I’m about to head to Paris on vacation, quite possibly there will be less blogging for the next couple of weeks. Here are a few Langlands-related items:

- The big Langlands-related event of the summer will be the July IHES Summer School on the Langlands Program. Laurent Fargues has posted notes for the summer school (written with Peter Scholze). They comment at the end that “it still remains to find an archimedean analog” of their work relating local Langlands and geometric Langlands on the Fargues-Fontaine curve.
- At Quanta, Alex Kontorovich has a popular introduction to the Langlands program.
- There’s a recent review by Michael Harris of expected properties of the local Langlands correspondence.
- There’s a program going on about automorphic forms at the Collège de France, featuring lectures by Ngô. Maybe I’ll stop by to hear one next week.
- I was sad to hear about the recent death of Joël Bellaïche, who had been a postdoc here at Columbia. David Hansen writes here about him and his work.

So I’m by no means an expert in p-adic geometry, but my understanding is that Scholze’s perfectoid spaces are very useful gadgets for relating problems in “mixed characteristic” (e.g., over the p-adic numbers Q_p) to problems in characteristic p. For perfectoid spaces, unlike in schemes, we have this “tilting functor” relating perfectoids over mixed characteristic fields to ones in positive characteristic. It’s often preferable to work in positive characteristic, as we have access to the Frobenius endomorphism in this case.

So very very broadly speaking—and perhaps somebody more knowledgeable could correct me on this—this is the reason perfectoid spaces and generalizations (diamonds, v-stacks etc.) are useful for studying local Langlands over non-Archimedean local fields—which are mixed characteristic. This is also the reason none of this machinery would work in the Archimedean case—there’s no tilting functor here.

@Topologist Guy: Yes, that’s a very good summary. But note that ~10 years ago very similar arguments were made that the “shtuka” techniques of the function field case can’t possibly be applied to p-adic fields, as there’s no Frobenius there (and you can’t take nontrivial self-products of Spec Q_p etc pp)… but yes, currently it’s completely unclear what the right structures over the real numbers would be. (Other than that it will involve some form of the twistor-P1 that’s strangely also featuring prominently in other contexts on this blog…)

Topologist Guy – definitely perfectoid technology is not relevant in the archimedean case, but there are many recent technological breakthroughs that are important in the Fargues-Scholze work that should be relevant there, in particular condensed mathematics.

Roughly speaking, the archimedean local Langlands conjecture in the spirit of Fargues-Scholze would relate derived categories of [_] sheaves on the stack of G-bundles on the twistor line and coherent sheaves on a stack of archimedean local Langlands parameters. Here G is a real reductive group, and the semistable G-bundles have as stabilizers inner forms of G. So sheaves on this stack restrict to give representations of these real groups, and this provides the mechanism to relate real groups and Langlands parameters.

The question is what kinds of sheaves? since we want to realize [admissible] infinite dimensional continuous representations of G, eg unitary reps, we need some notion of sheaves of topological vector spaces, and here one needs condensed (and in particular presumably liquid) technology to make sense of this and to formulate the matching notion of Langlands parameters.

In fact an archimedean local Langlands conjecture in much the same spirit appears in work of David Nadler and mine from 2007 (and will be reviewed in our IHES summer school contribution with Harrison Chen and David Helm, to be posted soon). The main differences as I understand them from the desired “real Fargues-Scholze” are that we work with parabolic bundles on the twistor line rather than just plain bundles (i.e. we are “tamely ramified” rather than “unramified”), old-fashioned constructible sheaves rather than a condensed version, and a fairly-close-to-traditional version of the space of Langlands parameter – realized as parabolic local systems on the twistor line.

On the semistable locus, rather than seeing pt/G (and inner forms) we see {complex flag manifold}/G, and the relation to admissible representations of G is not simply “take stalks” but rather “take maps to holomorphic functions” — this is the Kashiwara-Schmid cousin of Beilinson-Bernstein’s realization of Harish-Chandra modules for real groups.

So this is less direct than in the eventual expected story, in that we replace representations by their realization on flag manifolds (eg finite dimensional reps get replaced by their realization given by the Borel-Weil theorem). Also I can’t imagine that the picture with parabolic bundles is what you want eventually in a more global story, but it’s what we can do with old technology.