[Paper Review] Whittaker and Bessel functors for GSp_4

Sergey Lysenko|arXiv (Cornell University)|Oct 22, 2003

Advanced Algebra and Geometry7 references4 citations

TL;DR

This paper introduces Whittaker and Bessel functors for the symplectic group GSp₄ in the geometric Langlands program, generalizing Gaitsgory's approach for GLₙ. It defines a geometric Waldspurger category and proves a multiplicity-one result for Waldspurger models, establishing foundational tools for studying hyper-cuspidal sheaves and their Hecke actions on moduli stacks of GSp₄-bundles.

ABSTRACT

One of the important technical tools in Gaitsgory's proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence ([3]) is the theory of Whittaker functors for GL_n. We define Whittaker functors for GSp_4 and study their properties. In a sense, these functors correspond to the maximal parabolic subgroup of GSp_4, whose unipotent radical is not commutative. We also study similar functors corresponding to the Siegel parabolic subgroup of GSp_4, they are related with Bessel models for GSp_4 and Waldspurger models for GL_2. We define the Waldspurger category, which is a geometric counterpart of the Waldspurger module over the Hecke algebra of GL_2. We prove a geometric version of the multiplicity one result for Waldspurger models.

Motivation & Objective

• To extend the theory of Whittaker functors from GLₙ to GSp₄, a group with non-commutative unipotent radical in its maximal parabolic.
• To define and study Bessel functors associated with the Siegel parabolic subgroup of GSp₄, linking them to Waldspurger models for GL₂.
• To introduce the Waldspurger category as a geometric counterpart of the Waldspurger module over the Hecke algebra of GL₂.
• To establish a geometric multiplicity-one result for Waldspurger models, analogous to classical automorphic forms.
• To lay the groundwork for understanding the Hecke action on hyper-cuspidal and cuspidal sheaves in the geometric Langlands program for GSp₄.

Proposed method

• Constructs Whittaker functors for GSp₄ using the maximal parabolic subgroup with non-commutative unipotent radical, generalizing the GLₙ case.
• Defines the Waldspurger category as a full triangulated subcategory of the derived category of sheaves on the stack of GSp₄-bundles.
• Applies Fourier transform techniques in equivariant categories to relate sheaves on different groupoid structures, using the action of vector bundles on schemes.
• Uses the Casselman–Shalika formula in the context of ind-sheaves to describe Whittaker functionals, accounting for non-local finiteness.
• Applies Verdier duality and base change to show that the Fourier transform functor is t-exact and commutes with duality up to character inversion.
• Establishes a geometric multiplicity-one result by analyzing the kernel of the Whittaker functor and relating it to the vanishing of certain integrals over unipotent subgroups.

Experimental results

Research questions

• RQ1 How can Whittaker functors be generalized from GLₙ to GSp₄, especially given the non-commutativity of the unipotent radical in its maximal parabolic?
• RQ2 What is the geometric counterpart of the Waldspurger module for GL₂, and how does it relate to Bessel models for GSp₄?
• RQ3 How do the Hecke functors act on the subcategory of hyper-cuspidal sheaves in the derived category of GSp₄-bundles?
• RQ4 Can a geometric multiplicity-one result for Waldspurger models be established, analogous to the classical automorphic case?
• RQ5 What is the role of the orthogonal complement of the hyper-cuspidal category in the geometric Langlands correspondence for GSp₄?

Key findings

• The paper defines a geometric Waldspurger category as a full triangulated subcategory of the derived category of sheaves on the stack of GSp₄-bundles, which is preserved by Hecke functors.
• It proves a geometric multiplicity-one result for Waldspurger models, showing that the space of Whittaker functionals on hyper-cuspidal sheaves is one-dimensional.
• The hyper-cuspidal category D_hcusp(Bun_G) is characterized as the intersection of the kernels of all Whittaker functors, generalizing the classical notion of hyper-cuspidal automorphic forms.
• The Whittaker functor for GSp₄ is shown to be injective on the hyper-cuspidal category, analogous to the GLₙ case, despite the non-commutativity of the unipotent radical.
• The Casselman–Shalika formula is adapted to the geometric setting, involving nontrivial denominators due to the ind-object K_E not being locally finite on the Waldspurger stack.
• The Fourier transform functor is established as an equivalence between D(Y) and D^W(E^* ×_Z Y), with the quasi-inverse given by the pushforward along the projection, and this equivalence commutes with Verdier duality up to character inversion.