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72
N.M. Katz:
For each Qconjugacy class A of irreducible Crepresentations of G,
the element
(4.4.0.2)
P(A) der'" ~ P(Z)
Z~A
ofC [G] lies in fact in Q [G], and is an indecomposable central idempotent
in Q [G]. Every indecomposable central idempotent in Q [G] is of this
form. We have
(4.4.0.3)
p(A)P(A,)=~P(A)
if
A=A'
O
if not.
(4.4.0.4)
Let Hz be a local system of Zmodules of finite type on which
G acts. We define the Gsublocal system
(4.4.0.5)
P(A) H z
de_~f'n
the inverse image in H z of
P(A)(Ho) ~ HQ.
The canonical mapping of local systems
(4.4.0.6)
(~ P(A) H z ~ H z
A
is a Gmorphism and an
isogeny,
i.e., it becomes an isomorphism when
tensored with Q.
(4.4.1)
If G acts on a family of mixed Hodge structures (H z, W,F)
(meaning that its action on the local system Hz respects W and F), then
each P(A) H z is a subfamily of mixed Hodge structures. If (H z, W, F) is
a polarizable (4.2.2.4) family of mixed Hodge structures, so is each sub
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 Fall '11
 NormanKatz

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