[Paper Review] Geometric theta-lifting for the dual pair GSp_{2n}, GSO_{2m}

Sergey Lysenko|arXiv (Cornell University)|Feb 4, 2008

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper establishes a geometric realization of Langlands functoriality for the dual pair (GSp_{2n}, GSO_{2m}) over a smooth projective curve in characteristic >2, showing that the geometric theta-lifting functors commute with Hecke functors via an L-group morphism involving Arthur's SL₂. The key result is a construction of automorphic Hecke eigensheaves on Bun_{GSp_4} for certain endoscopic local systems, realizing a special case of the geometric Langlands correspondence.

ABSTRACT

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 corresponding morphisms of L-groups. As an application, we prove a particular case of the geometric Langlands conjectures for GSp_4. Namely, we construct the automorphic Hecke eigensheaves on Bun_{GSp_4} corresponding to the endoscopic local systems on X.

Motivation & Objective

• To extend the geometric Langlands program to similitude groups GSp_{2n} and GSO_{2m} that split over an étale double cover of a curve.
• To establish that geometric theta-lifting functors commute with Hecke functors via a morphism of L-groups involving Arthur's SL₂.
• To construct automorphic Hecke eigensheaves on Bun_{GSp_4} for endoscopic local systems, realizing a special case of the geometric Langlands conjecture.

Proposed method

• Define geometric theta-lifting functors F_G: D(Bun_H) → D(Bun_G) and F_H: D(Bun_G) → D(Bun_H) between derived categories of ℓ-adic sheaves on moduli stacks of G-torsors.
• Use the framework of metaplectic Weil representations and theta functionals to define the lifting functors in the unramified setting.
• Construct a morphism of L-groups H^L × G_m → G^L (for m ≤ n) or G^L × G_m → H^L (for m > n) that governs the Hecke compatibility.
• Employ the theory of Hecke functors and their actions on D(Bun_G) and D(Bun_H), showing compatibility via the L-group morphism.
• Utilize the action of groupoids on spaces of sheaves and the structure of graded modules over Hecke algebras to prove the main theorem.
• Apply a key lemma on invariants under the map f^{κ_G} to show that certain submodules coincide with the full Hecke algebra, establishing the eigensheaf property.

Experimental results

Research questions

• RQ1 Does the geometric theta-lifting functor for the dual pair (GSp_{2n}, GSO_{2m}) commute with Hecke functors under a suitable L-group morphism?
• RQ2 How can the geometric Langlands functoriality be realized for similitude groups that are not split over the base curve but over an étale double cover?
• RQ3 Can automorphic Hecke eigensheaves be constructed on Bun_{GSp_4} for endoscopic local systems via geometric theta-lifting?
• RQ4 What is the role of Arthur's SL₂ in the L-group morphism governing the Hecke compatibility of the theta-lifting functors?
• RQ5 How does the graded structure of the Hecke algebra and the action of the groupoid T relate to the eigensheaf property?

Key findings

• The geometric theta-lifting functor F_G: D(Bun_H) → D(Bun_G) commutes with Hecke functors when m ≤ n, via a morphism H^L × G_m → G^L involving Arthur's SL₂.
• For m > n, the functor F_H: D(Bun_G) → D(Bun_H) commutes with Hecke functors via G^L × G_m → H^L, establishing functoriality in both directions.
• In the case n = m, the L-group morphism is trivial on the G_m factor, simplifying the correspondence.
• The construction yields automorphic Hecke eigensheaves on Bun_{GSp_4} for certain endoscopic local systems, realizing a special case of the geometric Langlands conjecture.
• The proof relies on a key lemma showing that a graded submodule W of the Hecke algebra H_{Q(G)} is equal to H_G when it contains the degree-zero component H_G.
• The compatibility of Hecke actions is established through the interplay of groupoid actions on sheaf spaces and the invariance of the Weil representation under the L-group morphism.