I have a naive question on complex representations of finite groups of Lie type.
Let $\bf G$ be a reductive group (say connected, with connected center, for safety)
defined over a finite field $\mathbb F\_q$, and let $G={\bf G}(\mathbb F\_q)$. Assume first that $G$ is split over $\mathbb F\_q$, so there is a maximal torus $\bf T$ in $\bf G$ defined and split over $\mathbb F\_q$. Let $g \in T = {\bf T}(\mathbb F\_q)$ be a regular element.



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> Then is it true that if $\pi$ is an irreducible complex representation of $G$, of character $\chi = tr\ \pi$, then $\chi(g) \neq 0$ implies $\pi$ is a principal series ?
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By a principal series, one means of course a sub representation of $Ind\_B^G \psi$, where $B$ is a Borel containing $T$ and $\psi: T \rightarrow \mathbb C^\ast$ a one-dimensional character. 


A glance at the table of characters found for example in Fulton-Harris shows that this is true
for $GL\_2$, so my intuition tells me that it should be true, and probably easy, in general,
but the argument escapes me and I haven't seen this statement in the literature (e.g. in
Curtis' survey or Digne and Michel's book). 


Also if the answer is yes, then can one remove the assumption that $G$ is split (that is
we work with a general $\bf G$ as above, and take for $\bf T$ a quasi-split group, and ask the same question) ? 



Principal series of finite group of Lie type
So, I think the answer to your question is yes. This may not be the slickest proof but I think it works. Firstly let $\mathrm{pr}\_G$ be the projection map from the space of all class functions to the subspace of uniform functions, i.e. the subspace spanned by Deligne-Lusztig virtual characters, (see Digne and Michel - 12.11 and 12.12). Now for a pair $(\mathbf{S},\theta)$ consisting of an $F$-stable maximal torus and character $\theta$ of $\mathbf{S}^F$ let $c\_{\mathbf{S},\theta} \in \mathbb{C}$ be such that


$$\mathrm{pr}\_G(\chi) = \sum\_{}c\_{\mathbf{S},\theta}R\_{\mathbf{S}}^{\mathbf{G}}(\theta)$$


where this sum is taken up to conjugacy. The characteristic function of a semisimple element is a uniform class function, (see DM - Proposition 12.20), in particular we have $(\mathrm{pr}\_G(\chi))(g) = \mathrm{pr}\_G(\chi(g))$. Hence if $\chi(g)$ is non-zero this implies that


$$\mathrm{pr}\_G(\chi)(g) = \sum\_{}c\_{\mathbf{S},\theta}R\_{\mathbf{S}}^{\mathbf{G}}(\theta)(g) \neq 0$$


However $R\_{\mathbf{S}}^{\mathbf{G}}(\theta)(g) \neq 0$ implies that the conjugacy class of $g$ intersected with $\mathbf{S}^F$ is non-empty (see Carter - Proposition 7.5.3). As your element is regular this implies that $\mathbf{S}$ is rationally conjugate to your chosen torus $\mathbf{T}$. Now $c\_{\mathbf{S},\theta} \neq 0$ implies that $\chi$ occurs in $R\_{\mathbf{S}}^{\mathbf{G}}(\theta)$ with non-zero multiplicity, hence in $R\_{\mathbf{T}}^{\mathbf{G}}(\theta')$ with non-zero multiplicity where $(\mathbf{T},\theta')$ is any pair rationally conjugate to $(\mathbf{S},\theta)$. However this says that $\chi$ is in the principal series as $R\_{\mathbf{T}}^{\mathbf{G}}(\theta')$ is simply the Harish-Chandra induction of $\theta'$.


Note that this argument does not require that $F$ is split.


