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.

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 ?

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) ?