几何Waldspurger周期
Geometric Waldspurger periods
AI总结：
本文在几何朗兰兹纲领框架下研究对偶约化对的θ提升，定义了对应导出范畴间的函子，证明了GO_2与GL_2对偶对的几何朗兰兹函子性，并应用于周期计算，给出GSp_4上自守层的三种猜想构造。
摘要：
本文是在几何朗兰兹纲领框架下开展θ提升（或称Howe对应）相关研究的一项阶段性成果。我们在光滑射影曲线X上研究（非分歧的）对偶约化对H=GO_{2m}、G=GSp_{2n}（假设我们的H在X的一个平展双层覆盖Y上是分裂的）。设Bun_G和Bun_H分别为X上的G-旋子层与H-旋子层的叠。我们定义并研究了导出范畴D(Bun_G)与D(Bun_H)之间的函子F_G和F_H，它们是经典θ提升算子的几何对应物。主要结果之一是对偶对(H=GO_2, G=GL_2)的几何朗兰兹函子性，其中GO_2是乘法群从非平凡平展覆盖Y到X的直像。从D(Bun_H)到D(Bun_G)的函子F_G关于朗兰兹L-群的对应映射G^L\to H^L与Hecke算子交换。作为应用，我们计算了Bun_{GL_2}上尖点自守层的Waldspurger周期，以及从Bun_{GO_4}到Bun_{GSp_4}的θ提升的Bessel周期。基于这些计算，我们给出了Bun_{GSp_4}上某类自守层的三种猜想性构造（其中一种仅对D-模有意义）。
英文摘要：
This paper is a step towards a version of the theta-lifting (or Howe correspondence) in the framework of the geometric Langlands program. We consider the (unramified) dual reductive pair H=GO_{2m}, G=GSp_{2n} over a smooth projective curve X (our H is assumed to be split on an etale two-sheeted covering Y of X). Let Bun_G and Bun_H be the stack of G-torsors and H-torsors on X. We define and study functors F_G and F_H between the derived categories D(Bun_G) and D(Bun_H) that are analogs of the classical theta-lifting operators. One of the main results is the geometric Langlands functoriality for the dual pair (H=GO_2, G=GL_2), where GO_2 is the direct image of the multiplicative group from a nontrivial etale covering Y to X. The functor F_G from D(Bun_H) to D(Bun_G) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups G^L\to H^L. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on Bun_{GL_2} and Bessel periods of theta-lifts from Bun_{GO_4} to Bun_{GSp_4}. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on Bun_{GSp_4} (one of them makes sense for D-modules only).
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