Comparing local Langlands correspondences

At least six people have independently asked me some variant of the question:

What are the prospects for showing that the Fargues-Scholze construction of L-parameters is compatible with other constructions of the local Langlands correspondence?

In this post I’ll briefly lay out the answer as I see it.

For reductive groups G over finite extensions F/\mathbf{Q}_p, the situation is complicated, since the status of LLC is complicated.

  1. \mathrm{GL}_n and D_{1/n}^{\times}. Compatibility for these groups is known and already proved in Fargues-Scholze, and follows from the realization of local Langlands and local Jacquet-Langlands in the cohomology of the Lubin-Tate tower.
  2. Any inner form of \mathrm{GL}_n. Compatibility here is Theorem 1.0.3 in H.-Kaletha-Weinstein.
  3. \mathrm{SL}_n and inner forms. Compatibility should follow from the previous two points, but I guess it’s not completely trivial. Someone should write it down.
  4. \mathrm{GSp}_4 and \mathrm{Sp}_4, and their unique inner forms. Compatibility for these groups has been proved by my student Linus Hamann. His preprint should be available very soon, and I’ll write a detailed blog post about it at that time. The arguments here rely on a number of special features of the group \mathrm{GSp}_4.
  5. Split \mathrm{SO}_{2n+1} and closely related groups. Partial results here are definitely possible by extending Hamann’s arguments, but it’s not clear to me whether complete results can be expected. I’ll say more about this when I write about Hamann’s paper.
  6. Unitary groups. Partial results should be possible by combining some aspects of Hamann’s methods with recent works of Nguyen and Bertoloni-Meli–Nguyen.
  7. \mathrm{GSp}_{2n} and \mathrm{Sp}_{2n} and their inner forms, n>2. This seems out of reach.
  8. Even special orthogonal groups. I’m frankly confused about what’s going on here. Is there even an unambiguous LLC? In any case, this also seems hard.
  9. Exceptional groups. There’s no “other” LLC here. Go home. (OK, for G_2 there’s a very cool recent paper of Harris-Khare-Thorne.)
  10. General groups splitting over a tame extension, p not too small. Here Kaletha has given a general construction which attaches a supercuspidal L-packet to any supercuspidal L-parameter. Compatibility of this construction with Fargues-Scholze might be approachable by purely local methods, but it seems to require substantial new ideas. An extremely weak partial result – constancy on Kaletha’s packets of the FS map from reps to L-parameters – is probably within reach, using the main results in H.-Kaletha-Weinstein.

The key point in many of the above situations is the following. Let’s say a group G is accessible if it admits a geometric conjugacy class of minuscule cocharacters \mu such that
1. The pair (G,\mu) is totally Hodge-Newton reducible in the sense of Chen-Fargues-Shen.
2. Any L-parameter \varphi: W_F \to \phantom{}^L G can be recovered up to isomorphism from the composition r_{\mu} \circ \varphi. (In practice one asks for slightly weaker versions of this.)
3. The local Shimura varieties attached to the local Shimura datum (G,\mu,b) (with b \in B(G,\mu) the unique basic element) uniformize the basic locus in a global Shimura variety of abelian type.

For groups satisfying this condition, there is hope. Very roughly, condition 2. implies that the FS construction is incarnated in the cohomology of a single local Shimura variety, whose cohomology can also be tightly related to the cohomology of a global Shimura variety using conditions 1. and 3. One then needs to know enough about the cohomology of these global Shimura varieties, namely that it realizes the “other” LLC you care about. Of course, this short outline veils substantial technical difficulties.

It turns out that \mathrm{GL}_{n}, \mathrm{GU}_n, \mathrm{GSp}_4, and \mathrm{SO}_{2n+1} are all accessible, and this accounts for the definitive results in scenarios 1.-4. above and my optimism in scenarios 5.-6. On the other hand, \mathrm{GSp}_{2n} is not accessible for n>2, and neither is \mathrm{SO}_{2n} for n>3, and no exceptional groups are accessible. Hence my pessimism in scenarios 7.-9.

For reductive groups over finite extensions F/\mathbf{F}_{p}((t)), the situation is completely different. Here Genestier-Lafforgue have constructed a local Langlands correspondence for all groups, uniquely characterized by its compatibility with V. Lafforgue’s construction of global Langlands parameters. It is an extremely attractive problem to compare the Genestier-Lafforgue LLC with the Fargues-Scholze LLC. This should absolutely be within reach! After all, both constructions are realized in the cohomology of moduli spaces of shtukas, so the only “real” task should be to physically relate the moduli spaces of shtukas used by GL with those used by FS. This is probably not trivial: the spaces used by FS are local and totally canonical, while those used by GL seem to depend on a globalization and some auxiliary choices in a messy way. Nevertheless, I’d be surprised if this comparison is still an open problem two years from now.

One thought on “Comparing local Langlands correspondences”

  1. For inner forms of special linear groups, just compare FS, IX.6.1 and the other construction of L-packets (Hiraga-Saito for p-adic fields and Aubert-Baum-Plymen-Solleveld for the equal characteristic case) .

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