72 N.M. Katz: For each Q-conjugacy class A of irreducible C-representations of G, the element (188.8.131.52) 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 (184.108.40.206) p(A)P(A,)=~P(A) if A=A' O if not. (220.127.116.11) Let Hz be a local system of Z-modules of finite type on which G acts. We define the G-sub-local system (18.104.22.168) P(A) H z de_~f'n the inverse image in H z of P(A)(Ho) ~ HQ. The canonical mapping of local systems (22.214.171.124) (~ P(A) H z --~ H z A is a G-morphism 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 sub-family of mixed Hodge structures. If (H z, W, F) is a polarizable (126.96.36.199) family of mixed Hodge structures, so is each sub-
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