Geometric Eisenstein series: twisted setting. (English) Zbl 1387.11082

The author generalizes the quantum geometric Langlands correspondence (for roots of unity) from the case of split torus. It geometrizes a conjectural extension of the Langlands program to metaplectic groups proposed by Weissman. Let \(k\) be an algebraically closed field, \(G\) a simple connected group, \(X\) a smooth projective curve over \(k\), and \(\mathrm{Bun}_G\) the stack of \(G\)-torsors on \(X\). For some \(n\geq1\) let \(N=2h^\vee n\), where \({h}^\vee\) is the dual Coxeter number for \(G\), and introduce a \(\mu_N\)-gerb \(\widetilde{\mathrm{Bun}}_G\to Bun_G\) that comes from the canonical Brylinski-Deligne extension of \(G\) by \(K_2\). For the parabolic subgroup \(P\subset G\) and its Levi factor \(M\) one similarly introduces \(\widetilde{\mathrm{Bun}}_P\) and \(\widetilde{\mathrm{Bun}}_M\). For an injective character \(\zeta:\mu_N^{(k)}\to\overline{\mathbb{Q}}^*_l\) the author considers the derived category of étale \(\mathbb{Q}_l\) sheaves on which \(\mu_N^{(k)}\) acts by \(\zeta\). The Eisenstein series is defined as a functor \(\mathrm{Eis}:D_\zeta(\widetilde{\mathrm{Bun}}_G)\to\widetilde{\mathrm{Bun}}_M\) using a twisted version of \(\mathrm{Bun}_P\) intersection homology sheaf \(IC_\zeta\), and the author proves that Eis commutes with the Hecke functors with respect to the dual embedding, and that the formation of Eis is transitive for the diagram \(T\subset M\subset G\), where \(T\) is a maximal torus.
For \(G=\mathrm{SL}_2\) a partial description of the Fourier coefficients of Eis in terms of a known sheaf is given, which relates them to quantum groups. As an application, the author derives a formula for the first Whitakker coefficient of the metaplectic extension of \(\mathrm{SL}_2\), which turns out to be an \(l\)-adic analog of the space of conformal blocks of the Wess-Zumino-Witten model, and can be seen as a generalization of the central value of an abelian \(L\) function. In the case of \(X=\mathbb{P}^1\) new automorphic sheaves (“theta-sheaves”) are constructed and their Hecke property is proved.


11R39 Langlands-Weil conjectures, nonabelian class field theory
11G45 Geometric class field theory
14H60 Vector bundles on curves and their moduli
Full Text: DOI arXiv


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