Geometric metaplectic parameters

Abstract

The objective of this work is to understand metaplectic parameters of the geometric Langlands program. In the usual Langlands program, metaplectic parameters are given by Brylinski--Deligne data, i.e., central extensions of the reductive group $G$ by the second algebraic K-theory group $\mathbf K_2$. They admit a combinatorial description \cite{brylinski2001central}, making it possible to define metaplectic Langlands dual data \cite{weissman2015groups}.

In the geometric setting, we first show that Brylinski--Deligne data are equivalent to factorization line bundles on the affine Grassmannian (joint work with J.~Tao \cite{tao2019extensions}). Guided by the principle that factorization ``twisting agents'' should serve as geometric metaplectic parameters \cite{gaitsgory2018parameters}, we explain the notion of factorization gerbes in various sheaf-theoretic contexts. We prove that in the \'etale and analytic contexts, they admit combinatorial descriptions and are related to factorization line bundles via the first Chern class map.

The second half of this work addresses the de Rham context, where the sheaf theory is that of algebraic $\cal D$-modules. The na\"ive notion of gerbes in this context does not behave as expected, and we suggest the modified notion of factorization ``tame gerbes'' to serve as metaplectic parameters. We also relate factorization gerbes to factorization twistings which play the role of quantum parameters of the geometric Langlands program and discuss a natural compactification of the latter.