda^2)\over6}\)
\eqno\eqTag{eq-formula-for-r}
$$
(with $\b\in\R$) such that either a triflection in $r$ reduces
the height of $w$
or else $(\ell-\lambda)^2=-3$ and $r\perp w$. Since $v$ has
minimal height in its $\G $-orbit, the latter possibility must
apply. By \ecite{splag}{p.~121}, the covering norm of $\ee$ is $-3/2$
and its deep holes are the
halves of the vectors with norm divisible by $6$. Therefore the condition
$(\ell-\lambda)^2=-3$ for all $\lambda\in K_1\oplus K_2$ nearest
$\ell$
implies that after a translation we may suppose that each of
$\ell_1$ and $\ell_2$ is one-half of a norm $-6$ vector of
$\ee$. Furthermore, by the transitivity (theorem~\tag{thm-Eisenstein-E8}) of $\aut\ee$ on such
vectors, we may suppose that each is one-half of a specific one,
say $(\theta,\theta,0,0)$.
The argument above, applied to $L_0$ and $\G_0$ instead of $L$
and $\G $, rules out the existence of $w$, so any null vector of
$L_0$ is $\G_0$-equivalent to $\rho$. Since the stabilizers of
$\rho$ in $\G_0$ and $\aut L_0$ coincide, we have proven all our
claims regarding $\G_0$.
We now return to studying $\G $ and $L$. We have deduced that any
null vector of $L$ not $\G $-equivalent to a multiple of $\rho$
is instead $\G $-equivalent to a multiple of
$$
w=\({\theta\over2},{\theta\over2},0,0;\,{\theta\over2},{\theta\over2},0,0;\,
1,\a+{\theta\over2}\)
\eqno\eqTag{eq-formula-for-w}
$$
for some $\a\in\R$. (We have used the equality $\ell^2=-3$.)
Furthermore, we know that for each $\lambda\in K_1\oplus K_2$
with $(\ell-\lambda)^2=-3$, there is a root $r$ of the form
\eqtag{eq-formula-for-r} orthogonal to $w$. Taking $\lambda=0$, so that
$r=(0;\,0;\,1,n+\wbar)$ for some $n\in\Z$, the condition $r\perp w$
requires $\a\in{1\over2}+\Z$. Taking
$\lambda=(\w,-\wbar,0,1;\,0,0,0,0)$, so that
$r=(\lambda;1,n)$ for some $n\in\Z$, the condition $r\perp w$
requires $\a\in\Z$. We conclude that $w$ cannot exist, so that
any primitive null vector of $L$ is $\G $-equivalent to
$\rho$. The proof is completed by the equality of the stabilizers
of $\rho$ in $\G $ and $\aut L$.
\endproof
One can show that
$\(\ee\){}^3\oplus\cell\isomorphism\cll\oplus\cell$, so that the
three Lorentzian lattices we have considered in this section and
the previous one form a natural sequence. We have shown that
$\G $ appears on the lists of
\cite{mostow:monodromy-groups-on-the-complex-n-ball} and
\cite{thurston:shapes-of-polyhedra}, and we conjecture
that $\G_0$ also appears, as
$\def\d{\discretionary{}{}{}}\Gamma({5\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3},\d{\pi\over3})$.
The quotients of complex hyperbolic space by the various
discrete groups of
\cite{deligne:monodromy-of-hypergeometric-functions},
\cite{mostow:picard-lattices-from-half-inegral-conditions} and
\cite{mostow:monodromy-groups-on-the-complex-n-ball} are
constructed in terms of the moduli spaces of point-sets in
$\cp1$; it would be very interesting to find a moduli-space
interpretation for our quotient of $\ch{13}$.
Thurston~\cite{thurston:shapes-of-polyhedra} states that there
is an $\Eisenstein$-lattice $L$ in $\C^{1,9}$ invariant under
$\G $ whose points of positive norm (up to $\G $-equivalence)
may be identified with the triangulations of the sphere $S^2$ in
which each vertex meets at most $6$ triangles. Furthermore, the
number of triangles in a triangulation equals the norm of a
corresponding element of $L$. One can show that up to scaling
there is only one $\Eisenstein$-lattice invariant under $\G
$. Since the fewest number of triangles possible is $2$,
Thurston's lattice must be a copy of $\ee\oplus\ee\oplus\cell$
with all norms multiplied by $2/3$.
\smallskip
The following theorem (used in the proof of theorem~\tag{thm-ch9-and-ch5}) is
not new, but I have been unable to find a reference for it,
especially for the transitivity on norm $-6$ vectors.
\beginproclaim Theorem
{\Tag{thm-Eisenstein-E8}}.
$\aut\ee$ is generated by the triflections
$S_8,\ldots,S_{11}$ and acts transitively on lattice vectors of
norms $-3$ and $-6$.
\endproclaim
\beginproof{Proof:}
$\ee/\theta\ee$ may be regarded as a vector space $V$ over
$\F_3=\Eisenstein/\theta\Eisenstein$, and the reduction modulo
$\theta$ of the
Hermitian form yields a
symplectic form on $V$. No more than 3 roots may be congruent to
each other modulo $\theta$, because a tetrahedron of edge-length
3 cannot be inscribed in a sphere of radius $\sqrt3$. Therefore
the 240 roots represent all $3^4-1=80$ nontrivial elements of $V$. A
triflection in a root acts on $V$ by a symplectic transvection
in the corresponding element of $V$, and these generate the full
symplecic group $\Sp{4}{3}$. Since this group acts transitively
on the 80 classes we obtain the transitivity on roots. Any
element of $\aut\ee$ acting trivially on $V$ must carry each
root to a multiple of itself by a cube root of 1.
Since $\ee$ is not a direct sum of lower-dimensional lattices,
any such transformation must be a scalar. This shows that
$|\aut\ee|=|\Sp{4}{3}|\cdot3=155{,}520$. By the last entry of the
table on p.~133 of
\cite{coxeter:finite-groups-generated-by-unitary-reflections},
the triflections $S_8,\ldots,S_{11}$
generate a group of this order, so this group is $\aut\ee$.
None of the 2160 vectors of norm $-6$ lie in $\theta\ee$, so each
is congruent modulo $\theta$ to a root. By transitivity on
roots, there are $2160/80=27$ vectors of norm $-6$ in each class,
and to prove transitivity it suffices to prove transitivity on
one such class of 27, say those
congruent to $(\theta,0,0,0)$. The 27 vectors are the images of
$(0,0,\theta,\thetabar)$ under the group generated by cyclic
permutation of the last three coordinates and multiplication of
the $4$th coordinate by $\w$. The transitivity follows from this
description.
\endproof
\section{\Tag{sec-quaternions}. The quaternionic case}
All of the above constructions have analogues when the ring of
Eisenstein integers is replaced by the quaternionic ring
$\Hurwitz$ of Hurwitz
integers. In particular, the Leech and $E_8$ lattices may be
regarded as $\Hurwitz$-lattices, and we can use them to
construct finite-covolume reflection groups acting on the
quaternionic hyperbolic spaces $\hh7$,
$\hh5$ and $\hh3$. The Hurwitz integers are the elements of the
skew field $\H$ of quaternions that have the form $(a+b\i+c\j+d\k)/2$
with $a,b,c$ and $d$ being integers that are all congruent
modulo $2$. The left ideal $\ideal$ generated by $p=1-\i$ (or
equally well by any other norm $2$ element of $\Hurwitz$) is
two-sided.
The constructions may be described loosely as ``those of
sections~\tag{sec-background}--\tag{sec-ch9-and-ch5}
with $\theta$ replaced by $p$.'' In particular, if $L$ is an
$\Hurwitz$-lattice, $r\in L$ has norm $-2$, and all inner
products of $r$ with lattice vectors lie in $\ideal$, then
the $\xi$-reflections \eqtag{eq-formula-for-reflection} in $r$
are isometries of $L$ if $\xi\in\{\pm1,\pm\i,\pm\j,\pm\k\}$. In
this case we call $r$ a root of $L$; we set $\reflec{L}$ to be
the group generated by these reflections in roots of $L$. We
define $\qcell$ to be the $2$-dimensional $\Hurwitz$-lattice with
inner product matrix
$\smallmatrix{0}{\pbar}{\p}{0}$. If $\Lambda$ is an
$\Hurwitz$-lattice then as before we write elements of
$L=\Lambda\oplus\qcell$ in the form $(\lambda;\mu,\nu)$ and
define $\rho=(0;0,1)$ and $\height
v=\ip{\rho}{v}$. If $\lambda\in \Lambda\tensor\R$ and
$z\in\im\H$ then the translation
$$
\T{\lambda}{z}:
\eqalign{
(\ell;0,0)
&\mapsto(\ell;0,-\pbar^{-1}\ip{\lambda}{\ell})\cr
(0;1,0)
&\mapsto(\lambda;1,\pbar^{-1}(z-\lambda^2/2))\cr
(0;0,1)
&\mapsto(0;0,1)
\cr}
$$
is an isometry of $L\tensor\R$ preserving $\rho$. One may write these
transformations in matrix form in a manner similar to
\eqtag{eq-translation-matrix} and check that the relations
\eqtag{eq-product-of-translations}--\eqtag{eq-conjugate-of-translation}
hold. $\aut L$ contains a translation by $\lambda$ just if
$\lambda\in \Lambda\cap\Lambda'\ideal$.
We have an analogue of theorems~\tag{thm-got-translations} and
\tag{thm-got-all-translations}, with slightly weaker conclusions:
\beginproclaim Theorem
{\Tag{thm-got-translations-quaternionic-case}}.
Suppose $\Lambda$ is an $\Hurwitz$-lattice and
$L=\Lambda\oplus\qcell$. If $\aut L$ contains a translation
by $\lambda\in \Lambda$ then $\reflec{L}$ contains a translation
by $\lambda(\i-1)$. In particular, if $\Lambda$ is definite then
the stabilizer of $\rho$ in $\reflec{L}$ has finite index in the
stabilizer in $\aut L$.
\endproclaim
\beginproof{Proof:}
This is similar to the proof of
theorem~\tag{thm-got-translations}. Let $R_1$ and
$R_2$ be the $\i$-reflections in the roots $r_1=(1,\i)$ and
$r_2=(1,-1)$ of $\qcell$. Computation reveals that
$R_1R_2=-\i\T{0}{4\i}$, by which we mean the product of left
scalar multiplication by $-\i$ on $\qcell$ and the
translation $\T{0}{4\i}$. For any $\T{\lambda}{z}\in\aut L$,
$$
R_1R_2\T{\lambda}{z}R_2^{-1}R_1^{-1}=\T{\lambda \i}{-\i z\i}\;.
$$
(One must verify this by multiplying the matrices together
rather than following the proof of theorem~\tag{thm-got-translations}, because there
may be no concept of left-multiplication by scalars on
$\Lambda$.) Then $\reflec{L}$ contains
$$
\Tinv{\lambda}{z}R_1R_2\T{\lambda}{z}R_2^{-1}R_1^{-1}=
\T{\lambda(\i-1)}{-z-\i z\i+\i\lambda^2}\;.
$$
This proves the first claim; the second follows by taking
commutators to obtain central translations, as in the proof of
theorem~\tag{thm-got-all-translations}.
\endproof
The Leech lattice admits an action of the binary tetrahedral
group (the multiplicative group of the 24 units of $\Hurwitz$)
such that no nontrivial group element fixes any nontrivial
lattice vector. This action allows us to regard $\rll$ as a
6-dimensional $\Hurwitz$-lattice. Wilson
\cite{raw:quaternionic-leech} describes this lattice in
detail; the only facts we require about it are that suitably
scaled it has no roots, all inner products are divisible by
$\p$, and that its minimal vectors have norm $-4$. In Wilson's
description \cite{raw:quaternionic-leech}, the minimal norm of
the lattice is 8 and all inner products are divisible by $2p$,
as may be verified using the basis he gives on p.~453. By $\qll$
we mean his lattice with all inner products divided by $-2$.
There is also a quaternionic form of the $E_8$ lattice, denoted
$\qee$ and defined to be the set of pairs $(x_1,x_2)$ in
$\Hurwitz^2$ satisfying $x_1\congruent x_2\,(\ideal)$, under the
standard (negative-definite) Hermitian form
$\ip{(x_1,x_2)}{(y_1,y_2)}=-\bar{x}_1y_1-\bar{x}_2y_2$.
\beginproclaim Theorem
\Tag{thm-thm-quaternionic-reflection-groups}.
Let $\Lambda$ be one of the lattices $\qll$, $\qee\oplus\qee$
and $\qee$, and let $L=\Lambda\oplus\qcell$. Then $\reflec{L}$
has finite index in $\aut L$. If $\Lambda=\qee$
{\rm(}resp. $\qll${\rm)} then $\reflec{L}$ acts transitively on
the (1-dimensional) primitive null lattices of $L$
{\rm(}resp. those orthogonal to no roots of $L${\rm)}.
\endproclaim
\beginproof{Proof:}
This is very similar to the proof of theorem~\tag{thm-ch13}. By
theorem~\tag{thm-got-translations-quaternionic-case}, $\reflec{L}$ contains a subgroup of finite index
in the stabilizer of $\rho$ in $\aut L$. In light of this, it
suffices to prove the claims regarding the transitivity on null
lattices, and also that if $\Lambda=\qee\oplus\qee$ then $\reflec{L}$
acts with only finitely many orbits of primitive null lattices.
Take $\Lambda=\qll$. Suppose $v$ is a null vector of $L$ not
proportional to $\rho$ and let $w$ be its multiple of the form
$w=(\ell;1,\pbar^{-1}(\a-\ell^2/2))$ where $\ell\in
\Lambda\tensor\R$ and $\a\in\im\H$. Because the covering norm of
$\Lambda$ is $-2$, we may choose $\lambda\in \Lambda$ with
$(\ell-\lambda)^2\in[-2,0]$. Then we may find a root $r'$ of $L$
of the form $r'=(\lambda;1,\pbar^{-1}(\b-1-\lambda^2/2))$ with
$\b\in\im\H$. For any $n\in\im\ideal$, to be chosen later,
$$
r=r'+(0;0,n)=
\(\lambda;1,\pbar^{-1}(n+\b-1-\lambda^2/2)\)
\eqno\eqTag{eq-formula-for-r-quaternionic-case}
$$
is also a root
of $L$. Computation proves that
$$\eqalign{
\ip{r}{w}
&=-2\[{1\over2}+{1\over4}(\ell-\lambda)^2
-{\im\ip{\lambda}{\ell}+\a+\bbar\over2}+{n\over2}\]\cr
&=-2[a+b+n/2]
\cr}$$
where $a={1\over2}+{1\over4}(\ell-\lambda)^2\in[0,1/2]$ and $b$
is purely imaginary. Denoting by $w'$ the image of $w$ under
the $\i$-reflection in $r$, we have
$$
\height w'=p[1-(1-\i)(a+b+n/2)]\;.
$$
In order to have reduced the height of $w$ we need to choose
$n\in\im\ideal$ such that the
term in brackets has norm less than 1.
That is, we desire
$$
|1-\i|^2\left|{1\over1-\i}-(a+b+n/2)\right|^2<1\;,
$$
which is equivalent to
$$
\left|{1+\i\over2}-(a+b+n/2)\right|^2<{1\over2}
$$
and to
$$
\left|{1\over2}-a\right|^2+\left|{\i\over2}-b-n/2\right|^2<{1\over2}\;.
\eqno\eqTag{eq-foo}
$$
By our condition on $a$, the first term on the left lies in
$[0,1/4]$ and equals $1/4$ just if $a=0$. Since $\im\ideal$ is
a copy of the $D_3$ root lattice (it is spanned by $\i\pm\j$ and
$\j\pm\k$), p.~112 of \cite{splag} shows that the covering
radius of ${1\over2}\im\ideal$ is $1/2$. Therefore by choice of
$n/2$ we may suppose that the second term is bounded by $1/4$,
with equality just if $b$ is a deep hole of
$\i/2+{1\over2}\im\ideal$. All deep holes of $\im\ideal$ are
equivalent by translations by elements of $\ideal$, and $0$ is
such a deep hole. Therefore if $b$ is a deep hole of
$\i/2+{1\over2}\im\ideal$ we may take
$n/2=-b\in{1\over2}\im\ideal$. We have shown that for each
$\lambda\in \Lambda$ with $(\lambda-\ell)^2\in[-2,0]$, there is
a root $r$ of $L$ of the form
\eqtag{eq-formula-for-r-quaternionic-case} such that either the
$\i$-reflection in $r$ reduces the height of $w$, or else $a=0$
and $b=-n/2$, in which case $(\lambda-\ell)^2=-2$ and $r\perp
w$.
For $\Lambda=\qll$, this shows that if $v$ is a null vector of
$L$ orthogonal to no roots, then by repeated reflections its
height may be reduced
to $0$, so that $v$ is equivalent under $\reflec{L}$ to a multiple of $\rho$.
For $\Lambda=\qee$, the same argument proves
that the height of {\it any} null vector may be reduced to $0$,
since the covering norm of $\qee$ is $-1$. For
$\Lambda=\qee\oplus\qee$, the argument shows that if the height
of $w$ cannot be reduced by an element of $\reflec{L}$ then
$\ell$ is a deep hole of $\Lambda$. Modulo translations there
are only finitely many possibilites for $\ell$.
For each such $\ell$ the condition that $w$ be orthogonal to a
root of the form \eqtag{eq-formula-for-r-quaternionic-case} for
each $\lambda$ nearest $\ell$ imposes an
integrality condition on $\a$; modulo central translations there
are only finitely many possibilities for $\a$. This proves that
$\reflec L$ acts
with only finitely many orbits on the primitive null
sublattices of $L$.
\endproof
\section{\Tag{sec-aut-forms}. Automorphic forms}
Borcherds \cite{borcherds:aut-forms-singularities-grassmannians}
has developed machinery for constructing automorphic forms on
the symmetric spaces for the orthogonal groups $O(m,n)$. That
is, for an even $\Z$-lattice $M$ of signature $(m,n)$
and a suitable modular form $F$ on the usual upper half-plane,
he constructs an $(\aut M)$-invariant function on the Grassmannian
$G(M)$
of maximal-dimensional positive-definite subspaces of
$M\tensor\R$. Furthermore, he explicitly describes the
singularities of this function
in terms of the lattice $M$ and the
Fourier coefficients of $F$. In this section we use his results
to obtain automorphic forms on complex and quaternionic
hyperbolic spaces for the symmetry groups of the various
Lorentzian lattices we have considered.
We begin with the complex case. We define the real form of an
$\Eisenstein$-lattice to be the underlying $\Z$-module, equipped
with the bilinear form given by the real part of the Hermitian
form. The real forms of $\cll$, $\ee$ and $\cell$, with inner
products multiplied by $2/3$, are even unimodular $\Z$-lattices. The
evenness follows because all norms of lattice vectors are
divisible by $3$, and the unimodularity from that of the Leech
and $E_8$ lattices (as $\Z$-lattices) together with a
computation of the determinant of the inner product matrix of a
$\Z$-basis for the real form of $\cell$. Borcherds' machinery
simplifies dramatically in the unimodular case, and we obtain
the following theorem.
\beginproclaim Theorem
{\Tag{thm-automorphic-forms-complex-case}}.
Suppose $\Lambda=\ee$ {\rm(}resp. $\ee\oplus\ee$, $\cll${\rm)} and
$L=\Lambda\oplus\cell$. Then there is a holomorphic automorphic form
$\Psi_0$ on $\ch5$ {\rm(}resp. $\ch9$, $\ch{13}${\rm)} of weight $84$
{\rm(}resp. $44$, $4${\rm)} for a one-dimensional representation of $\aut
L$ taking values among the cube roots of unity. Furthermore,
$\Psi_0$ vanishes exactly on the subspaces orthogonal to roots
of $L$, and these zeros have multiplicity one.
\endproclaim
\beginproof{Proof:}
Suppose $\Lambda=\ee$. Then the real form $L^\R$ of $L$ is a
copy of $\II 2,{10}$ (the selfdual even $\Z$-lattice of
signature $(2,10)$) with all norms multiplied by $3/2$. Let $F$
be the modular form
$$
F(\tau)=E_4^2(\tau)/\Delta(\tau)=
\sum_{n\geq-1}c(n)q^n=q^{-1}+504+73764q+\cdots
\eqno\eqTag{eq-e4squared-over-delta}
$$
of weight $-4$ for $\SL{2}{\Z}$, where $q=e^{2\pi\i\tau}$ and
$\tau$ lies in the upper half-plane. Then theorem~13.3 of
\cite{borcherds:aut-forms-singularities-grassmannians} provides a meromorphic automorphic form $\Psi$ on
$G(\II2,{10})$ of weight $c(0)/2=252$ for some one-dimensional
character of $\aut \II2,{10}$. Furthermore, the only zeros and
poles of $\Psi$ lie along the divisors orthogonal to those
$\lambda\in \II2,{10}$ with $\lambda^2<0$, and are zeros of
order
$$
\sum_{\textstyle{%
x\in\R^+
\atop
x \lambda\in\II2,{10}}}
c(x^2 \lambda^2/2)\;.
$$
That is, $\Psi$ has a simple zero along the divisor orthogonal
to each norm $-2$ vector of $\II2,{10}$ and no other zeros or
poles; this implies that $\Psi$ is holomorphic. By definition,
$\ch5$ is the space of all positive-definite complex lines in
$L\tensor\R$, and is obviously a subspace of the Grassmannian
$G(\II2,{10})$. By restricting $\Psi$ to $\ch5$ we obtain an
automorphic form for $\aut L$. The orthogonal complement of a
root of $L$ meets $\ch5$ in the obvious complex hyperplane. The
orthogonal complements of $\lambda\w$ and $\lambda\wbar$ meet
$\ch5$ in this same hyperplane, so the zeros of the restriction
of $\Psi$ to $\ch5$ are precisely the hyperplanes orthogonal to
roots of $L$, with multiplicity $3$.
If $r$ is a root of $L$ then the real form of the
$\Eisenstein$-span of $r$ , with inner products multiplied by
$2/3$, is a copy of the $A_2$ lattice. The three real
reflections in the 6 roots of $A_2$ generate the symmetric group
$S_3$, and the $120^\circ$ rotations are commutators in
$S_3$. This proves that the triflections in $r$ are commutators
in $\aut\II2,{10}$. Since the triflections generate $\aut L$,
we find that $\Psi$ transforms according to the trivial
character of $\aut L$. A cube root $\Psi_0$ of the
restriction of $\Psi$ is automorphic
with respect to a character of $\aut(L)$ taking values among the
cube roots of unity. Finally, $\Psi_0$ obviously has simple
zeros along each mirror, and weight
$252/3=84$.
If $\Lambda=\ee\oplus\ee$ or $\cll$ then our assertions follow
from the argument above, with $F$ replaced by
$$\displaylines{
F(\tau)=E_4(\tau)/\Delta(\tau)=q^{-1}+264+8244q+\cdots
\rlap{\qquad or}\cr
F(\tau)=1/\Delta(\tau)=q^{-1}+24+324q+\cdots\;,
\cr}$$
respectively. In the case $\Lambda=\cll$, to prove that $\Psi$
is invariant under $\aut L$, one must use the fact that $\aut L$
is generated by triflections together with the perfect group
$\aut\cll$. ($\aut\cll$ is perfect because it is the universal
central extension of a simple group.)
\endproof
Now we consider the quaternionic case. As before, the real forms
of $\qll$, $\qee$ and $\qcell$ are even unimodular lattices
(this time, no rescaling is required).
Borcherds obtained the form $\Psi$ used above by
exponentiating another function
$\Phi$ which has logarithmic singularities along the mirrors. In
the quaternionic case, the relevant real lattices are
$\II4,{4n}$, and the analogue of $\Phi$ has poles rather than
logarithmic singularities.
\beginproclaim Theorem
{\Tag{thm-automorphic-forms-quaternionic-case}}.
Let $\Lambda=\qee$ {\rm(}resp. $\qee\oplus\qee$, $\qll${\rm)} and let
$L=\Lambda\oplus\qcell$. Then there is a function $\Phi$ on $\hh3$ {\rm(}resp. $\hh5$, $\hh7${\rm)}
that is invariant under $\aut L$ and real-analytic except at its
singularities. The set on which $\Phi$ is singular is the
union of the mirrors orthogonal to roots of $L$.
Along the mirror orthogonal to a root $r$, the singularity has
type
$$
{12\over\pi}{w^2\over|\ip{w}{r}|^2}\;,
$$
where $w$ is a vector in $L\tensor\R$ representing a point $v$
of hyperbolic space near $r^\perp$.
\endproclaim
\beginproof{Proof:}
Suppose $\Lambda=\qee$, so that $L^\R$ is isometric to $\II4,{12}$.
We take $F(\tau)=E_4^2(\tau)/\Delta(\tau)$,
as in \eqtag{eq-e4squared-over-delta}. Then the function $\Phi$
on $G(L^\R)$ defined in
\ecite{borcherds:aut-forms-singularities-grassmannians}{\S6}
(with the polynomial $p$ set to $1$) is real
analytic except at its singularities.
If $v_0$ is a maximal-dimensional positive-definite
subspace of $L^\R\tensor\R$ then for $v\in G(L^\R)$ near $v_0$,
$\Phi$ has a singularity of type
$$
\sum_{\textstyle{%
\lambda\in L^\R\cap v_0^\perp
\atop
\lambda\neq0}}
c(\lambda^2/2)\(2\pi \lambda_{v^+}^2\)^{-1}\;,
$$
where $\lambda_{v^+}$ is the projection of $\lambda$ to the
subspace $v$. Any $\lambda$ appearing in the sum satisfies
$\lambda^2<0$, so the only $\lambda$ for which the term in the
sum is nonzero have norm $-2$. We now restrict to
$\hh3$, with $v_0$ a generic point of $r^\perp$ (\ie
orthogonal to no elements of $L$ except multiples of $r$),
represented by a vector $w_0$ of $L\tensor\R$ of positive
norm. The sum extends over the 24 roots $\lambda$ (the unit
scalar multiples of $r$) orthogonal to $w_0$. For $v$ a point of
$\hh3$ near $v_0$, represented by a vector $w$,
$\lambda_{v^+}$ is
the projection $\lambda$ to $w\H$
with respect to the Hermitian form, and
$\lambda_{v^+}^2=|\ip{w}{\lambda}|^2/w^2$.
Therefore the singularity near $v_0$ has type
$$
\sum_{u}c(-1){w^2\over2\pi |\ip{w}{ru}|^2}
$$
where the sum extends over the units $u$ of $\Hurwitz$. Since
$c(-1)=1$ and
the 24 terms in the sum all coincide, we have proven our claims
in the case $\Lambda=\qee$.
If $\Lambda=\qee\oplus\qee$ or $\qll$ then one can repeat the
proof above, with $F=E_4/\Delta$ or $F=1/\Delta$ respectively.
\endproof
In the introduction we mentioned the analogy between the two
series of groups we have studied here and the series of real
hyperbolic reflection groups discussed by Conway
\cite{jhc:26dim}, namely the groups of the lattices
$$
\II1,9,\quad
\II1,{17}
\hbox{\quad and\quad}
\II1,{25}\;.
$$
The analogy extends even to the construction of automorphic
forms. In each of the real, complex and quaternionic cases,
there are automorphic forms constructed from the modular forms
$E_4^2/\Delta$, $E_4/\Delta$ and $1/\Delta$. (See examples~10.7
and~12.2 of \cite{borcherds:aut-forms-singularities-grassmannians} for the real case.) Borcherds has observed
(private communication and \ecite{borcherds:aut-forms-singularities-grassmannians}{\S12}) that there seems to
be a very close (maybe one-to-one?) correspondence between nice
reflection groups on real, complex and quaternionic hyperbolic
spaces and automorphic forms with singularities exactly along
reflection hyperplanes. The examples of this paper strengthen
this correspondence.
In an earlier paper \cite{allcock:oh2} we constructed two
reflection groups acting on the octave hyperbolic plane $\oh2$,
and it is natural to wonder whether the analogies extend to this
case. That they might is suggested by the fact that one of the
groups may be regarded as the automorphism group of the
``lattice'' $\oee\oplus\smallmatrix0110$, where $\oee$ is the
$E_8$ lattice regarded as a ``1-dimensional lattice'' over a
suitable discrete subring $\ocint$ of the skew field $\O$ of
octaves. This is obviously analogous to the first term in each
of the series of lattices over $\Z$, $\Eisenstein$ and
$\Hurwitz$.
\section{References}
% Do not tamper with the following line
% ---begin bibentries---
\bibitem{allcock:oh2}
D.~J. Allcock.
Reflection groups on the octave hyperbolic plane.
Preprint, 1997.
\bibitem{dja:complex-reflection-groups}
D.~J. Allcock.
New complex and quaternion-hyperbolic reflection groups.
Preprint, 1996.
\bibitem{borcherds:aut-forms-singularities-grassmannians}
R.~E. Borcherds.
Automorphic forms with singularities on {G}rassmannians.
Preprint, 1997.
\bibitem{jhc:26dim}
J.~H. Conway.
The automorphism group of the 26-dimensional even unimodular
{L}orentzian lattice.
{\it Journal of Algebra}, 80:159--163, 1983.
Reprinted in \cite{splag}.
\bibitem{jhc:leech-radius}
J.~H. Conway, R.~A. Parker, and N.~J.~A. Sloane.
The covering radius of the {L}eech lattice.
{\it Proceedings of the Royal Society of London}, A380:261--90, 1982.
Reprinted in \cite{splag}.
\bibitem{splag}
J.~H. Conway and N.~J.~A. Sloane.
{\it Sphere Packings, Lattices, and Groups}.
Springer-Verlag, 1988.
\bibitem{coxeter:finite-groups-generated-by-unitary-reflections}
H.~S.~M. Coxeter.
Finite groups generated by unitary reflections.
{\it Abh. Math. Sem. Univ. Hamburg}, 31:125--135, 1967.
\bibitem{deligne:monodromy-of-hypergeometric-functions}
P.~Deligne and G.~D. Mostow.
Monodromy of hypergeometric functions and non-lattice integral
monodromy.
{\it Pub. I.H.E.S.}, 63:5--90, 1986.
\bibitem{feit:unimodular-Eisenstein}
W.~Feit.
Some lattices over ${Q}(\sqrt{-3})$.
{\it Journal of Algebra}, 52:248--263, 1978.
\bibitem{mostow:picard-lattices-from-half-inegral-conditions}
G.~D. Mostow.
Generalized {P}icard lattices arising from half-integral conditions.
{\it Pub. I.H.E.S.}, 63:91--106, 1986.
\bibitem{mostow:monodromy-groups-on-the-complex-n-ball}
G.~D. Mostow.
On discontinuous action of monodromy groups on the complex $n$-ball.
{\it Jour. A.M.S.}, 1:555--586, 1988.
\bibitem{thurston:shapes-of-polyhedra}
W.~P. Thurston.
Shapes of polyhedra.
Geometry Supercomputer Project, University of Minnesota, research
report GCG 7, 1987.
\bibitem{vinberg:nonexistence-crystallographic-reflection-groups}
E.~B. Vinberg.
The nonexistence of ctystallographic reflection groups in
{L}obachevskii spaces of high dimension.
{\it Funct. Anal. Appl.}, 15:128--130, 1981.
\bibitem{raw:quaternionic-leech}
R.~A. Wilson.
The quaternionic lattice for $2{G}_2(4)$ and its maximal subgroups.
{\it Journal of Algebra}, 77:449--466, 1982.
\bibitem{raw:complex-leech}
R.~A. Wilson.
The complex {L}eech lattice and maximal subgroups of the {S}uzuki
group.
{\it Journal of Algebra}, 84:151--188, 1983.
% ---end bibentries---
% Do not tamper with the previous line.
\bye