[Paper Review] Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian

Ryan Reich|arXiv (Cornell University)|Dec 28, 2010

Advanced Algebra and Geometry12 references4 citations

TL;DR

This paper establishes a twisted, factorizable geometric Satake equivalence using symmetric factorizable gerbes as a generalization of both the classical geometric Satake equivalence and the twisted version by Finkelberg–Lysenko. It shows that perverse sheaves twisted by such gerbes categorically realize representations of a dual group constructed from quadratic forms on the weight lattice, unifying the factorizable and twisted settings via a unified framework based on gerbes and t-structures.

ABSTRACT

We prove a generalization of the twisted geometric Satake equivalence of Finkelberg--Lysenko in the context of the factorizable grassmannian of a reductive group G relative to a smooth curve X, similar to Gaitsgory's generalization in "On de Jong's conjecture" of the Satake equivalence itself. In order to express the dependence of this result on factorizability, we consider a certain 2-category of gerbes on this grassmannian, which we call "sf gerbes"; the theorem then concerns perverse sheaves twisted by such gerbes. This notion of twisting differs from, but in special cases is equivalent to, that considered by Finkelberg--Lysenko. We give a classification of these gerbes in terms of quadratic forms on the coweight lattice of G and gerbes on X. In addition, we introduce the technical device of universal local acyclicity (ULA) into the proof of the Satake equivalence, using it in combination with Beilinson's nearby cycles gluing of perverse sheaves to reduce all of the main constructions in the Mirkovic--Vilonen proof of the equivalence to the ULA case, where they have simple descriptions whose properties have simple proofs. We also give a geometric proof that the category of spherical perverse sheaves on the ordinary affine grassmannian is semisimple.

Motivation & Objective

• To unify the geometric Satake equivalence, the factorizable Satake equivalence (Gaitsgory), and the twisted Satake equivalence (Finkelberg–Lysenko) into a single framework.
• To generalize the notion of twisting from roots of unity to gerbes, particularly symmetric factorizable gerbes on the Beilinson–Drinfeld Grassmannian.
• To show that the category of twisted perverse sheaves on the affine Grassmannian is equivalent to the category of representations of a dual group constructed from quadratic forms on the weight lattice.
• To provide a new proof of the tensor category structure of twisted Satake categories using a formalism compatible with Mirković–Vilonen’s methods.
• To establish a connection between these twisted Satake equivalences and the quantum Langlands correspondence via duality of gerbes on dual groups.

Proposed method

• Introduce symmetric factorizable gerbes as a generalization of twisting by roots of unity, parameterized by quadratic forms on the weight lattice of a reductive group G.
• Define a twisted derived category and twisted t-structures on the Beilinson–Drinfeld Grassmannian using gerbes, preserving the properties needed for perverse sheaf theory.
• Use the formalism of equivariant twisted objects and multiplicative structures on gerbes to ensure compatibility with convolution and fusion operations.
• Construct the twisted fiber functor on the absolute Grassmannian and prove semisimplicity of the twisted Satake category via convolution and d´evissage techniques.
• Apply Tannakian duality to identify the dual group associated with a given quadratic form, showing that it is isomorphic to the Langlands dual when the form is trivial.
• Establish a duality between gerbes on G and LG via a nondegenerate bilinear form, leading to an equivalence of twisted Satake categories that realizes a local quantum Langlands correspondence.

Experimental results

Research questions

• RQ1 How can the geometric Satake equivalence be generalized to include both factorizability and arbitrary gerbe twists?
• RQ2 What is the precise class of gerbes that can be used to twist the Satake equivalence in a way that preserves the tensor category structure?
• RQ3 How do symmetric factorizable gerbes on the Grassmannian relate to quadratic forms on the weight lattice of G?
• RQ4 Can the twisted Satake equivalence be recovered and generalized using a uniform framework that includes both the classical and Finkelberg–Lysenko cases?
• RQ5 What is the relationship between the gerbe-twisted Satake categories for G and its Langlands dual LG, and how does this relate to the quantum Langlands correspondence?

Key findings

• The category of G-twisted spherical perverse sheaves on the Beilinson–Drinfeld Grassmannian is equivalent to the category of representations of a reductive group ˇGQ constructed from a quadratic form Q on the weight lattice of G.
• The dual group ˇGQ associated with a quadratic form Q is isomorphic to the Langlands dual LG if and only if Q(λ) = (−1)^⟨2ρ,λ⟩, which corresponds to the critical twist in the Beilinson–Drinfeld setting.
• The gerbe associated with the quadratic form Q(λ) = (−1)^⟨2ρ,λ⟩ is trivial as a gerbe but nontrivial as a symmetric factorizable gerbe, and it corresponds to the critical twist used in [BD].
• For a nondegenerate C-valued W-invariant bilinear form b, the dual group ˇGQ associated with Q(λ) = exp(πi b(λ,λ)) is isomorphic to the Langlands dual of LG under the same form, establishing a duality between gerbes on G and LG.
• The twisted Satake categories Sph(Gn) and Sph(LGn) associated with dual gerbes are equivalent, realizing a local version of the quantum Langlands correspondence.
• The proof establishes that the twisted Satake category is semisimple and that simple objects correspond to irreducible representations of ˇGQ, with convolution preserving the tensor structure.