Moduli spaces of principal \(F\)-bundles.

*(English)*Zbl 1070.14026Let \(X/\mathbb F_{q}\) 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].

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].

Reviewer: Paul Gunnells (Amherst)