Kazhdan’s theory loosely states that the complex representation theory of the group G($F$), where G is a split connected reductive group over $\mathbb{Z}$ and $F$ is a non-archimedean local field of characterstic $p$, can be viewed as a “limit” of the complex representation theories of the groups G($F’$), where $F’$ varies over non-archimedean local fields of characteristic 0 with residue characteristic $p$. A similar theory for representations of the Galois group Gal($F_s/F$) is due to Deligne. In this talk we will review this theory, discuss some applications of this theory to the local Langlands correspondence, and some ingredients in generalizing the work of Kazhdan and some variants of it to non-split groups.

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