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\proclaim Coxeter groups, Lorentzian lattices, and K3 surfaces.
\hfill 25 June, 27 July 1998
Internat. Math. Res. Notices 1998, no. 19, 1011--1031.
Richard E. Borcherds,
\footnote{$^*$}{ Supported by a Royal Society
professorship and an NSF grant.}
D.P.M.M.S.,
16 Mill Lane,
Cambridge,
CB2 1SB,
England.
e-mail: reb@dpmms.cam.ac.uk
www home page www.dpmms.cam.ac.uk/\~{}reb
\bigskip
\proclaim Contents.
1.~Introduction.
2.~Notation and statement of main theorem.
3.~Proof of main theorem.
4.~The structure of $\Gamma_{\Omega}$.
5.~Examples.
\proclaim
1.~Introduction.
The main result of this paper describes the normalizer
$N_{W_\Pi}(W_J)$ of a finite parabolic subgroup $W_J$ of a (possibly
infinite) Coxeter group $W_\Pi$. More generally we describe
$N_{W_\Pi.\Gamma_\Pi}(W_J)$ where $\Gamma_\Pi$ is a group of diagram
automorphisms of the Coxeter diagram $\Pi$ of $W_\Pi$. By taking $\Pi$
to be Conway's Coxeter diagram of the reflection group of $II_{1,25}$
we compute the automorphism groups of some Lorentzian lattices and K3
surfaces.
In the case when $W_\Pi$ is a finite Coxeter group (and $\Gamma_\Pi =
1$) the normalizer of $W_J$ has been described by Howlett [H]. His
result states that $N_{W_\Pi}(W_J)$ is a split extension $W_J.W'_J$,
where $W'_J=W_{\Omega}.\Gamma_{\Omega}$ is in turn a split extension
with $W_{\Omega}$ a Coxeter group and $\Gamma_{\Omega}$ a more
mysterious group acting on $\Omega$. Howlett showed by case by case
analysis that if $\Pi$ is connected (and $W_\Pi$ is finite) then
$\Gamma_{\Omega}$ is an elementary abelian 2-group
and is a subgroup of $Aut(J)$.
When $W_\Pi$ is infinite the normalizer $N_{W_\Pi}(W_J)$ has a similar
structure, except that the group $\Gamma_{\Omega}$ can be more
complicated. Although there is still a canonical map from
$\Gamma_\Omega$ to $Aut(J)$, the kernel can be non trivial, though it
has finite cohomological dimension. The kernel is trivial in the case
of finite $W_\Pi$ considered by Howlett because any finite group of
finite cohomological dimension must be trivial. For example, the case
when $J=A_1$, $\Gamma_\Pi=1$ has been done by Brink [Br], who showed
that $\Gamma_\Omega$ is a free group (and therefore has cohomological
dimension at most 1). We will extend Brink's result to Coxeter
diagrams of arbitrary finite reflection groups. More precisely we
construct a category $Q_4$ using $\Pi$ and $J$ and prove that the
classifying space of this category is a classifying space of the group
$\Gamma_{\Omega}$. The main point about this category $Q_4$ is that it is
often finite and can often be written down explicitly,
in which case we can easily read off a presentation of $\Gamma_\Pi$.
For example, if $J=A_1$ we show that the
classifying space of this category $Q_4$ is 1-dimensional, so its
fundamental group is free and we recover Brink's result.
After writing this paper I discovered that Brink and Howlett
had previously announced a related description
of the normalizer of a parabolic
subgroup of a Coxeter group; see [B-H], and
see example 2.8 for the relation between their
result and theorem 2.7.
For later applications we need some generalizations as follows. First
of all, instead of calculating the normalizer of $W_J$ in a Coxeter group
$W_\Pi$, we calculate the normalizer in an extension
$W_\Pi.\Gamma_\Pi$, where $\Gamma_\Pi$ is a group of diagram
automorphisms. Secondly, we sometimes want to compute not the full
normalizer, but a subgroup with image contained in some subgroup
$\Gamma_J$ of $Aut(J)$. Thirdly, we sometimes want to vary the choice
of the Coxeter group $W_{\Omega}$, which we do by varying
a certain normal subgroup $R$ of $\Gamma_J$. For example, in calculating the
automorphism groups of $K3$ surfaces we take $W_{\Omega}$ to be
generated by reflections of norm $-2$ vectors rather than by all
reflections, so we take $R=1$.
Section 2 contains a statement of the main result (theorem 2.7)
describing a classifying category for the group $\Gamma_\Omega$,
and section 3 contains the proof of this result. Section 4 contains
some more information about the structure of $\Gamma_\Omega$.
In section 5 we give some applications of theorem 2.7, and
in particular
show how to describe the automorphism groups of some Lorentzian lattices
by embedding them in $II_{1,25}$ and using the description of
$Aut(II_{1,25})$ in [C]. The idea of studying Lorentzian lattices
by embedding them as orthogonal complements of root lattices in
$II_{1,25}$ comes from Conway and Sloane ([C-S]).
Work of I. Piatetski-Shapiro and I. R. Shafarevich [P-S]
shows that there is a map from the automorphism group of a K3 surface
to the group of automorphisms of its Picard lattice
modulo the group generated by reflections of norm $-2$ vectors
which has finite kernel and co-finite image,
so in practice if we want to describe the automorphisms of a K3
surface the main step is to calculate the automorphism group of its
Picard lattice.
Kondo showed in [K] that the automorphism groups of some
K3 surfaces could be studied by embedding their Picard lattice
as the orthogonal complement of a root lattice in $II_{1,25}$.
We use Kondo's idea to describe the automorphism groups
of some K3 surfaces in terms of combinatorics of the Leech lattice.
In particular we reprove some results of
Vinberg [V] on the ``most algebraic'' K3 surfaces and extend them to the
``next most algebraic'' K3 surface. Kondo showed in [K] that the
automorphism group of the Kummer surface of a generic genus 2 Jacobian
was generated by the classically known automorphisms together with some
new automorphisms found by Keum [Ke], and we show how to use Kondo's results
to describe the structure of this group. Kondo and Keum [K-K] have recently
proved similar results for some Kummer surfaces associated to the products
of two elliptic curves.
Kondo recently found another mysterious connection between
automorphism groups of K3 surfaces and Niemeier lattices [K98],
and used this to give a short proof of Mukai's classification [Mu] of
the finite groups that act on K3 surfaces.
I would like to thank
I. Cherednik, I. Grojnowski, R. B. Howlett,
J. M. E. Hyland, S. Kondo, U. Ray, and G. Segal
for their help.
\proclaim
2.~Notation and statement of main theorem.
This section states the main result (theorem 2.7) describing normalizers of
parabolic subgroups of Coxeter groups.
We recall some basic definitions about Coxeter systems.
For more about them see [Hi] or [Bo].
A pair $(W,S)$ is called a {\bf Coxeter system} if
$W$ is a group with a subset $S$ such that $W$ has the presentation
$$\langle s:s\in S|(ss')^{m_{ss'}}=1 \hbox{ when $m_{ss'}<\infty$}\rangle$$
where $m_{ss'}\in
\{1,2,3,\ldots,\infty\}$ is the order of $ss'$, and $m_{ss'}=1$ if and
only if $s=s'$. A diagram automorphism of $S$ is an automorphism of
the set $S$ that extends to an automorphism of the group $W$, and
$Aut(S)$ means the group of diagram automorphisms of $S$. We say that
$(W,S)$ is {\bf irreducible} if $S$ is not a union of two disjoint
commuting subsets. The number of elements of $S$ is called its {\bf
rank}. The Coxeter system is called {\bf spherical} if $W$ has finite
order. The irreducible spherical Coxeter diagrams are $A_{n}$ $(n\ge
1)$, $B_n=C_n$ $(n\ge 2)$, $D_{n}$ $(n\ge 4)$, $E_6$, $E_7$, $E_8$,
$F_4$, $G^{(n)}_2=I_2(n)$ $(n\ge 5)$, $H_3$, and $H_4$. It is also
sometimes useful to define the Coxeter diagrams $B_1=C_1=A_1$,
$D_3=A_3$, $D_2=A_1^2$, $E_5=D_5$, $E_4=A_4$, $E_3=A_2A_1$,
$G_2^{(4)}=B_2=C_2$, $G_2^{(3)}=A_2$, $G_2^{(2)}=A_1^2$.
If $(W_\Pi,\Pi)$ is a Coxeter system then we write
$V_\Pi$ for the (possibly infinite dimensional) real
vector space with a basis of elements $e_s$
for $s\in \Pi$, and put a symmetric bilinear form on $V_\Pi$ by
defining
$$(e_s,e_{s'})= 2\cos(\pi/m_{ss'}).$$
Note that we normalize the roots $e_s$ so that they have norm $(e_s,e_s)=-2$
rather than 1; this is done to be consistent with the usual conventions
in algebraic geometry.
The Coxeter group $W_\Pi$ acts
on $V_\Pi$ with the element $s\in \Pi\subseteq W_\Pi$ acting as
the reflection $v\mapsto v+(v,e_s)e_s$ in the
hyperplane $e_s^\perp$. Any subgroup $\Gamma_\Pi$ of $Aut(\Pi)$
acts on $V_\Pi$ by
permutations of the elements $e_s$, so we get an action of
$W.\Gamma_\Pi$ on $V_\Pi$, and hence on the dual space $V_\Pi^*$.
We write $\Delta^+$ for the set of positive roots of $W_\Pi$.
We define the fundamental
chamber $C_\Pi\subseteq V_\Pi^*$ of $W_\Pi$ by
$$C_\Pi=\{x\in V_\Pi^*|x(r)\ge 0 \hbox{ for all } r\in \Pi
\hbox{ and } x(r) > 0 \hbox{ for almost all }
r\in \Delta^+ \}.$$
(Recall that ``almost all'' means ``all but a finite number of''.)
A theorem due
independently to Tits and Vinberg
states that no two distinct points of $C_\Pi$ are conjugate under
$W_\Pi$, and the subgroup of $W_\Pi$ fixing all points of some subset $A$ of
$W_\Pi$ is generated by the reflections in the faces of $C_\Pi$ containing
$A$. In particular $W_\Pi$ acts simply transitively on the conjugates
of $C_\Pi$. The union $W_\Pi(C_\Pi)$ of all conjugates of $C_\Pi$
under $W_\Pi$ is given by
$$W_\Pi(C_\Pi)=\{x\in V_\Pi^*| x(r) > 0 \hbox{ for almost all
} r\in \Delta^+ \}.$$ In particular $W_\Pi(C_\Pi)$ is
convex and closed under multiplication by positive real numbers.
If $x\in W_\Pi(C_\Pi)$ then the set of roots vanishing on $x$
is a finite root system.
Note that $W_\Pi(C_\Pi)$ is usually slightly smaller than the Tits cone,
which is defined in the same way except that we omit the condition
that $x(e_s) > 0$ for all but a finite number of $ s$ in the definition
of the fundamental domain. The Tits cone can be thought of
as obtained from $W_\Pi(C_\Pi)$ by ``adding some boundary components''.
The reason for
using $W_\Pi(C_\Pi)$ rather than the Tits cone is that
the cone $W_\Pi(C_\Pi)$ has the property that the subgroup
of $W_\Pi$ fixing any vector of it is finite.
We fix a spherical subset $J$ of $\Pi$.
In particular we get a
spherical Coxeter system $(W_J,J)$.
Suppose $K$ is an isometry of Coxeter diagrams
from $J$ into $\Pi$.
We write $W_K$ for the finite reflection group
generated by $K(J)$.
There is a natural homomorphism
$p:N_{W_\Pi.\Gamma_\Pi}(W_K)\mapsto Aut(J)$. We let
$N_{W_\Pi.\Gamma_\Pi}(W_K;\Gamma_J)$ be the subgroup
of elements whose image is in a subgroup $\Gamma_J$ of $Aut(J)$.
We are interested in describing the group
$N_{W_\Pi.\Gamma_\Pi}(W_J;\Gamma_J)$.
Most of the time we take $\Gamma_J=Aut(J)$
in which case $N_{W_\Pi.\Gamma_\Pi}(W_J;\Gamma_J) =N_{W_\Pi.\Gamma_\Pi}(W_J)$,
but it is occasionally useful to use other values of $\Gamma_J$;
see example 5.7 and theorem 4.1.
We define $W'_K$ to be the subgroup of $N_{W_\Pi.\Gamma_\Pi}(W_K;\Gamma_J)$
mapping $K(J)$ to itself.
\proclaim Lemma 2.1. $N_{W_\Pi.\Gamma_\Pi}(W_K;\Gamma_J) = W_K.W'_K$.
Proof. This follows immediately from the fact that
$W_K$ acts simply transitively on the Weyl chambers of $W_K$, and
$W'_K$ is the subgroup of $N_{W_\Pi.\Gamma_\Pi}(W_K;\Gamma_J)$
fixing a Weyl chamber of $W_K$.
This proves lemma 2.1.
The group $W'_K$ acts on the subspace $V_\Pi^{*W_K}$ of $V_\Pi^*$ of
all vectors fixed by $W_K$. We now construct a reflection group
$W_{\Omega_K}$ acting on $V_\Pi^{*W_K}$. We choose a normal subgroup
$R$ of $\Gamma_J$. (The subgroup $R$ is used to control the
reflection group $W_{\Omega}$ defined below. Often we want
$W_{\Omega}$ to be as large as possible and we take $R=\Gamma_J$, but
sometimes we want to take a smaller $W_{\Omega}$; see examples 5.3,
5.4 and 5.5.) We define the group $W_{\Omega_K}$ to be the subgroup
of $W_\Pi\cap W'_K$ generated by elements $w\in W_\Pi\cap W'_K$ such
that $w$ acts on $V_\Pi^{*W_K}$ as a reflection and acts on $J$ as an
element of $R$. We define $\Omega_K$ to be the Coxeter diagram of
$W_{\Omega_K}$ and $C_{\Omega_K}$ to be its fundamental chamber. If
$K$ is the identity map from $J$ to $J\subseteq \Pi$ then we write
$W_\Omega$, $C_\Omega$, and $\Omega$ instead of $W_{\Omega_K}$,
$C_{\Omega_K}$, and $\Omega_K$. Note that $W_{\Omega_K}$ is obviously
contained in the inverse image of $R$ in $W_\Pi\cap W'_K$, but can be
much smaller; see for example the discussion of $D_4$ in example 5.7.
We define the group $\Gamma_{\Omega_K}$ to be the subgroup of $W'_K$
of elements $w$ with $w(C_{\Omega_K})=C_{\Omega_K}$.
\proclaim Lemma 2.2. The group $W'_K$ is
a semidirect product $W'_K=W_{\Omega_K}.\Gamma_{\Omega_K}$.
Proof.
We first show that the group $W_{\Omega_K}$ acts faithfully on $V_\Pi^{*W_K}$.
More generally we will show that if $w\in W_\Pi\cap W'_K$ acts
trivially on $V_\Pi^{*W_K}$, then $w=1$. To see this we
observe that $w$ fixes the point $ x\in C_\Pi\cap
V_\Pi^{*W_K}$ such that $x(e_s)=0$ if $s\in K(J)$ and $x(e_s)=1$ if
$s\notin K(J)$.
Therefore $w$ is in the subgroup $W_K$ of $W_\Pi$ generated by the
simple reflections of $W_\Pi$ fixing $x$. On the other hand
$w$ maps $K(J)$ into itself as $w\in W'_K$.
This implies that $w=1$ because $1$ is the only element of
$W_K$ mapping $K(J)$ into itself. This
proves that the group $W_{\Omega_K}$ acts faithfully on $V_\Pi^{*W_K}$.
Lemma 2.2 now follows from the fact that $W_{\Omega_K}$ acts simply
transitively on the conjugates of $C_{\Omega_K}$ under $W'_K$, and
$\Gamma_{\Omega_K}$ is the stabilizer of $C_{\Omega_K}$. This proves
lemma 2.2.
Warning: the group $\Gamma_{\Omega_K}$ need not act faithfully on
$V_\Pi^{*W_K}$ (though it does act faithfully on $V_\Pi^{*W_K}\times J$).
We define a {\bf classifying category} of a group $\Gamma$ to be a
category whose geometric realization is a classifying space for
$\Gamma$. (Recall from [Q] that the geometric realization of
a category is a space with a 0-cell for each object and an $n$-cell for
each sequence of $n$ composable morphisms if $n>0$.)
For example, the category with one object whose morphisms
are the elements of $\Gamma$ (with composition given by group
multiplication) is a classifying category for $\Gamma$.
We have more or less reduced the problem of describing
$N_{W_\Pi.\Gamma_\Pi}(W_J; \Gamma_J)$ to that of describing
$\Gamma_{\Omega_K}$. (The Coxeter diagram $\Omega$ of $W_\Omega$
can be described once we know $\Gamma_\Omega$.)
The main theorem of this paper describes
$\Gamma_{\Omega_K}$ by giving an explicit classifying category for
it. To define this category we need some more definitions.
Suppose that $S$ is the Coxeter diagram of a finite reflection group
$G$ of a finite dimensional vector space with no vectors fixed by $G$.
Fix a Weyl chamber $C$ of $G$, so that the walls of $C$ correspond to
the points of $S$. Then there is a unique element $\sigma_S$ of $G$
taking $C$ to $-C$ called the opposition involution. The
involution $-\sigma_S$ acts on the the Coxeter diagram $S$, and its
action on $S$ does not depend on the choice of $C$. This action can be
described as follows. The points of the Coxeter diagram correspond to
the simple roots of $C$. This set of roots is the same as the set of
simple roots of $\sigma_S(C)=-C$ multiplied by $-1$.
Hence $-\sigma_S$ acts on this set of
simple roots, in other words on the Coxeter diagram $S$. The involution
$-\sigma_S$ of $S$ can be described explicitly as follows. On diagrams
of type $A_1$, $B_n=C_n$, $D_{2n}$, $E_7$, $E_8$, $F_4$, $G^{(2n)}_2$,
$H_3$, $H_4$ for any $n\ge 1$ the involution $-\sigma_S$ is
the trivial automorphism of $S$, while for diagrams of types
$A_{n+1}$, $D_{2n+1}$, $E_6$, $G_2^{(2n+1)}$ for $n\ge 1$ the involution
$-\sigma_S$ is the unique nontrivial automorphism of the Coxeter
diagram $S$. Finally if the diagram $S$ is a union of connected
components then $-\sigma_S$ acts on each connected component as
described above.
Suppose that $J$ and $S$ are Coxeter diagrams. Suppose that $K$ and
$K'$ are two isometries from $J$ into $S$. We define $K$ and $K'$ to
be {\bf adjacent} if there is a point $s$ of $S$ not in $K(J)$ such that
$K(J)\cup s$ is spherical and $\sigma_{K(J)\cup s}\sigma_{K(J)}$ takes
$K$ to $K'$. If $K$ is adjacent to $K'$ then $K'$ is adjacent to
$K$. We define two isometries $K$, $K'$ of $J$ into $S$ to be {\bf
associate} if there is a sequence of isometries $K=K_1$, $K_2,\ldots,
K_n=K'$ from $J$ into $S$ such that $K_i$ and $K_{i+1}$ are adjacent
for $1\le i