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I have kind of a soft question. I've studied the basics of L. Lafforgue's proof of function field Langlands for GLn, and its use of the moduli of shtukas with two legs, and the cocharacters $[1,0,\ldots, 0]$ and $[0,0,\ldots,-1]$. I vaguely know that V. Lafforgue's approach to the general reductive group involves higher legs; does it also involve other cocharacters? For instance, when would moduli of shtukas, even just with one leg, with cocharacter $[2,0]$, say, be useful? Or are they easily described in terms of the moduli of shtukas with minuscule cocharacter?

This question is also partly motivated by the fact that locally, shtukas with one leg of cocharacter $[1,1,\ldots, 1, 0,0, \ldots, 0]$ are closely analogous to moduli of $p$-divisible groups via the Dieudonne functor. I'd like a similar intuition/idea for other dominant cocharacters.

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Yes, in general you need to consider all cocharacters.

$\mathrm{GL}_n$ has the special property that the dominant coweights are all sums of minuscule cocharacters, i.e. ones of the form $[1,\cdots,1,0,\cdots,0]$ or $[0, \ldots, 0, -1,\ldots, -1]$. (For a general group, a dominant cocharacter is minuscule if its pairing with any positive root is at most $1$).

In terms of representations, this says that all irreducible algebraic representations can be built inside tensor products of exterior powers of the standard representation and its dual. A manifestation of this fact is the linear algebra theorem that a semisimple element of $\mathrm{GL}_n(\overline{\mathbf Q_\ell})$ is determined up to conjugation by the coefficients of its characteristic polynomial. It's true more generally for a reductive group $G$ over an algebraically closed field $k$ that a conjugacy class of semisimple elements $g \in G(k)$ is determined by the trace of $g$ acting on each irreducible representation. So if I want to construct a Galois representation into $G(\overline{\mathbf Q_\ell})$ (up to semisimplification), I need to know the traces on all irreps.

On the automorphic side, the image of the minuscule representations (representations with minuscule highest weight) under the Satake isomorphism $\mathrm {Rep}(\mathrm{GL}_n) \rightarrow \mathscr{H}$ generates the spherical Hecke algebra $\mathscr{H}$, so all Hecke eigenvalues of an eigenform are determined by these special elements.

Putting this together, to verify the (automorphic to Galois direction of the) Langlands conjecture for $\mathrm{GL}_n$, given a Hecke eigenform $f$, it suffices to construct a Galois representation such that the coefficients of the characteristic polynomial of Frobenius elements at each point are the eigenvalues of $f$ for the elements corresponding to minuscule characters.

For general groups $G$, the minuscule characters do not generate the dominant weights. Thus, in order to make the Langlands correspondence unique up to semisimplification, we must match the eigenvalues of $f$ under the Hecke operators corresponding to any representation $V$ of $\widehat{G}$ with traces of Frobenius elements acting on $V$ through $\widehat{G}$.

Loosely speaking, V. Lafforgue's approach to this is to construct a Galois representation for each $V$ by studying the cohomology of shtukas with legs bounded by the corresponding cocharacter of $G$. (There's the further complication that there are no shtukas with one leg and thus we have to consider tuples of representations/cocharacters and verify a ton of tricky compatibilities coming from adding or subtracting legs - this is one of the principal difficulties of his paper).-


On shtukas for $\mathrm{GL}_n$ for more general cocharacters:

We like minuscule cocharacters because the corresponding modifications of vector bundles are parametrized by the usual Grassmannian, which is a smooth projective variety. This means we don't need any perverse sheaves or intersection cohomology!

The fact that any dominant cocharacter of $\mathrm{GL}_n$ is a sum of minuscule ones lets us (locally) write an arbitrary modification of vector bundles as a sequence of modifications by minuscule cocharacters. If $\mu = \sum n_i \mu_i$ with $\mu_i$ minuscule, there is a resolution of the Schubert cell for $\mu$ in the affine Grassmannian by the "iterated affine Grassmannian" which parametrizes sequences of $\mu_i$ modifications. This space can be constructed as an iterated bundle of usual Grassmannians: in particular, it is smooth.

We can pull this back to an analogous resolution of the stack of shtukas with a leg bounded by $\mu$. One can prove that this map doesn't change the IC cohomology, so we don't get new Galois representations (I think this is right but I'm not 100% sure).


Aside: mixed characteristic

The fact that minuscule cocharacters generate everything for $\mathrm{GL}_n$ is essential to the proof of the local Langlands correspondence in that case! For similar reasons to above, this means that we can construct the required Galois representations by studying the cohomology of Lubin-Tate space.

$p$-divisible groups are "local shtukas with minuscule cocharacter" or "$p$-adic Hodge structures of weight $1$". The minuscule property is exactly what lets us parametrize these by an honest rigid-analytic space/formal scheme.

Recently, Scholze and Fargues have proposed an interpretation of more general shtukas in mixed characteristic, in the hopes that they could carry over Lafforgue's ideas to the $p$-adic setting. This uses Scholze's theory of diamonds, and involves a lot of very exotic geometry. (Meanwhile, local Langlands has been proved for many $p$-adic groups using different methods. The Scholze-Fargues program gives a very nice conceptual framework for the problem, but there are many fundamental foundational problems to be solved before it has any hope of working).

Also, I should mention that in the number field setting, Shimura varieties are some vague analogue of moduli of shtukas with minuscule legs - a minuscule cocharacter is part of the data defining the Shimura variety. This is a fundamental obstruction to "cohomologically" constructing a Langlands correspondence for general groups over number fields (never mind the fact that there's no hope of handling e.g. Maass forms with these sorts of algebro-geometric methods).

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  • $\begingroup$ thanks very much for the detailed response! one more naive question: could you elaborate why there are "no shtukas with one leg"? in what setting is this true? $\endgroup$ – xir May 7 at 14:27
  • $\begingroup$ maybe you mean for GLn? i can see that, at least (reduce to the case of line bundles via determinants and then it's just basic facts about positivity), but it seems like there are shtukas with one leg for, say, PGLn. take for instance simply the natural ideal sheaf inclusion O(-p)^n -> O^n for some rational point p; this is a shtuka for PGLn, no? $\endgroup$ – xir May 8 at 15:59
  • $\begingroup$ For $\mathrm{GL}_n$, this should come from the fact that pulling back a vector bundle $\mathscr{E}$ on $X_S$ by $F = 1 \times \mathrm{Frob}_S$ doesn't change the degree, so an everywhere-defined morphism $F^*\mathscr{E} \rightarrow \mathscr{E}$ that only vanishes at one point must be an isomorphism. For general groups, I think you can just use the above argument via the Tannakian description of torsors. $\endgroup$ – dorebell May 8 at 18:48
  • $\begingroup$ as i said, i agree for GLn. i'm not sure i understand the second part, though i think i get why my example above doesn't work; do you have a reference? $\endgroup$ – xir May 8 at 22:16
  • $\begingroup$ I can't find a reference right now but the argument is: The category of $G$-torsors is the same thing as the category of faithful exact tensor functors from the category of representations of $G$ into vector bundles. So a $G$-shtuka is the same thing as a collection of $GL(V)$-shtukas for each representation of $V$ which are compatible with tensor products etc. Thus if I had a $G$-shtuka with one leg, all of the corresponding $\GL(V)$ shtukas would be trivial, and thus the original $G$-shtuka must be as well. $\endgroup$ – dorebell May 8 at 22:38

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