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It seems that the answer to your question is no, based on a quick look at the character table of $\mathrm{Sp}_4(\mathbb{F}_q)$ worked out by Bhama Srinivasan in Trans. Amer. Math. Soc. 131 (1968), available online here. In her notation, the regular semisimple elements in a split maximal torus belong to the classes labelled $B_3(i,j)$. The two unipotent characters $\theta_1, \theta_3$ don't occur as constituents of the character induced from the trivial character of the split Borel subgroup (hence are not in the principal seris), but they do take nonzero values at such regular semisimple elements.

However, your question "almost" has an affirmative answer, which makes finite general linear groups tricky for experimental purposes (their characters are too well-behaved). See for instance Prop. 7.5.3 in Roger Carter's 1985 book Finite Groups of Lie Type, based on the Deligne-Lusztig theory. (Srinivasan's special case predates the 1976 Deligne-Lusztig paper but along with Green's work on finite general linear groups helped to motivate the later developments.)

EDIT: Sorry, the characters I mentioned are not actually unipotent, so more care is needed in interpreting Srinivasan's table. Stay tuned.

[REVISION]: After looking closer at the literature related to Deligne-Lusztig's Annals of Math. 103 (1976) paper, I'm more persuaded that the answer to your question is yes. In their paper, see in particular the formula (7.6.2) describing the value of an arbitrary irreducible character at a regular semisimple element. This applies in particular to such an element lying in the split maximal torus, but is more general.

I got misled at first while trying to understand some of the earlier published examples in rank 2; there it's tricky to work out which of the characters are in the principal series. The special cases are well worth studying, but preferably translated into the Deligne-Lusztig framework (as presented in the large book by Carter and the smaller one by Digne-Michel). As I mentioned in my earlier attempt at an answer, the character table of $\mathrm{Sp}_4(\mathbb{F}_q)$ was worked out by Bhama Srinivasan in Trans. Amer. Math. Soc. 131 (1968), available online here. In her notation, the regular semisimple elements in a split maximal torus belong to the classes labelled $B_3(i,j)$. (There are a handful of minor errors in the character table, mostly related to parameters. But note that the degree of $\theta_1$ is $\frac{1}{2} q^2 (1+q^2)$. It does lie in the principal series.) The characters of finite groups of type $G_2$ were worked out in a 1974 conference paper by Chang and Ree. .

Another related result based on Deligne-Lusztig theory which I quoted earlier only shows why your question "almost" has an affirmative answer: .see Prop. 7.5.3 in Roger Carter's 1985 book Finite Groups of Lie Type. (Srinivasan's special case predates the 1976 paper but along with Green's work on finite general linear groups helped to motivate the later developments.) In any case, I hope my reference above to the D-L paper is more helpful.

It seems that the answer to your question is no, based on a quick look at the character table of $\mathrm{Sp}_4(\mathbb{F}_q)$ worked out by Bhama Srinivasan in Trans. Amer. Math. Soc. 131 (1968), available online here. In her notation, the regular semisimple elements in a split maximal torus belong to the classes labelled $B_3(i,j)$. The two unipotent characters $\theta_1, \theta_3$ don't occur as constituents of the character induced from the trivial character of the split Borel subgroup (hence are not in the principal seris), but they do take nonzero values at such regular semisimple elements.

However, your question "almost" has an affirmative answer, which makes finite general linear groups tricky for experimental purposes (their characters are too well-behaved). See for instance Prop. 7.5.3 in Roger Carter's 1985 book Finite Groups of Lie Type, based on the Deligne-Lusztig theory. (Srinivasan's special case predates the 1976 Deligne-Lusztig paper but along with Green's work on finite general linear groups helped to motivate the later developments.)

It seems that the answer to your question is no, based on a quick look at the character table of $\mathrm{Sp}_4(\mathbb{F}_q)$ worked out by Bhama Srinivasan in Trans. Amer. Math. Soc. 131 (1968), available online here. In her notation, the regular semisimple elements in a split maximal torus belong to the classes labelled $B_3(i,j)$. The two unipotent characters $\theta_1, \theta_3$ don't occur as constituents of the character induced from the trivial character of the split Borel subgroup (hence are not in the principal seris), but they do take nonzero values at such regular semisimple elements.

However, your question "almost" has an affirmative answer, which makes finite general linear groups tricky for experimental purposes (their characters are too well-behaved). See for instance Prop. 7.5.3 in Roger Carter's 1985 book Finite Groups of Lie Type, based on the Deligne-Lusztig theory. (Srinivasan's special case predates the 1976 Deligne-Lusztig paper but along with Green's work on finite general linear groups helped to motivate the later developments.)

EDIT: Sorry, the characters I mentioned are not actually unipotent, so more care is needed in interpreting Srinivasan's table. Stay tuned.

It seems that the answer to your question is no, based on a quick look at the character table of $\mathrm{Sp}_4(\mathbb{F}_q)$ worked out by Bhama Srinivasan in Trans. Amer. Math. Soc. 131 (1968), available online here. In her notation, the regular semisimple elements in a split maximal torus belong to the classes labelled $B_3(i,j)$. The two unipotent characters $\theta_1, \theta_3$ don't occur as constituents of the character induced from the trivial character of the split Borel subgroup (hence are not in the principal seris), but they do take nonzero values at such regular semisimple elements.

However, your question "almost" has an affirmative answer, which makes finite general linear groups tricky for experimental purposes (their characters are too well-behaved). See for instance Prop. 7.5.3 in Roger Carter's 1985 book Finite Groups of Lie Type, based on the Deligne-Lusztig theory. (Srinivasan's special case predates the 1976 Deligne-Lusztig paper but along with Green's work on finite general linear groups helped to motivate the later developments.)

It seems that the answer to your question is no, based on a quick look at the character table of $\mathrm{Sp}_4(\mathbb{F}_q)$ worked out by Bhama Srinivasan in Trans. Amer. Math. Soc. 131 (1968), available online here. In her notation, the regular semisimple elements in a split maximal torus belong to the classes labelled $B_3(i,j)$. The two unipotent characters $\theta_1, \theta_3$ don't occur as constituents of the character induced from the trivial character of the split Borel subgroup (hence are not in the principal seris), but they do take nonzero values at such regular semisimple elements.

However, your question "almost" has an affirmative answer, which makes finite general linear groups tricky for experimental purposes (their characters are too well-behaved). See for instance Prop. 7.5.3 in Roger Carter's 1985 book Finite Groups of Lie Type, based on the Deligne-Lusztig theory. (Srinivasan's special case predates the 1976 Deligne-Lusztig paper but along with Green's work on finite general linear groups helped to motivate the later developments.)