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Moduli spaces of principal $$F$$-bundles. (English) Zbl 1070.14026
Let $$X/\mathbb F_{q}$$ be a smooth, geometrically connected, projective curve with field of rational functions $$F$$. Let $$\mathbb A$$ be the adele ring of $$F$$, $$\Gamma_{F}$$ the absolute Galois group of $$\bar F/ F$$, and $$l$$ a fixed prime not dividing $$q$$. Let $$G/\mathbb F_{q}$$ be a split reductive group with dual group $$\hat G/\bar \mathbb Q_{l}$$. Then the Langlands correspondence predicts that to every pair $$(\pi ,\omega)$$, where $$\pi$$ is a tempered cuspidal representation with finite order central character and $$\omega$$ is an algebraic representation of $$\hat G$$, one can attach an $$l$$-adic representation $$\rho_{\pi ,\omega}$$ of $$\Gamma_{F}$$ with $$L$$-function equal to that of $$(\pi ,\omega)$$. The correspondence is known to be true when $$G=\text{GL}_{n}$$, thanks to work of L. Lafforgue [Invent. Math. 147, 1–241 (2002; Zbl 1038.11075)].
This setup can be generalized as follows. For $$n\geq 1$$, let $$F^{(n)}$$ be the function field of the product $$X^{n}$$, and let $${\bar \omega} = (\omega_{1},\dots ,\omega_{n})$$ be an $$n$$-tuple of algebraic representations of $$\hat G$$. Then the Langlands correspondence also predicts the existence of a certain $$l$$-adic representation $$\rho_{\pi ,{\bar \omega}}$$ of $$\Gamma_{F^{(n)}}$$. Moreover, one hopes that $$\rho_{\pi ,{\bar \omega}}$$ can be motivically realized, which means that there should exist an algebraic variety $$\mathcal X_{\pi ,{\bar \omega}}/F^{(n)}$$ such that $$\rho_{\pi ,{\bar \omega}}$$ is a subquotient of its cohomology.
The goal of the paper under review is to present a candidate $$\mathcal X_{{\bar \omega}}$$ for a space whose cohomology realizes all of the representations $$\rho_{\pi ,{\bar \omega}}$$, in the sense that all the relevant representations are hoped to appear (with multiplicities) in a certain subquotient of the intersection cohomology of $$\mathcal X_{{\bar \omega}}$$ [Conjecture 2.35]. The construction of $$X_{{\bar \omega}}$$ is too complicated to describe here; we merely indicate that it is a closed substack of a certain “twisted” global affine Grassmannian over $$X^{n}$$. To support his conjecture, the author verifies it in the cases considered by Drinfeld, and (in a certain sense) in the cases considered by Lafforgue [Theorem 2.37].

##### MSC:
 14G35 Modular and Shimura varieties 14H60 Vector bundles on curves and their moduli 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11G45 Geometric class field theory 11R39 Langlands-Weil conjectures, nonabelian class field theory
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