Character Sheaves on Tori over Local Fields

Abstract

Let $\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field $k$ and ring of integers $\breve{O}$. Let $T$ be a torus defined over $\breve{K}$. Let $L^{+}T$ denote the connected commutative pro-algebraic group over $k$ obtained by applying the Greenberg functor to the connected Néron model of $T$ over $\breve{O}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $\ddot{I}{€}_1(L^{+}T)$. We then study multiplicative local systems (or character sheaves) on $L^{+}T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^{+}T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence.

32 pages