Abstract

Let$G$be a connected split reductive group over a finite field$\mathbb{F}_{q}$and$X$a smooth projective geometrically connected curve over$\mathbb{F}_{q}$. The$\ell$-adic cohomology of stacks of$G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of$X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

Highlights

  • IntroductionLet X be a smooth projective geometrically connected curve over a finite field Fq. We denote by F its function field, by A the ring of adèles of F and by O the ring of integral adèles

  • Let G be a connected split reductive group over a finite field Fq and X a smooth projective geometrically connected curve over Fq

  • The converse direction follows from the following fact: any non-zero image of the constant term morphism along a proper parabolic subgroup P with Levi quotient M is supported on the components indexed by a cone in the lattice of the cocharacters of the center of M

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Summary

Introduction

Let X be a smooth projective geometrically connected curve over a finite field Fq. We denote by F its function field, by A the ring of adèles of F and by O the ring of integral adèles. The converse direction follows from the following fact: any non-zero image of the constant term morphism along a proper parabolic subgroup P with Levi quotient M is supported on the components indexed by a cone in the lattice of the cocharacters of the center of M It generates an infinite-dimensional vector space under the action of the Hecke algebra of M. We define the cuspidal cohomology HGj,,Icu,Wsp ⊂ HGj ,I,W as the intersection of the kernels of the constant term morphisms for all proper parabolic subgroups. This construction was suggested by Vincent Lafforgue. For any stack (respectively scheme) (for example ChtG,N,I,W and GrG,I,W ), we consider only the reduced substack (respectively subscheme) associated to it

Parabolic induction diagram of stacks of shtukas
1.1.12 We have the morphism of paws
1.1.16 We have a morphism of prestacks
Quotient by Ξ
Harder–Narasimhan stratification of stack of shtukas
The following diagram is commutative
Cohomology of stacks of shtukas
Constant term morphisms and cuspidal cohomology
More on cohomology groups
We define a morphism
Contractibility of deep enough horospheres
We define
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