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Unformatted text preview: bi, b > 0, consider the lattice Λτ in Cn spanned by the vectors ei and τ ei . This is a Ginvariant
lattice and every Ginvariant lattice is obtained in this way. Moreover Λτ = Λτ if
and only if τ and τ belong to the same orbit of the modular group PSL(2, Z) which
acts on the upper halfplane {z = a + bi ∈ C : b > 0} by the M¨bius transformations
o
z → (az + b)/(cz + d). The linear part G is a ﬁnite group of automorphisms of the
compex torus (C/Λτ )n with the orbit space (C/Λτ )n /G isomorphic to a weighted
4 Also G(6, 6, n) as pointed out in [50].
is the quotient of Cn \ {0} by the action of C∗ deﬁned in coordinates by (z1 , . . . , zn ) →
(λq1 z1 , . . . , λqn zn ).
5 This 18 IGOR V. DOLGACHEV projective space. In the case when Gr is the Weyl group of a simple simply connected Lie group H , this quotient is naturally isomorphic to the moduli space of
principal H bundles on the elliptic curve C/Z + τ Z (see [44]).
Remark 3.5. If Γ is a real crystallographic group in aﬃne space E n , then its comn
plexiﬁcation is a complex noncrystallographic reﬂection group in EC . Every comn
plex noncrystallographic reﬂection group in EC is obtained in this way (see [93],
2.2).
We will discuss later a construction of complex crystallographic reﬂection groups
n
in HC for n ≤ 9. The largest known dimension n for which such groups exist is 13
([3]). It is believed that these groups occur only in ﬁnitely many dimensions.
4. Quadratic lattices and their reflection groups
4.1. Integral structure. Let Γ be an orthogonal linear reﬂection group in a real
vector space V of dimension n equipped with a nondegenerate symmetric bilinear
form of signature (n, 0) or (n − 1, 1). We assume that the intersection of its reﬂection
hyperplanes is the origin. We say that Γ admits an integral structure if it leaves
invariant a free abelian subgroup M ⊂ V of rank n generating V . In other words,
there exists a basis (e1 , . . . , en ) in V such that n Zei is Γinvariant.
i=1
A linear reﬂection group admitting an integral structure is obviously a discrete
subgroup of the orthogonal group O (V ) and hence acts discretely on the corresponding space of constant curvature S n−1 or H n−1 (because the isotropy subgroups are
compact subgroups of O (V )). Thus Γ is a reﬂection group of S n−1 or H n−1 . By
a theorem of Siegel [104], the group O(M ) = {g ∈ O (V ) : g (M ) = M } is of ﬁnite
covolume in the orthogonal group O(V ); hence Γ is of ﬁnite covolume if and only
if it is of ﬁnite index in O(M ).
Let H ⊂ V be a reﬂection hyperplane in Γ and let eH be an orthogonal vector to
H . In the hyperbolic case we assume that H deﬁnes a hyperplane in H n−1 ; hence
(eH , eH ) > 0. The reﬂection rH is deﬁned by
r H ( x) = x − 2(x, αH )
eH
(αH , αH ) for some vector αH proportional to eH . Taking x from M we obtain that the
x,α
vector (2(H ,αH )) αH belongs to M. Replacing αH by proportional vector, we may
α
H
assume that αH ∈ M and also that αH is a pri...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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