Let $\mathcal{C}\_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$.


If this is a Tannakian category, it has an associated Tannakian fundamental group $G^{\mathrm{aut}}\_F$. What is it and how is it related to $G$ itself (for example, for $G = \operatorname{GL}\_n$)? In other words, what does Tannakian duality recover in this case?


EDIT: In light of Bugs Bunny's answer below, if we consider the group of endomorphisms of the forgetful functor from $\mathcal{C}\_{\mathrm{aut}}(G, F)$ (with completed tensor product) to the category of Hilbert spaces, how is the group related to $G$?



Tannakian fundamental group of automorphic representations
It is not a tannakian category. The issue is the tensor product. Let $V$ and $W$ be automorphic representations. The algebraic tensor product $V\otimes W$ is no longer automorphic. You need some kind of completion $V\widehat{\otimes}W$ but, according to David Loeffler, the completion is too big to be automorphic.


