Let $P_{\alpha}$ be the principal series representation of $GL_n(\mathbb{F}_q)$, where $\alpha = ( \alpha_1, \alpha_2, \cdots, \alpha_n)$ and $\alpha_i : \mathbb{F}_q^* \rightarrow \mathbb{C}^*$.

The character value of $P_{\alpha}$ on the unipotent conjugacy class ( Jordan form has eigenvalues $x$) are of the form $f(q) \prod_i \alpha_i(x) $, where $f$ is a polynomial in $q$. Is it possible to say what is the degree of this polynomial ( or maybe a non trivial bound on its degree)?

(p.s. There is already an explicit formula for the character value of the cuspidal representations on unipotent conjugacy classes, and if we do a bases change to $\mathbb{F}_{q^n} $, the cuspidal representation becomes a principal series, hence I'm inclined to think that the degrees match for the two of them, but don't know how to proceed with this argument.)

(p.p.s. if $\alpha_i = \alpha_j$ for all $i,j$, then the principal series is reduced to a direct sum of unipotent representations, and if one can say anything about the maximum possible degree of the polynomial occurring in unipotent representation for each conjugacy class, then that gives a non-trivial bound. This should give a bound on the degrees occurring in a given unipotent class, and here again I don't know how to prove it for all unipotent representations)

principal seriesis most often used for the family of representations induced from 1-dimensjonal representaton of a Borel subgroup, here the upper (or lower) triangular matrices. (Carter's 1985 book has a more general viewpoint based on the Deligne-Lusztig construction.) $\endgroup$ – Jim Humphreys Nov 4 '19 at 16:13