In this paper we first review the setting for the geometric Langlands functoriality and establish a result for the `backward' functoriality functor. We illustrate this by known examples of the geometric theta-lifting. We then apply the above result to obtain new Hecke eigen-sheaves. The most important application is a construction of the automorphic sheaf for G=GSp_4 attached to a G^L-local system on a curve X such that its standard representation is an irreducible local system of rank 4 on X.
We prove the classification of discrete automorphic representations of GSp$_4$ explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp$_4$ and Sp$_4$ .
In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of Levi subgroups of $GSpin$, by constructing Langlands parameters for general generic irreducible admissible representations of $GSpin $ from the one for generic irreducible supercuspidal representations of its Levi subgroups.
We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ being the base change of a stable cuspidal packet of the quasi-split unitary group associated to the quadratic extension $E/F$, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of $\mathrm{GL}_{n,X'}$ is nonvanishing if and only if the corresponding local system $E$ on $X'$ is conjugate self-dual with respect to an étale double cover $X'/X$ of curves,…
Deligne constructed a remarkable local system on $\bP^1-\{0,\infty\}$ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct l-adic Kloosterman sheaves for…
This article is a companion to several works of the author and others on the arithmetic of automorphic forms for GSp(4), and their associated L-functions and Galois representations. These works require, at various points, an input from smooth representation theory over p-adic local fields: the computation of values of the unique trilinear form on an unramified representation of GSp(4) x GL(2) x GL(2) invariant under the diagonal GL(2) x GL(2), for various different input data. In this note, we carry out these local computations, using a general formula for Shintani functions due to Gejima.
Let X be a smooth projective connected curve over an algebraically closed field k of positive characteristic. Let G be a reductive group over k, γbe a dominant coweight for G, and E be an \ell-adic \check{G}-local system on X, where \check{G} denotes the Langlands dual group. Let \Bun_G be the moduli stack of G-bundles on X. Under some conditions on the triple (G,γ,E) we propose a conjectural construction of a distinguished E-Hecke automorphic sheaf on \Bun_G. We are motivated by a construction of automorphic forms suggested by Ginzburg, Rallis and Soudry in [6,7]. We also generalize Laumon's …
Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that …
Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. We show that for m\le n (respectively, for m>n) the theta-lifting functor from D(Bun_H) to D(Bun_G) (respectively, from D(Bun_G) to D(Bun_H)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL_2 of Arthur. So, they realize the geometric Langlands functoriality for the corresp…
Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bun_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bun_G(X)$. We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of $X$. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli $Loc_{G^\vee}(X)$ of dual group local systems. In particular, we establish the "automorphic to Galois" direction in the Betti Geometric Langlands corr…
In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\textrm{GL}_n$-local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as $\check{G}$-local systems, for a classical group $\check{G}$. This article aims to realize the geometric Langlands correspondence for these $\check{G}$-local systems. We study this problem from two aspects. In the first approach, we define the hypergeometric automorphic data for a classical group $G$ i…
We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp(4) to GL(4) follows from our result as a special case.
Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Let Bun_{Mp_2} be the stack of metaplectic bundles on X of rank 2. In this paper we study the derived category of genuine l-adic sheaves on Bun_{Mp_2} in the framework of the quantum geometric Langlands. We describe the corresponding Whittaker category, develop the theory of geometric Eisenstein series and calculate the most non-degenerate Fourier coefficients of these Eisenstein series. The existing constructions of automorphic sheaves for GL_n are based on using Whittaker sheaves. Our calculations lea…
The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of $G$-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if $G$ is an abe…
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are t-exact on the category of tempered D-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for GL_n. We also show that Whittaker coefficient functors are t…
We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when $G$ …
The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years ago, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence to the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in partic…
We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack Bun_G(X) o…
