On the extended Whittaker category

Abstract

Let $G$ be a connected reductive group, with connected center, and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let ${\mathrm{Bun}}_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a "Fourier transform" functor, called ${\mathsf{coeff}}_{G,\mathsf{ext}}$, from the DG category of $\mathfrak{D}$-modules on ${\mathrm{Bun}}_G$ to a certain DG category $\mathcal{W}h(G,\mathsf{ext})$, called the \emph{extended Whittaker category}. Combined with work in progress by other mathematicians and the author, this construction allows to formulate the compatibility of the Langlands duality functor ${\mathbb{L}}_G:{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LocSys}}_{\check{G}})\to \mathfrak{D}({\mathrm{Bun}}_G)$ with the Whittaker model. For $G=G{L}_n$ and $G=PG{L}_n$, we prove that ${\mathsf{coeff}}_{G,\mathsf{ext}}$ is fully faithful. This result guarantees that, for those groups, ${\mathbb{L}}_G$ is unique (if it exists) and necessarily fully faithful.

To appear in Selecta Mathematica

References in corpus (5)

• Higher Topos Theory
• Loop group actions on categories and Whittaker invariants
• Sheaves of categories and the notion of 1-affineness
• On the extended Whittaker category
• D-modules on Spaces of Rational Maps and on other Generic Data

Cited by in corpus (7)

• Loop group actions on categories and Whittaker invariants
• On the extended Whittaker category
• Sheaves of categories with local actions of Hochschild cochains
• Recent progress in geometric Langlands theory
• Tempered D-modules and Borel-Moore homology vanishing
• On the geometric Ramanujan conjecture
• The category of singularities as a crystal and global Springer fibers