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Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations: $$ t\mapsto \begin{pmatrix} \xi(t) & \\ & \xi'(t)\end{pmatrix} $$ for quasi-characters $\xi, \xi':F^{\times} \to \mathbb{C}^{\times}$, or $$ t\mapsto \xi(t) \begin{pmatrix} 1& v(t)\\&1\end{pmatrix} $$ for some quasi-character $\xi$ and a valuation map $v:F^{\times} \to \mathbb{Z}$.

I want to know the complete classification of every (finite dimensional) admissible representation of $F^{\times}$ (including reducible ones). It seems that the above representations like Jordan blocks, so maybe all the finite dimensional representations look like the above ones. Is this true?

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  • $\begingroup$ With $F = \Bbb{Q}_p$ you can look at $\rho(p), \rho(\zeta_{p-1}), \rho(1+p)$ and (multiplying by a character $F^\times \to \Bbb{C}^\times$) assume they are of determinant $1$, in some basis $\rho(\zeta_{p-1}),\rho(1+p)$ are diagonal, $\rho(\zeta_{p-1})^{p-1} = I, \rho(1+p)^{p^m} = I$ and $\rho(p)$ is upper triangular and commutes with them $\endgroup$ – reuns Mar 28 at 23:47
  • $\begingroup$ $F^{\times}$ is abelian, so irreducible representations are just quasicharacters, which are one-dimensional. $\endgroup$ – Peter Humphries Mar 29 at 11:33

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