admissible characters for $PGL_{2}(F)$ What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?
 A: Character formulas are not known in all generality (i.e. for all reductive $p$-adic groups and  all representations). However in the case of ${\rm GL}(2,F)$ you can find much information in the literature. For instance :
a) A character formula is given for the principal series of ${\rm GL}(2,F)$ in Jacquet, Langlands, Automorphic forms on ${\rm GL}(2)$, Chap. 1, Prop. 7.6. 
b) The character of the Steinberg representation is constant on the set of regular elliptic elements, where it takes value $-1$. 
c) Certain values of characters of supercuspidal representation have been computed by e.g. P. Kutzko and Pantoja: 
Kutzko, Philip Character formulas for supercuspidal representations of ${\rm GL}(l)$,$l$ a prime. Amer. J. Math. 109 (1987), no. 2, 201–221.
Kutzko, Phil; Pantoja, José Character formulas for supercuspidal representations of the groups ${\rm GL}_2$, ${\rm SL}_2$. Comm. Algebra 26 (1998), no. 6, 1679–1697.
A: I know little about literature. But theortically, parabolically induced representations have their characters being parabolic induction of the character from the Levi (see e.g. Sec. 13 of Kottwitz' article in 2003 Clay summer school proceeding). This includes the case of the Steinberg one, since we then know the sum of the character of the trivial one and $St$ (in particular, the result that Paul discussed).
On other other hand, when the residue field has charateristic $p$ odd, supercuspidal representations of $PGL_2(F)$ is given (in a bijective way) by a choice of an elliptic torus in $PGL(N)$ (which correspond to one of the three degree $2$ extension $E/F$) together with an suitably "generic" character on $E^{\times}/F^{\times}$. See Howe's tamely ramified supercuspidal representations of $GL_n(F)$, this is known fairly well for $n$ coprime to $p$.
If the character on $E^{\times}/F^{\times}$ is tamely ramified, i.e. trivial on $1+p_E$, then the representation is of depth zero and the character can be expressed in terms of the character formula of Deligne-Lusztig for $PGL_2(k)$, $k$ residue field. Otherwise the character is mainly given by the Fourier transform of some elliptic element in the Lie algebra of $PGL_2(F)$, which the Kottwitz article (Sec. 5) also provide an algorithm to compute.
I mainly learn the character computation from the character formula article of Adler and Spice, but that is for general groups and I don't think that is a good place to look up if one only want $GL(2)$ or $GL(n)$.
