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There is a classiﬁcation of crystallographic reﬂection groups in EC (due to
3
V. Popov [93]).
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First observe that any a ∈ EC deﬁnes a surjective homomorphism g → g from
¯
the aﬃne group to the linear group of the corresponding complex linear space
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V . We write any x ∈ EC in the form x = a + v, for a unique v ∈ V , and get
3 According to [50] some groups are missing in Popov’s list. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 17 g (a + v ) = g (a) + g (v ). This deﬁnition of g does not depend on the choice of a. In
¯
¯
particular, choosing a on a reﬂecting aﬃne hyperplane H , we see that g is a linear
¯
reﬂection which ﬁxes H − a. This implies that the image of a crystallographic
reﬂection subgroup Γ of Cn U(n) is a ﬁnite reﬂection subgroup of U(n).
Theorem 3.2. Let G be a ﬁnite irreducible reﬂection group in U(n). Then the
following properties are equivalent:
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(i) there exists a complex reﬂection group Γ in EC with linear part G;
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(ii) there exists a Ginvariant lattice Λ ⊂ EC of rank 2n;
(iii) the number of the group G in Table 2 is 1, 2(m = 2, 3, 4, 6), 3(m = 2, 3, 4, 6),
4, 5, 8, 12, 24 − 29, 31 − 37. If G is not of type G(4, 2, n), n ≥ 4 (number 2), or GL(2, 3) (type 12) or EN4
(number 31),4 then Γ is equal to the semidirect product Λ G. In the exceptional
cases, Γ is either the semidirect product or some nontrivial extension of G with
normal subgroup Λ.
A table in [93] describes all possible lattices and the extensions for each G as
above.
Recall from Theorem 3.1 that the algebra of invariant polynomials of a ﬁnite
complex reﬂection group Γ in Cn is a polynomial algebra. This can be restated
as follows. One considers the induced action of Γ in Pn−1 (C) and the orbit space
Pn−1 (C)/Γ which exists as a projective algebraic variety. Now the theorem asserts
that this variety is isomorphic to a weighted projective space P(q1 , . . . , qn ),5 where
the weights are equal to the degrees of free generators of the invariant algebra. The
following is an analog of Theorem 3.1 for aﬃne complex crystallographic groups
due to BernsteinShwarzman [11] and Looijenga [75].
Theorem 3.3. Assume that the linear part of a complex crystallographic group
Γ is a complexiﬁcation of a real ﬁnite reﬂection group W . Then the orbit space
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EC /Γ exists as an algebraic variety and is isomorphic to a weighted projective space
P(q0 , . . . , qn ), where the weights are explicitly determined by W .
It is conjectured that the same is true without additional assumption on the
linear part.
Example 3.4. Let G be a ﬁnite complex reﬂection group arising from the complexiﬁcation of a real reﬂection group Gr . Any such group is realized as the linear
part of a complex crystallographic group Γ in aﬃne space and Γ is the semidirect
product of G and a Ginvariant lattice. Suppose Gr is of ADE type. Let e1 , . . . , en
be the norm vectors of the Coxeter polytope in Rn . For any τ = a +...
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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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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