G-shtuka - Knowlpedia

Let $X/\mathbb F_q$ be a smooth projective curve, let $G$ be a connected reductive group, and let $I$ be a finite set. A $G$ -shtuka with legs $(x_i)_{i\in I}$ over $S$ consists of a $G$ -bundle $\mathcal G$ on $X\times S$ and an isomorphism

\varphi: \mathcal G\big|_{(X\times S)\setminus\bigcup_{i\in I}\Gamma_{x_i}} \xrightarrow{\ \sim\ } {}^\tau\mathcal G \big|_{(X\times S)\setminus\bigcup_{i\in I}\Gamma_{x_i}}, \qquad {}^\tau\mathcal G=({\rm id}_X\times{\rm Frob}_S)^*\mathcal G,

whose relative position at each leg is bounded by prescribed dominant coweight data. Here ${\rm Frob}_S$ is the absolute Frobenius endomorphism of $S$ .

Level and boundedness data

A level structure along a finite subscheme $N\subset X$ , disjoint from the legs, trivializes the bundle compatibly with $\varphi$ . If a representation $W$ of the dual group $\widehat G^I$ is used instead of a tuple of coweights, geometric Satake supplies the corresponding bound and intersection complex.

Because the stack is generally not of finite type, one also uses Harder–Narasimhan truncations and a quotient by a lattice in the adelic center.

Iterated modifications

For an ordered partition $I=I_1\sqcup\cdots\sqcup I_k$ , the Frobenius isomorphism can be factored as a chain

\mathcal G_0\dashrightarrow\mathcal G_1 \dashrightarrow\cdots\dashrightarrow \mathcal G_k={}^\tau\mathcal G_0,

where the $j$ -th modification occurs at the legs in $I_j$ . This form defines partial Frobenius operations.

Cohomological structure

The intersection cohomology of stacks of bounded $G$ -shtukas carries commuting Hecke and partial-Frobenius actions. Fusion or coalescence identifies the cohomology when legs merge. These operations provide the creation, Galois-action, and annihilation maps used in excursion operators.

References

Yakov Varshavsky, “Moduli spaces of principal $F$ -bundles,” Selecta Mathematica 10 (2004), 131–166. DOI.
Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” Definitions 0.2 and 0.6. arXiv.