By Sophie Morel

ISBN-10: 0691142939

ISBN-13: 9780691142937

This e-book reviews the intersection cohomology of the Shimura types linked to unitary teams of any rank over Q. as a rule, those types usually are not compact. The intersection cohomology of the Shimura style linked to a reductive crew G incorporates commuting activities of absolutely the Galois workforce of the reflex box and of the crowd G(Af) of finite adelic issues of G. the second one motion could be studied at the set of complicated issues of the Shimura type. during this booklet, Sophie Morel identifies the Galois action--at solid places--on the G(Af)-isotypical parts of the cohomology. Morel makes use of the strategy built by way of Langlands, Ihara, and Kottwitz, that's to check the Grothendieck-Lefschetz mounted aspect formulation and the Arthur-Selberg hint formulation. the 1st challenge, that of making use of the fastened aspect formulation to the intersection cohomology, is geometric in nature and is the article of the 1st bankruptcy, which builds on Morel's prior paintings. She then turns to the group-theoretical challenge of evaluating those effects with the hint formulation, while G is a unitary team over Q. purposes are then given. specifically, the Galois illustration on a G(Af)-isotypical element of the cohomology is pointed out at just about all areas, modulo a non-explicit multiplicity. Morel additionally supplies a few effects on base switch from unitary teams to normal linear teams.

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**Extra resources for On the Cohomology of Certain Non-Compact Shimura Varieties**

**Sample text**

Define a cohomological correspondence ! uC : ( j T g )∗ (iC1 TC1 )! LC1 −→ T 1 (iC2 TC2 )! 1), and the direct image with compact support is defined by duality). Finally, write −1 NC = [K(2) N : h2 K h2 ∩ Nnr (Af )]. 3 The coefficient uC1 ,C2 in the above matrix is equal to NC uC , C where the sum is taken over the set of C ∈ CP such that Tg (C ) = C1 and T1 (C ) = C2 . 5). 5 of [P2], and it is true as well for the Shimura varieties considered here. 7. 5. , that G is not an orthogonal group). Fix an algebraic closure F of Fp .

UC : ( j T g )∗ (iC1 TC1 )! LC1 −→ T 1 (iC2 TC2 )! 1), and the direct image with compact support is defined by duality). Finally, write −1 NC = [K(2) N : h2 K h2 ∩ Nnr (Af )]. 3 The coefficient uC1 ,C2 in the above matrix is equal to NC uC , C where the sum is taken over the set of C ∈ CP such that Tg (C ) = C1 and T1 (C ) = C2 . 5). 5 of [P2], and it is true as well for the Shimura varieties considered here. 7. 5. , that G is not an orthogonal group). Fix an algebraic closure F of Fp . Let V ∈ Ob RepG .

Then, for every z ∈ (RE/Q Gm,Q ), µ(z) = (z, 1)Ip 0 0 (1, z)Iq . 2 Let p ∈ {1, . . , n}. Define a cocharacter µp : Gm,E −→ GE by µp (z) = (z, 1)Ip 0 (1, z)In−p . 2 PARABOLIC SUBGROUPS Let G be a connected reductive algebraic group over Q. Fix a minimal parabolic subgroup P0 of G. Remember that a parabolic subgroup of G is called standard if it contains P0 . Fix a Levi subgroup M0 of P0 . Then a Levi subgroup M of G will be called standard if M is a Levi subgroup of a standard parabolic subgroup and M ⊃ M0 .

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