Outline of the proof of the geometric Langlands conjecture for \(\mathrm{GL}(2)\). (English. French summary) Zbl 1406.14008
Bost, Jean-Benoît (ed.) et al., De la géométrie algébrique aux formes automorphes (II). Une collection d’articles en l’honneur du soixantième anniversaire de Gérard Laumon. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-806-0/pbk). Astérisque 370, 1-112 (2015).
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Let \(X\) be an irreducible smooth projective curve over an algebraically closed field of characteristic \(0\), let \(G\) be a reductive group over \(k\), let \(\check{G}\) be its Langlands dual, regarded as an algebraic group over \(k\). Let \(\mathrm{Bun}_{G}\) be the algebraic stack of \(G\)-bundles on \(X\), let \(\mathrm{LocSys}_{\check{G}}\) be the derived algebraic stack of \(\check{G}\)-local systems on \(X\). The author outlines a proof of the “categorical” geometric Langlands conjecture for \(G=\mathrm{GL}_{2}\): There exists an equivalence of DG categories \[ \mathbb{L}_{G}:\mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}})\to \mathrm{D-Mod}(\mathrm{Bun}_{G}), \] compatible with the geometric Satake equivalence and the Eisenstein construction.
The idea is best illustrated by the diagram: \[ \begin{tikzcd} \mathrm{Glue}(\check{G})_{\mathrm{spec}} \ar[r, "{\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}}"] &\mathrm{Whit}^{\mathrm{ext}}(G,G) \\ \mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}}) \ar[r,"{\mathbb{L}_{G}}"] \ar[u, "{\mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}})}"] &\mathrm{D-Mod}(\mathrm{Bun}_{G})\ar[u, "{\mathrm{Coeff}_{G,G}^{\mathrm{ext}}}A" '] \\ \mathrm{QCoh}(\mathrm{Op}(\check{G})_{\lambda^{I}}^{\mathrm{glob}}) \ar[u, "{(v_{\lambda^{I}})_{*}}"] \ar[r, equal] &\mathrm{QCoh}(\mathrm{Op}(\check{G})_{\lambda^{I}}^{\mathrm{glob}})\ar[u, "{q\mathrm{-Hitch}_{\lambda^{I}}}" ']\ . \end{tikzcd} \]
On the spectral side, the DG category \(\mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}})\) is an enlargement of \(\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})\), the idea is to get compatibility with the Eisenstein series construction. The DG category \(\mathrm{Glue}(\check{G})_{\mathrm{spec}}\) is a gluing of \(\mathrm{F}_{\check{P}}\mathrm{-mod}(\mathrm{QCoh}(\mathrm{LocSys}_{\check{P}}))\), where \(\mathrm{F}_{\check{P}}\) is the monad induced by the vector fields on \(\mathrm{LocSys}_{\check{P}}\) along the derived fiber of the morphism \(\mathrm{LocSys}_{\check{P}}\to \mathrm{LocSys}_{\check{G}}\). The functor \[ \mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}}): \mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}}) \to \mathrm{Glue}(\check{G})_{\mathrm{spec}} \] is a gluing of analogues of the constant term functor for automorphic forms. In a joint work with D. Arinkin, they prove that the functor is fully faithful for all reductive groups \(G\).
On the geometric side, the DG category \(\mathrm{Whit}^{\mathrm{ext}}(G,G)\) is a gluing of the “degenerate” Whittaker categories \(\mathrm{Whit}(G,P)\). The functor \[ \mathrm{Coeff}_{G,G}^{\mathrm{ext}}: \mathrm{D-Mod}(\mathrm{Bun}_{G})\to \mathrm{Whit}^{\mathrm{ext}}(G,G) \] is an enhancement of the familiar Fourier-Whittaker expansion of an automorphic form. It is conjectured to be fully faithful (Theorem for the group \(G=\mathrm{GL}_{n}\)).
The functor \(\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\) is a gluing of functors \[ \mathbb{L}_{G,P}^{\mathrm{Whit}}:\mathrm{F}_{\check{P}}\mathrm{-mod}(\mathrm{QCoh}(\mathrm{LocSys}_{\check{P}}))\to \mathrm{Whit}(G,P). \] Definition of the functors \(\mathbb{L}_{G,P}^{\mathrm{Whit}}\) can be reduced to that of \(\mathbb{L}_{G,G}^{\mathrm{Whit}}\), the latter is defined by spreading over \(\mathrm{Ran}(X)\). More precisely, there exists a fully faithful functor \[ \mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})\to \mathrm{Rep}(\check{G})_{\mathrm{Ran}(X)}, \] and the category \(\mathrm{Whit}(G,G)\) is canonically equivalent to a variant of \(\mathrm{Rep}(\check{G})_{\mathrm{Ran}(X)}\) (Basically this is the Casselman-Shalika formula), and \(\mathbb{L}_{G,G}^{\mathrm{Whit}}\) is composition of the two functors. By construction, the functors are compatible with the action of the monoidal category \(\mathrm{Rep}(\check{G})_{\mathrm{Ran}(X)}\), i.e. the geometric Satake equivalence. Finally, it is stated as a “quasi-Theorem” that the functor \(\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\) is fully faithful.
Now that the functors \(\mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}})\), \(\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\), \(\mathrm{Coeff}_{G,G}^{\mathrm{ext}}\) are fully faithful, it is enough to compare the essential images of \[ \mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\circ \mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}})\quad \text{ and }\mathrm{Coeff}_{G,G}^{\mathrm{ext}} \] in order to establish the categorical Langlands correspondence. By induction, it boils down to comparing images of irreducible local systems on the spectral side and that of cuspidal forms on the geometric side. This is accomplished by generalizing Beilinson-Drinfeld’s construction of Hecke eigensheaves on \(\mathrm{Bun}_{G}\) associated to special \(\check{G}\)-local systems on \(X\) called opers. That is the heart of the work and it is the last piece in the commutative diagram above.
For the entire collection see [Zbl 1319.14004].
Let \(X\) be an irreducible smooth projective curve over an algebraically closed field of characteristic \(0\), let \(G\) be a reductive group over \(k\), let \(\check{G}\) be its Langlands dual, regarded as an algebraic group over \(k\). Let \(\mathrm{Bun}_{G}\) be the algebraic stack of \(G\)-bundles on \(X\), let \(\mathrm{LocSys}_{\check{G}}\) be the derived algebraic stack of \(\check{G}\)-local systems on \(X\). The author outlines a proof of the “categorical” geometric Langlands conjecture for \(G=\mathrm{GL}_{2}\): There exists an equivalence of DG categories \[ \mathbb{L}_{G}:\mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}})\to \mathrm{D-Mod}(\mathrm{Bun}_{G}), \] compatible with the geometric Satake equivalence and the Eisenstein construction.
The idea is best illustrated by the diagram: \[ \begin{tikzcd} \mathrm{Glue}(\check{G})_{\mathrm{spec}} \ar[r, "{\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}}"] &\mathrm{Whit}^{\mathrm{ext}}(G,G) \\ \mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}}) \ar[r,"{\mathbb{L}_{G}}"] \ar[u, "{\mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}})}"] &\mathrm{D-Mod}(\mathrm{Bun}_{G})\ar[u, "{\mathrm{Coeff}_{G,G}^{\mathrm{ext}}}A" '] \\ \mathrm{QCoh}(\mathrm{Op}(\check{G})_{\lambda^{I}}^{\mathrm{glob}}) \ar[u, "{(v_{\lambda^{I}})_{*}}"] \ar[r, equal] &\mathrm{QCoh}(\mathrm{Op}(\check{G})_{\lambda^{I}}^{\mathrm{glob}})\ar[u, "{q\mathrm{-Hitch}_{\lambda^{I}}}" ']\ . \end{tikzcd} \]
On the spectral side, the DG category \(\mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}})\) is an enlargement of \(\mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})\), the idea is to get compatibility with the Eisenstein series construction. The DG category \(\mathrm{Glue}(\check{G})_{\mathrm{spec}}\) is a gluing of \(\mathrm{F}_{\check{P}}\mathrm{-mod}(\mathrm{QCoh}(\mathrm{LocSys}_{\check{P}}))\), where \(\mathrm{F}_{\check{P}}\) is the monad induced by the vector fields on \(\mathrm{LocSys}_{\check{P}}\) along the derived fiber of the morphism \(\mathrm{LocSys}_{\check{P}}\to \mathrm{LocSys}_{\check{G}}\). The functor \[ \mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}}): \mathrm{IndCoh}_{\mathrm{Nilp}_{\check{G}}^{\mathrm{glob}}}(\mathrm{LocSys}_{\check{G}}) \to \mathrm{Glue}(\check{G})_{\mathrm{spec}} \] is a gluing of analogues of the constant term functor for automorphic forms. In a joint work with D. Arinkin, they prove that the functor is fully faithful for all reductive groups \(G\).
On the geometric side, the DG category \(\mathrm{Whit}^{\mathrm{ext}}(G,G)\) is a gluing of the “degenerate” Whittaker categories \(\mathrm{Whit}(G,P)\). The functor \[ \mathrm{Coeff}_{G,G}^{\mathrm{ext}}: \mathrm{D-Mod}(\mathrm{Bun}_{G})\to \mathrm{Whit}^{\mathrm{ext}}(G,G) \] is an enhancement of the familiar Fourier-Whittaker expansion of an automorphic form. It is conjectured to be fully faithful (Theorem for the group \(G=\mathrm{GL}_{n}\)).
The functor \(\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\) is a gluing of functors \[ \mathbb{L}_{G,P}^{\mathrm{Whit}}:\mathrm{F}_{\check{P}}\mathrm{-mod}(\mathrm{QCoh}(\mathrm{LocSys}_{\check{P}}))\to \mathrm{Whit}(G,P). \] Definition of the functors \(\mathbb{L}_{G,P}^{\mathrm{Whit}}\) can be reduced to that of \(\mathbb{L}_{G,G}^{\mathrm{Whit}}\), the latter is defined by spreading over \(\mathrm{Ran}(X)\). More precisely, there exists a fully faithful functor \[ \mathrm{QCoh}(\mathrm{LocSys}_{\check{G}})\to \mathrm{Rep}(\check{G})_{\mathrm{Ran}(X)}, \] and the category \(\mathrm{Whit}(G,G)\) is canonically equivalent to a variant of \(\mathrm{Rep}(\check{G})_{\mathrm{Ran}(X)}\) (Basically this is the Casselman-Shalika formula), and \(\mathbb{L}_{G,G}^{\mathrm{Whit}}\) is composition of the two functors. By construction, the functors are compatible with the action of the monoidal category \(\mathrm{Rep}(\check{G})_{\mathrm{Ran}(X)}\), i.e. the geometric Satake equivalence. Finally, it is stated as a “quasi-Theorem” that the functor \(\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\) is fully faithful.
Now that the functors \(\mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}})\), \(\mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\), \(\mathrm{Coeff}_{G,G}^{\mathrm{ext}}\) are fully faithful, it is enough to compare the essential images of \[ \mathbb{L}_{G,G}^{\mathrm{Whit}^{\mathrm{ext}}}\circ \mathrm{Glue}(\mathrm{CT}_{\mathrm{spec}}^{\mathrm{enh}})\quad \text{ and }\mathrm{Coeff}_{G,G}^{\mathrm{ext}} \] in order to establish the categorical Langlands correspondence. By induction, it boils down to comparing images of irreducible local systems on the spectral side and that of cuspidal forms on the geometric side. This is accomplished by generalizing Beilinson-Drinfeld’s construction of Hecke eigensheaves on \(\mathrm{Bun}_{G}\) associated to special \(\check{G}\)-local systems on \(X\) called opers. That is the heart of the work and it is the last piece in the commutative diagram above.
For the entire collection see [Zbl 1319.14004].
Reviewer: Zongbin Chen (Beijing)
MSC:
| 14D24 | Geometric Langlands program (algebro-geometric aspects) |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14H60 | Vector bundles on curves and their moduli |
