Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?).

From what I have been able to gather from several lecture videos by Gaitsgory (such as Gaitsgory - Singular support of coherent sheaves), the categorical-geometric Langlands Correspondence $G$ is supposed to be an equivalence of categories: $$\mathsf{DMod}\left(\mathsf{Bun}_G(X)\right) \cong \mathsf{QCoh}(\mathsf{LocSys}^{{}^LG}(X))$$ wherein $\mathsf{DMod}\left(\mathsf{Bun}_G(X)\right)$ is the ($\infty$-)category of D-modules on the moduli stack $\mathsf{Bun}_G(X)$ of $G$-bundles on $X$, and $\mathsf{QCoh}(\mathsf{LocSys}^{{}^LG})$ is the ($\infty$-)category of quasi-coherent modules on the stack of ${}^L G$-equivariant local systems on $X$, where ${}^L G$ is the Langlands dual of $G$.

My questions are then:

  1. Is this the correct statement? Specifically, I would like to understand if we need to replace $\mathsf{QCoh}$ with $\mathsf{IndCoh}$ (see part II of A study in derived algebraic geometry, Volume I: Correspondences and duality by Gaitsgory and Rozenblyum for an introduction to ind-coherent sheaves), and why (or why not)?
  2. What is a standard reference that I can consult? Most references that I have been able to find (by Gaitsgory, Laumon, etc.) are about the Langlands Correspondence for $\operatorname{GL}_2$, $\operatorname{GL}_n$, or more general kinds of reductive groups, and I don't think I have what it takes yet to extract the special case of tori from these papers.

1 Answer 1


In case $G=T$ is a torus and $G^\vee=T^\vee$ is the dual torus, the geometric Langlands conjecture — or “categorical geometric class field theory” (for a smooth projective curve $C$ over $\mathbb C$) becomes a theorem, and has the form you stated: a derived equivalence $$\mathsf{DMod}(\mathsf{Bun}_T)\simeq \mathsf{QCoh}(\mathsf{LocSys}_{T^\vee}).$$ It is due (in a form at least very close to this) to Laumon ("Transformation de Fourier Generalisée") and Rothstein ("Sheaves with Connection on Abelian Varieties"), who independently extended the Fourier–Mukai transform from coherent sheaves on an abelian variety to D-modules.

Indeed $\mathsf{Bun}_T$ is the product of a lattice, an abelian variety and $\mathrm{pt}/T$. $\mathsf{LocSys}$ is the product of the rigidified scheme of flat connections, a factor $\mathrm{pt}/T^\vee$ from automorphisms and a constant derived factor $\mathfrak t^*[-1]$. The lattice matches with the stacky factor $\mathrm{pt}/T^\vee$, while the $\mathrm{pt}/T$-factor matches with the constant derived structure of $\mathsf{LocSys}$. The "interesting" part is the abelian variety and the dual rigidified space of flat connections, which is the Laumon–Rothstein theorem.

The reason why you see $\mathsf{QCoh}$ rather than $\mathsf{IndCoh}$ is the manner in which you treat the stacky factor on $\mathsf{Bun}$, ie by what you mean by D-modules on a stack — you can change ("renormalize") the definition to get all of $\mathsf{IndCoh}$. (You can see it from Koszul duality, lining up usual $D$-modules on $\mathrm{pt}/T$ with quasicoherent rather than ind-coherent modules over the Koszul dual exterior algebra.) For general $G$ you can match up D-modules with ind-coherent sheaves with nilpotent singular support (which in the abelian case recovers $\mathsf{QCoh}$), as developed in the (first) paper Singular support of coherent sheaves and the geometric Langlands conjecture of Arinkin–Gaitsgory. Or you can "renormalize" D-modules to get something that conjecturally recovers all of $\mathsf{IndCoh}$.

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    $\begingroup$ I tried to edit in links to papers, but I couldn't find one by Laumon called "Transformation de Fourier généralisée". Could it be "Transformation de Fourier homogène"? ALso, I think there is a word 'from' missing in "comes how you treat" in the last paragraph. $\endgroup$
    – LSpice
    Commented Jun 6, 2021 at 21:27
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    $\begingroup$ Thanks @LSpice, added the (recent) arxiv link (for some reason this was never published) $\endgroup$ Commented Jun 6, 2021 at 21:59
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    $\begingroup$ @DatMinhHa For 1: concretely it means the relevant category of sheaves is decomposed as a tensor product, so you can approach the duality one factor at a time. More conceptually, we're just doing a fancy form of Pontrjagin/Carier duality and the abelian group in question is a product of three abelian groups, so we can do the duality factor-wise. $\endgroup$ Commented Jun 7, 2021 at 18:48
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    $\begingroup$ Very roughly, the question is whether we define the big category of D-modules on a quotient stack by first looking at equivariant compact objects (coherent D-modules) and then generating by colimits, or just looking at equivariant big objects - this latter is the standard definition, but has the feature that some coherent D-modules on pt/G (like the structure sheaf) are not compact. See the paper of Drinfeld-Gaitsgory "Some finiteness conditions.." $\endgroup$ Commented Jun 7, 2021 at 18:51
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    $\begingroup$ This matches very precisely what we mean by the derived category of modules for an exterior algebra (the relevant one being the homology of G, or exterior algebra arising from derived structure on dual side) -- there's a straightforward definition, where perfect modules are compact, and then we could instead declare all finite dim modules (eg the augmentation) to be compact. The latter (here and above) is what's called (by Gaitsgory et al) renormalization and what leads to IndCoh. $\endgroup$ Commented Jun 7, 2021 at 18:53

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