This preview shows page 1. Sign up to view the full content.
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 ndimensional 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 ndimensional complex projective space equipped with the standard FubiniStudy metric. It is a homogeneous space PU(n + 1)/U(n).
n
• HC = {z ∈ Cn : z  < 1}, the ndimensional 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 ndimensional 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 ntorus)
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....
View
Full
Document
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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

Click to edit the document details