
TL;DR
This paper constructs an extended Whittaker category and a Fourier transform functor for $G$-bundles on a curve, establishing key properties for classical groups and linking to the Langlands duality conjecture.
Contribution
It introduces the extended Whittaker category and a Fourier transform functor, proving full faithfulness for $GL_n$ and $PGL_n$, and relates to Langlands duality.
Findings
The functor $ ext{coeff}_{G, ext{ext}}$ is fully faithful for $GL_n$ and $PGL_n$.
The construction supports the compatibility of Langlands duality with Whittaker models.
Guarantees uniqueness and faithfulness of the Langlands duality functor for classical groups.
Abstract
Let be a connected reductive group, with connected center, and a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let denote the stack of -bundles on . In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a "Fourier transform" functor, called , from the DG category of -modules on to a certain DG category , 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 with the Whittaker…
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