# Classification of (finite dimensional) admissible representations of $F^{\times} = \mathrm{GL}(1, F)$.

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?

• 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 – reuns Mar 28 at 23:47
• $F^{\times}$ is abelian, so irreducible representations are just quasicharacters, which are one-dimensional. – Peter Humphries Mar 29 at 11:33