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Let $F$ be a $p$-adic field, and $C_0^\infty(F^\times)$ the space of smooth compactly supported functions on $F^\times$. Under the regular action of $F^\times$ on $C_0^\infty(F^\times)$, I believe we have the decomposition $$ C_0^\infty(F^\times)=\bigoplus\chi $$ where $\chi$ is an irreducible representation of $F^\times$, namely one dimensional representation.

Now, which $\chi$ appears in this decomposition? Only unitary characters? Also, is there any natural basis element in $C_0^\infty(F^\times)$ that corresponds to $\chi$?

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  • $\begingroup$ This is just the $p$-adic theory of the Mellin transform. $\endgroup$ Mar 19, 2023 at 22:07

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No smooth character $\chi$ of $F^\times$ embeds as a sub-$F^\times$-module of $C_c^\infty (F^\times )$ (for an obvious matter of support : smooth characters of $F^\times$ never vanish so do not have compact support).

On the other hand any smooth character $\chi$ of $F^\times$ is a quotient of the $F^\times$-module $C_c^\infty (F^\times )$. Indeed the following linear map is onto and $\chi$-equivariant :

$$ C_c (F^\times )\longrightarrow {\mathbb C} $$ $$ f\mapsto \int_{F^\times} f(x)\chi^{-1} (x)\, d^\times x $$ where $d^\times x$ is a Haar measure.

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