Generalized Eichler-Shimura Map
Let G/ℤ(p) for ℤ(p)=ℚ∩ℤ p be a connected reductive group split over ℤ p (see Section 4.4.3 for the definition of split algebraic groups). We first prove semi- simplicity of the commutative Hecke algebra acting on the nearly ordinary cohomology group H q n.ord (X(S), L) inside the interior cohomology H ! q (X(S),L)for a modular variety X(S) associated with an arbitrary p-power level open compact subgroup S of G(A∞), where H ! q is the image of the compactly supported cohomology group inside the standard cohomology group. Here the locally constant or coherent sheaf L of ℚ-vector spaces on X(S) is associated with a rational representation of G twisted by a finite-order character. After dealing with topological cohomology groups, we relate as Hecke modules the topological and the coherent cohomology groups via the generalized Eichler-Shimura map (which shows the semi-simplicity of the Hecke algebra acting on the coherent cohomology and topological cohomology). Although we have assumed that G is split over ℤ p , the argument works equally well for a connected reductive G smooth quasi-split over ℤ p .
KeywordsCohomology Group Compact Subgroup Hermitian Form Congruence Subgroup Maximal Compact Subgroup
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