In the first part of the talk, we will discuss the main statement of local class field theory that describes the abelian extensions of a non-archimedean local field $F$ in terms of the arithmetic of the field $F$. Then we will discuss the statement of the local Langlands conjectures, a vast generalization of local class field theory, that gives a (conjectural) parametrization of the irreducible complex representations of $G(F)$, where $G$ is a connected, reductive group over $F$, in terms of certain Galois representations. We will then discuss a philosophy of Deligne and Kazhdan that loosely says that to obtain such a parametrization for representations of $G(Fâ)$, with $Fâ$ of characteristic $p$, it suffices to obtain such a parametrization for representations of $G(F)$ for all local fields $F$ of characteristic $0$. In the second half of the talk, we will mention some instances where the Deligne-Kazhdan philosophy has been applied successfully to obtain a Langlands parametrization of irreducible representations of $G(Fâ)$ in characteristic $p$ and focus on some recent work on variants/generalizations of the work of Kazhdan.

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Last updated: 12 Apr 2024