# Character values of principal series representations of $GL_n(\mathbb{F}_q)$

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)

• The standard source for these characters is usually the paper by J.A. Green in the Transactions of AMS )1955), which is freely available online; or the version in Ian Macdonald's book (latest edition). In any case, the term 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.) – Jim Humphreys Nov 4 '19 at 16:13
• P.S. The tag 'algebraic-groups' is probably more suitable here than the tag 'lie-groups', since the methods used are most often algebraic though suggested by Harish-Chandra's analytic methods. – Jim Humphreys Nov 4 '19 at 16:17
• thank you for the suggestions @JimHumphreys – Sagars Nov 5 '19 at 6:42

Thus the degree of f(q) is equal to the dimension of the Springer fibre, which is known. So if u has Jordan type λ, then $$\operatorname{deg}(f)=\sum_i {\lambda^t_i\choose 2}$$