##
**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)].

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)].

Reviewer: Vasyl I. Andriychuk (Lviv)

### 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) |