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 $\chi$?