Is Howe's construction of tame supercuspidal representations independent of additive character? 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.
 A: The only real use of $\chi$ is to identify Moy–Prasad quotients with their character lattices; but notice that this is done twice, first to produce from $\theta$ an element $y$ (on p. 442), then to produce from this element $y$ a character $\theta$ of a 'wider' but 'deeper' group (in Lemma 12 on p. 450).  These two identifications are inverse in an obvious sense; in particular, their composition is insensitive to the particular choice of $\chi$ used to make them.
By the way, a much more modern perspective on all this is Yu's "Construction of tame supercuspidals" paper (MR); but that can be a bit intimidating, so you might want to start with Adler's paper "Refined anisotropic K-types …" (MR), which I believe inspired Yu.  Even if you know for sure that you will only ever want to consider representations of general linear groups, please see Reimann (not Riemann!)'s paper "Representations of tamely ramified $p$-adic division and matrix algebras" (MR), which, if I'm remembering correctly, compares the differing normalisations of the Weil representation appearing in various Howe-inspired constructions.  (Yu also goes into considerably more detail than Howe on this point; see, for example, p. 601 of the cited paper, where he says:  "In the literature, people often write:  then we have homomorphisms $K \to \operatorname{Sp}(V)$ and $K \ltimes H \to \operatorname{Sp}(V) \ltimes H$, and we can pull back the Weil representation (after choosing a central character).  This is not true.")
A: Another paper that is worth noting in this regard is one of Waldspurger: 
Waldspurger, J.-L. Algèbres de Hecke et induites de représentations cuspidales, pour GL(N), J. Reine Angew. Math. 370 (1986), 127–191.
He extends the results of Howe about as far as you can without getting into types ala Bushnell/Kutzko. He also provides details on some tricky calculations, particularly with regard to extensions, that are not in Howe. 
