Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.

First, let us formulate the theorem of Harish-Chandra in a little more precise manner: it is a priori obvious that the character of $V$ is well-defined as a distribution. Now the theorem says that this distribution is given by a locally $L^1$-function which is well defined and is locally constant on an open dense subset of $G$ (but there is no good way to define on the whole of $G$). For example for principal series the function will be well defined on the open subset of regular semi-simple elements. What you can prove is that if such an element $g$ is not split, then the value of the character of an unramified principal series representation at $g$ is just equal to $0$. If $g$ is split, then up to conjugacy it lies in the standard split torus and the character is equal to the Weyl group average of the original character $\chi:T\to {\mathbb C}^*$ from which the principal series representation was induced (up to the standard "$\rho$-shift").

Although you didn't mention them, it seems appropriate to bring up the so called ‘reducible principal series’—more precisely, the irreducible components of a full induced representation off a Borel. These were computed for the case of $\operatorname{SL}_2$ by Paul Sally's student Stephen Franklin. As far as I know, the thesis was never published; but the formula is announced in Sally–Shalika 3. There must be other explicit calculations out there in this setting, but I don't know them.

The trace of a principal series representation can be easily computed from the main result of "Computation of Certain Induced Characters of p-Adic Groups" by van Dijk, Math. Ann. 199 229-240 (1972), doi: 10.1007/BF01429876, eudml. It gives you a formula to compute the character of a parabolically induced character based on that of the inducing character. It is of course supported on conjugates of Levi, and for $\gamma$ belonging to the Levi M, the character of the induced representation at gamma equals the weighted sum of characters of the inducing representation at W(G, M)-conjugates of $\gamma$, the weight being an appropriate discriminant factor. The formula also occurs somewhere in Kazhdan's "Cuspidal geometry of p-adic groups".

In particular, in the p-adic case if you induce $(\mu_1, \mu_2)$ to $GL_2(K)$, the character is supported on the conjugates of diagonals, and the value of the character at $(\lambda_1, \lambda_2)$ is equal to something like (I might get some factor wrong) $\displaystyle\frac{\mu_1(\lambda_1) \mu_2(\lambda_2) + \mu_1(\lambda_2) \mu_2(\lambda_1)}{|\lambda_1 - \lambda_2|}$. Here the $|\lambda_1 - \lambda_2|^2$ factor is the reason why it does not extend nicely to $GL_2(K)$, and is also why it is not bounded. However, if you multiply it by $|\lambda_1 - \lambda_2|$, which is what is denoted $|D(g)|^{1/2}$ where $g = (\lambda_1, \lambda_2)$ then you do get a nice function on $GL_2(K)$ (it extends nicely to the non-regular set).

Note that the computation of characters of subquotients of principal series is a totally different ballgame. These guys are not supported on conjugates of the Levi. I doubt anyone has ventured seriously into that problem.