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O(M/M ) = O(E8 ) = W (E8 ) has characteristic polynomial (1 + t + t2 )4 . It follows
from the classiﬁcation of conjugacy classes in W (E8 ) that w is the product of the
Coxeter elements in the sublattice A4 of E8 . Its centralizer is a subgroup of index
2 in the wreath product Σ4 Σ4 of order 31104. This group is a ﬁnite complex
reﬂection group L4 (No. 32 in the list). The centralizer of g is the unique complex
crystallographic group in aﬃne space of dimension 5 with linear part L4 .
10. Complex ball quotients
10.1. Hypergeometric integrals. Let S be an ordered set of n +3 distinct points
z1 , . . . , zn+3 in P1 (C). We assume that (zn+1 , zn+2 , zn+3 ) = (0, 1, ∞). Let U = P1 \
S and γ1 , . . . , γn+3 be the standard generators of π1 (U ; u0 ) satisfying the relation
γ1 · · · γn+3 = 1. We have a canonical surjection of the fundamental group of U to
the group A = (Z/dZ)n+3 /∆(Z/dZ) which deﬁnes an ´tale covering V → U with
the Galois group A. The open Riemann surface V extends A-equivariantly to a
compact Riemann surface X (z ) with quotient X (z )/A isomorphic to P1 (C).
Let µ = (m1 /d, . . . , mn+3 /d) be a collection of rational numbers in the interval
(0, 1) satisfying
d n+3 mi = k ∈ Z.
i=1 They deﬁne a surjective homomorphism
χ : A → C∗ , γi → e2π
¯ √ −1mi /d , where γi is the image of γi in A.
The following computation can be found in  (see also ).
Lemma 10.1. Let H 1 (X (z ), C)χ be the χ-eigensubspace of the natural representation of the Galois group A on H 1 (X (z ), C). Then
dim H 1 (X (z ), C)χ = n + 1.
Let Ω(X (z )) be the space of holomorphic 1-forms on X (z ) and Ω(X (z ))χ be the
χ-eigensubspace of A in its natural action on the space Ω(X (z )). Then
dim Ω(X (z ))χ = k − 1, 52 IGOR V. DOLGACHEV where k is deﬁned in (10.1).
Recall that H 1 (X (z ), C) = Ω(X (z )) ⊕ Ω(X (z )), so we get
H 1 (X (z ), C)χ = Ω(X (z ))χ ⊕ Ω(X (z ))χ ,
and we can consider the hermitian form in H 1 (X (z ), C)χ induced by the skewsymmetric cup-product
H 1 ( X ( z ) , C) × H 1 ( X ( z ) , C) → H 2 ( X ( z ) , C) ∼ C.
The signature of the hermitiain form is equal to (k − 1, n − k + 2).
Now let us start to vary the points z1 , . . . , zn in P1 (C) but keep them distinct
and not equal to 0, 1 or ∞. Let U ⊂ P1 (C)n be the corresponding set of parameters.
Its complement in (P1 (C)n ) consists of N = n + 3n hyperplanes Hij : zi − zj = 0
and Hi (0) : zi = 0, Hi (1) : zi = 1, Hi (∞) : zi = ∞. Fix a point z (0) ∈ U . For
each of these hyperplanes H consider a path which starts at z (0) , goes to a point
on a small circle normal bundle of an open subset of H , goes along the circle, and
then returns to the starting point. The homotopy classes s1 , . . . , sN of these paths
generate π1 (U ; z (0 ).
It is not diﬃcult to construct a ﬁbration over U whose ﬁbres are the curves
X (z ). This deﬁnes a local coeﬃcient system...
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