Taking x from m we obtain that the x vector 2h h h

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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 G-invariant lattice and every G-invariant 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 Γ-invariant. 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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