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Geometric Eisenstein series. (English) Zbl 1046.11048
Let $$X$$ be a curve over a finite field $$\mathbb{F}_q$$, $$G$$ be a reductive group over the function field on $$X$$. Let $$T, B$$ and $$P$$ be respectively a Cartan subgroup, a Borel subgroup and a parabolic subgroup of $$G$$. The authors define a functor $$\text{Eis}= \text{Eis}_T^G: \text{Sh}(\text{Bun}_T) \to \text{Sh}(\text{Bun}_G)$$, where $$\text{Sh}(\text{Bun}_G)$$ is the derived category of constuctible sheaves on the stalk $$\text{Bun}_G$$ of $$G$$-bundles on $$X$$, discuss the relationships of this functor with Drinfeld’s and Laumon’s compactifications on $$\text{Bun}_B$$, and investigate its properties. Section 1 of the paper is devoted to the definition of Drinfeld’s compactifications.
The main results of the paper are stated in Section 2. Let $$\text{Rep}( \check{G})$$ be the category of finite-dimensional representations of the Langlands dual group $$\check{G}$$. Let $$\Lambda$$ be the covering lattice of $$T$$, and $$\Lambda_G^+$$ be the semi-group of dominant coweights. It is proved in the paper (Theorem 2.1.5) that for $$x\in X$$ and $$\mathcal{S}\in \text{Sh}( \text{Bun}_T)$$ there exists a functorial isomorphism $H_G^\lambda \circ \text{Eis}( \mathcal{S}) \simeq\oplus_{\mu\in\Lambda} ( \text{Eis}\boxtimes \text{id})\circ H_T^\mu( \mathcal{S})\otimes V^\lambda(\mu),$ where $$\text{Eis}\boxtimes \text{id}$$ denotes the corresponding functor $$\text{Sh}( \text{Bun}_T\times X)\to \text{Sh}( \text{Bun}_G\times X)$$, and $$H_T^\mu$$ denotes the corresponding Hecke functor (introduced by the authors) for the group $$T$$.
Next, the authors single out the subcategory of $$\text{Sh}(\text{Bun}_T)$$ consisting of so-called regular sheaves (i.e. such that for any positive coroot $$\alpha$$ and for the corresponding projection map $$T\to T/ \mathbb{G}_m$$ one has $$\mathfrak{f}_!^\alpha( \mathcal{S})=0$$, $$\mathfrak{f}^\alpha: \text{Bun}_T \to \text{Bun}_{T/ \mathbb{G}_m}$$ being the induced map of stacks), define the action $$w\cdot \mathcal{S}$$ of the Weyl group $$W$$ of $$G$$ on $$\text{Bun}_T$$, and prove (Theorem 2.1.8) that for a regular $$\mathcal{S}\in \text{Sh}( \text{Bun}_T)$$ there exists a functorial isomorphism $$\text{Eis}(w\cdot \mathcal{S}) \simeq \text{Eis}( \mathcal{S})$$ which is a counterpart for the functor $$\text{Eis}$$ of the classical functional equation for Eisenstein series. This isomorphism is compatible with the isomorphism of Theorem 2.1.5 above. The functional equation $$\text{Eis}(w\cdot \mathcal{S}) \simeq \text{Eis}( \mathcal{S})$$ of Theorem 2.1.8 does not contain the $$L$$-factors entering in the classical functional equation. This feature is explained by Theorem 2.2.12, which asserts that $$L$$-function is incorporated in the definition of $$\text{Eis}$$. Another principal result of the paper (Theorem 2.3.10) asserts that $$\text{Eis}_T^G \simeq \text{Eis}_M^G\circ \text{Eis}_T^M$$. Note that the proof of the latter result is quite complicated.
The technique developed in the paper allows the authors to prove a special case of Langlands conjecture: namely, for an unramified irreducible representation of $$\pi_1(X)$$ into $$\check{G}$$, such that $$\pi_1(X)^{geom}$$ maps to $$\check{T}\subset \check{G}$$, there exists an unramified automorphic form on $$G_\mathbb{A}$$ ($$\mathbb{A}$$ is the ring of adèles of the function field on $$X$$) which corresponds to this representation in the sense of Langlands.
In Section 3 the authors prove that an Eisenstein series functor commutes with Hecke functors in the principal case (i.e. in case of a Borel subgroup of $$G$$); the obtained results are generalized in Section 4 to the case of a general reductive subgroup. Here the investigation is based on the Lusztig-Drinfeld-Ginzburg-Mircovich-Vilonen theory of spherical perverse sheaves on the affine Grassmannian $$\text{Gr}_G$$.
Section 5 contains the important technical results about the local acyclity. The definition of local acyclity introduced in this section differs from usual one, but is equivalent to it; the proof of equivalence is given in Appendix B. The key result of this section is Theorem 5.1.4 which asserts that the stack $$\overline{\text{Bun}}_P$$ (resp. $$\widetilde{\text{Bun}}_P$$) is locally acyclic with respect to the natural projection $$\mathfrak{q}: \overline{\text{Bun}}_B \to \text{Bun}_T$$ (resp. $$\mathfrak{q}_P: \widetilde{\text{Bun}}_B\to \text{Bun}_M$$). Here $$\overline{\text{Bun}}_P$$ is the Drinfeld compactification of $$\text{Bun}_P$$, and $$\widetilde{\text{Bun}}_P$$ is an another compactification of $$\text{Bun}_P$$ constructed by the authors. Theorem 5.1.4 implies that the functor $$\text{Eis}$$ commutes with Verdier duality. Moreover, it is a key tool in the proof of Theorem 2.3.10 on the composition of $$\text{Eis}$$.
In Section 6 and 7 the authors investigate some stratifications on the stacks $$\overline{\text{Bun}}_B$$ and $$\widetilde{\text{Bun}}_P$$ needed to prove the isomorphism $$\text{Eis}_T^G \simeq \text{Eis}_M^G\circ \text{Eis}_T^M$$. In particular, in Section 7 it is shown that the map $$\text{Bun}_{B,P}\to \widetilde{\text{Bun}}_P$$ is stratified-small in the sense of Goresky-MacPherson. This last result allows the authors to finish the proof of $$\text{Eis}_T^G \simeq \text{Eis}_M^G\circ \text{Eis}_T^M$$. Finally, in Section 7 the proof of the functional equation is completed.
The paper concludes with two appendices. Appendix A is devoted to proving an important auxiliary result on the reduction of $$\check{G}$$-local system by using an argument of G. Prasad, and in Appendix B it is shown that the definition of local acyclicity from Section 5 of the paper is equivalent to the definition given in [P. Deligne, Théorèmes de finitude en cohomologie $$l$$-adique, SGA $$4\frac{1}{2}$$, Lecture Notes in Mathematics 569, 233–261 (1977; Zbl 0349.14013)].

##### MSC:
 11G45 Geometric class field theory 11F60 Hecke-Petersson operators, differential operators (several variables) 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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