Geometric theta-lifting for the dual pair GSp_2n, GSO_2m

2008/02/04 by Sergey Lysenko, Lysenko, Sergey · 1 citation
Mathematics · #11R39 #14H60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0802.0457

openalex publication_date 2008/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≤ 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 SL₂ 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 GSp₄. Namely, we construct the automorphic Hecke eigensheaves on Bun_GSp4 corresponding to the endoscopic local systems on X.

Citations

• Langlands' Functoriality and the Weil Representation

Cited by

• Gauge Theory and Langlands Duality

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