N n a reection group of conite volume in ec or hc is

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Unformatted text preview: n are the groups G(2, 2, n). Finally, the groups of type I2 (m) are the groups G(m, m, 2). These groups are distinguished from other groups by the property that one of invariant polynomials is of degree 2. 16 IGOR V. DOLGACHEV 3.3. Complex crystallographic reflection groups. A complex analog of a space of constant curvature is a simply connected complex K¨hler manifold of constant a holomorphic curvature (complex space form ). There are three types of such spaces (see [60]): n • EC , the n-dimensional affine space equipped with the standard hermitian form |z |2 = |zi |2 . It is a homogeneous space (Cn U(n))/U(n); • Pn (C), the n-dimensional complex projective space equipped with the standard Fubini-Study metric. It is a homogeneous space PU(n + 1)/U(n). n • HC = {z ∈ Cn : |z | < 1}, the n-dimensional complex hyperbolic space. The n hermitian metric on HC is defined by 1 1 − |z |2 n n zi dzi + zi dzi ) + (1 − |z | ) ¯ ¯ 2 i=1 dzi dzi . ¯ i=1 They are simply connected hermitian complex homogeneous manifolds of dimension n with isotropy subgroups equal to the unitary group U(n). The complex hyperbolic space has a model in complex projective space Pn (C) equal to the image of the subset C = {(z0 , z1 , . . . , zn ) ∈ Cn+1 : −|z0 |2 + |z1 |2 + . . . + |zn |2 < 0}. The unitary group U(n + 1) of the hermitian form −|z0 |2 + |z1 |2 + . . . + |zn |2 of n signature (n, 1) acts transitively on HC with isotropy subgroup U(n). It defines a transitive action of PU(n, 1) = U(n, 1)/U(1) with isotropy subgroups isomorphic to U(n). n n Let XC be an n-dimensional complex space form. A reflection in XC is a holomorphic isometry whose set of fixed points is a hypersurface. A reflection group is a discrete group of holomorphic automorphisms generated by reflections. A reflection n n group Γ of XC = Pn (C) (resp. HC ) can be centrally extended to a reflection subgroup of U(n) (resp. U(n, 1)) such that over every reflection lies a linear reflection. n A reflection group of EC is a discrete subgroup of Cn U(n) which is generated by affine reflections. It can be considered as a linear reflection group in a complex vector space V of dimension n + 1 equipped with a hermitian form of Sylvester n signature (t+ , t− , t0 ) = (n, 0, 1). We take for EC the affine subspace of the dual ∗ ∗ linear space V of linear functions φ ∈ V satisfying φ(v ) = 1, where v is a fixed nonzero vector in V ⊥ . The corresponding linear space is the hyperplane {φ ∈ V ∗ : φ(v ) = 0}. n n A reflection group Γ of cofinite volume in EC or HC is called a complex crysn n tallographic group (affine, hyperbolic). If XC = P (C), it is a finite group defined n n by a finite linear complex reflection group in Cn+1 . If XC = EC , then Γ leaves n n invariant a lattice Λ ⊂ C of rank 2n (so that C /Λ is a compact complex n-torus) and Lin(Γ) is a finite subgroup of U(n). This implies that Γ is also cocompact. n If XC = Pn (C), then Γ, being a discrete subgroup of a compact Lie group PU(n + 1), is finite and cocompact....
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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