Unformatted text preview: omy group of (X, x0 )
and ΓG be the G-equivariant monodromy group. Then
ΓG = Γ ∩ AutZ[G] (M ).
Consider the natural action of G on the jacobian algebra (8.2) via its action on
the partial derivatives of the function f .
Theorem 9.2 (C.T.C. Wall ). Assume (X, x0 ) is an isolated hypersurface
singularity deﬁned by a holomorphic function f : Cn+1 → C. Let Jf be its jacobian
algebra. There is an isomorphism of G-modules,
MC = Jf ⊗ detG ,
where detG is the one-dimensional representation of G given by the determinant.
Example 9.3. Let f (z1 , z2 , z3 ) = z1 z2 + z3 k be a simple surface singularity of type
A2k−1 . Consider the action of the group G = Z/2Z by (z1 , z2 , z3 ) → (z1 , z2 , −z3 ).
We take 1, z3 , . . . , z3 k−2 to be a basis of the jacobian algebra. Thus we have k in2
variant monomials 1, z3 , . . . , z3 k−2 and k − 1 anti-invariant monomials z3 , . . . , z3 k−3 .
It follows from Theorem 9.2 that MC is the direct sum of the k − 1-dimensional
invariant part M+ and the k-dimensional anti-invariant part M− . We have O(M ) = W (A2k−1 ) (τ ), where τ is the nontrivial symmetry of the Coxeter diagram of type A2k−1 . In fact it
is easy to see that the semi-direct product is the direct product. Let α1 , . . . , α2k−1
be the fundamental root vectors. We have τ (αi ) = α2k−i ; hence dim M τ = k. This
shows that the image σ of the generator of G in O(M ) is not equal to τ . In fact, it
must belong to W (A2k−1 ). To see this we use that all involutions in W (M ) ∼ Σ2k
are the products of at most k transpositions; hence their ﬁxed subspaces are of
dimension ≤ k − 1. The group O(M ) is the product of W (M ) and 1; thus all
involutions in O(M ) \ W (M ) have ﬁxed subspaces of dimension ≤ k − 1. The
conjugacy class of σ ∈ W (A2k−1 ) ∼ Σ2k is determined by the number r of disjoint
transpositions in which it decomposes. We have dim M σ = 2k − r − 1. Thus σ
is conjugate to the product of k disjoint transpositions. It follows from the model
of the lattice A2k−1 given in (2.1) that σ is conjugate to the transformation αi →
−α2k−i . Hence the sublattice M− is generated by βi = αi + α2k−i , i = 1, . . . , k − 1,
and βk = αk . We have
⎪−4 if i = j = k,
⎨−2 if i = j = k,
(βi , βj ) =
if |i − j | = 1,
if |i − j | > 1. 50 IGOR V. DOLGACHEV Comparing this with Example 4.1 we ﬁnd that M = N (2), where N is the lattice
deﬁning an integral structure for the reﬂection group of type Bk . In other words,
the reﬂections rβi generate the Weyl group of type Bk .
Similar construction for a symmetry of order 2 of singular points of types Dk+1 ,
n ≥ 4 (resp. E6 , resp. D4 ), leads to the reﬂection groups of type Bk (resp. F4 ,
resp. G2 = I2 (6)). The Milnor lattices obtained as the invariant parts of the Milnor
lattice of type A2k−1 and Dk+1 deﬁne the same reﬂection groups, but their lattices
are similar but not isomorphic. Figure 14 shows how the Coxeter diagrams of types
Bk , F4...
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