Statement of local geometric Langlands

Question

A precise statement of the global geometric Langlands conjecture is well-known.

However, I am unable to find a statement of the local Langlands conjecture. Does anyone have a modern statement or a reference to such? In particular, the statement should account for temperedness issues.

Answer 1

There is no precise formulation of local geometric Langlands in the literature, but the rough outline is known and goes back to the papers of Frenkel-Gaitsgory starting with https://arxiv.org/abs/math/0508382 and further refined by Gaitsgory and collaborators, see https://arxiv.org/abs/1601.05279. The state of the art at the time (mostly about the quantum version, which is actually better developed) is laid out in the lectures at the Paris winter school https://sites.google.com/site/winterlanglands2018/home.

[BTW you write "geometric Langlands is well known" and I assume you mean the de Rham version, but there are also the Betti version and the restricted version, which is the intersection of the other two but has the advantage of making sense over any field -- in particular is the only one currently relevant to the Langlands program over function fields. Really one would like to know what the restricted version of local GL is, I believe Sam Raskin and others have thought about it, but nothing is written AFAIK.]

In any case the rough idea is you categorify the local Langlands correspondence, which (these days) very roughly relates smooth representations of a group over a local field with (ind-)coherent sheaves on the stack of local Langlands parameters (well the spectral side as I state it is way bigger but there's a way to enlarge the automorphic side to correct that, which is unnecessary once we categorify). So a categorical analog is the following:

$\bullet$ There's an equivalence of ($\infty$,2)-categories between [de Rham] categorical representations of the loop group $LG$ of a reductive group $G$ and sheaves of categories over the stack $Conn_{G^\vee}(D^\star)$ of local Langlands parameters.

Automorphic side: a [de Rham] categorical representation of a group means an algebraic action of the group for which the Lie algebra action has been trivialized (this is a good categorified analog of a smooth representation, where matrix coefficients are locally constant), or equivalently module category for the monoidal category of D-modules on the group under convolution. Examples include categories of D-modules on G-spaces and the category of reps of the Lie algebra. Here the group in question is the infinite-dimensional LG so one has to deal with a lot of infinite-dimensional complications, but these are I believe largely understood now (see e.g. papers of Beraldo and Raskin about the theory of Whittaker models in this setting).

Spectral side: the stack $Conn_{G^\vee}$ of local Langlands parameters means flat $G^\vee$ connections on the punctured disc. By a "sheaf of categories" one might initially mean module categories for $QC(Conn_{G^\vee})$, or quasicoherent sheaves of categories on the stack (the two are the same thanks to a surprising result of Raskin https://arxiv.org/abs/1511.01378) [Edit: according to the "History" section of that paper, the lack of such an identification is a reason the local geometric Langlands conjecture wasn't formulated earlier.]

But just as in global geometric Langlands quasicoherent sheaves are not the final answer, and that's where we run into the cutting edge (the reason why the conjecture hasn't appeared in print yet). The notion of quasicoherent sheaf of categories is "too smooth" - for example when calculating its categorical trace you end up with QC not ind-coherent sheaves on the singular space given by its inertia stack. So we're running into the categorified analog of the relation distinction between perfect complex and coherent sheaf (or cohomology vs homology or function vs distribution or...) This problem is also familiar in topological field theory, in trying to define boundary conditions for Rozansky-Witten theory of a cotangent bundle.

So to properly formulate the spectral side of the conjecture you need a notion of "ind-coherent sheaf of categories" and a microlocal understanding thereof (so you can define "nilpotent singular support" in this setting). There are lectures of Arinkin and Gaitsgory presenting such a notion, but more recently (2021) the Berkeley PhD thesis work of German Stefanich gives a thorough development of this notion of "2IndCoh".

Anyway to conclude I believe the correct statement of local geometric Langlands is an equivalence of 2-categories

$$D(LG)-mod \simeq 2IndCoh_{\mathcal N}(Conn_{G^\vee}(D^\star))$$

(plus of course a load of compatibilities, with Hecke operators, with Whittaker normalization, and more.)