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Geometrization of trace formulas. (English) Zbl 1319.22015

The aim of this paper is to develop geometric methods for applying trace formulas to the Langlands program for global function fields. The idea of functoriality principle, roughly speaking, asserts that for each homomorphism \({}^LH\to {}^LG\) of the Langlands dual groups of two reductive algebraic groups \(H\), \(G\) over a global field \(F\) (assuming \(G\) quasi-split), there exists a transfer of automorphic representations from \(H(\mathbb{A})\) to \(G(\mathbb{A})\), where \(\mathbb{A}\) is the adele ring of \(F\).
In [E. Frenkel et al., Ann. Sci. Math. Qué. 34, No. 2, 199–243 (2010; Zbl 1267.11113)] and earlier works of Langlands, an approach of using trace formulas to prove functoriality was proposed as follows. One should use the trace formula to decompose the traces of a family of integral operators as a sum over those groups \(H\) that by functoriality give automorphic representations of \(G\). By comparing orbital integrals on \(G\) and \(H\) for suitable test functions, one should be able to get identities on the spectral side of the trace formula, then finally prove functoriality.
The present paper makes first steps towards analyzing the orbital integrals for the function field case, that is, the function field \(F\) of an algebraic curve \(X\) over a finite field \(\mathbb{F}_q\). The paper consists of two parts.
The first part is concerned with the geometrization of the orbital integral side of the trace formula. The word “geometrization” refers to the Grothendieck faisceaux-fonctions dictionary. It is shown that the averaging operator \(K_{d,\rho}\) considered in [loc. cit.] can be obtained from a perverse sheaf \(\mathcal{K}_{d,\rho}\) on an algebraic stack over Bun\(_G\), the moduli stack of \(G\)-bundles on \(X\). Hence in this case the orbital integral side is expressed as the cohomology of a complex of sheaves on the moduli stack \(\mathcal{M}_G\) of \(G\)-Higgs bundles on \(X\), which is closely related to but different from the Hitchin moduli stack used by one of the authors to prove the Fundamental Lemma in [B. C. Ngô, “Le lemme fondamental pour les algèbres de Lie” Publications IHES 111, 1–169 (2010) arXiv:0801.0446]. Based on this interpretation the authors formulate a precise conjecture relating the cohomologies of two such moduli stacks for \(G=\text{SL}_2\) and \(H\) a one-dimensional twisted torus.
The second part of the paper is concerned with the geometrization of the spectral side of the trace formula. The authors’ main observation is that the answer could be given by the categorical geometric Langlands correspondence, which is conjectured by Beilinson and Drinfeld. This conjecture, roughly speaking, states an equivalence between derived categories of coherent sheaves on the moduli stack Loc\(_{{}^LG}\) of \({}^LG\)-local systems on a complex curve \(X\) and \(\mathcal{D}\)-modules on the moduli stack Bun\(_G\) mentioned above. Here every geometric object is over \(\mathbb{C}\). The authors propose a conjectural geometrization in this framework, which follows from the conjectural categorical geometric Langlands correspondence. Finally the relative geometric trace formulas are discussed as well, which are necessary to this program.
This paper is written clearly and organized very well. People working on the geometric Langlands program should benefit a lot from it.

MSC:

22E57 Geometric Langlands program: representation-theoretic aspects
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
14D24 Geometric Langlands program (algebro-geometric aspects)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 1267.11113
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References:

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