Let $F$ be a $p$-adic field.

In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math **73** (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the following data:

A tamely ramified extension $F'/F$ of degree $n$, and

An 'admissible' character $\psi: F'^\times \to \mathbb{C}^\times$.

See also Moy's thesis "Local Constants and the Tame Langlands Correspondence", (Am. J. Math **108** (1986)).

Throughout, Howe has chosen an additive character $\chi: F^+ \to \mathbb{C}^\times$ but suppresses the dependence on $\chi$ throughout (Moy does the same). My question is: does the representation $\pi_{\psi}$ constructed depend on choice of $\chi$, up to isomorphism?

In the case $n = 2$, Schmidt gives a slightly different construction of tame supercuspidal representations; see "Some remarks on local newforms for GL(2)", (J. Ramanujan Math. Soc. **17** (2002)). This construction also depends on an additive character; Schmidt says the independence of $\pi_\psi$ on the additive character $\chi$ is 'clear'. I haven't checked the details, and I won't claim that the independence is obvious to me, but this gives me hope that the construction should be independent of the additive character. Nonetheless, I haven't spoken to anyone that has seen this fact written down and I'm curious if this is known, or even true.