Zeroes of characters of general linear group induced from certain characters of parabolic subgroups

Question

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which certain characters vanish. In particular, first let us consider $G = \GL(4,q)$ and the partition $(3,1)$ of $4$. Let $x_1$ be a cuspidal character of $\GL(3,q)$ and $x_2$ be a linear character of $\GL(1,q)$. If $x_1\otimes x_2$ denotes the tensor product of $x_1$ and $x_2$, then it is an irreducible character of the Levi complement $L_{(2,1)}$ which can be lifted to an irreducible character of the parabolic subgroup $P_{(3,1)}$. Clearly, the induced character $\Ind(x_1\otimes x_2)$ of $\GL(4,q)$ is irreducible. Then I am interested to know that on what type of conjugacy classes it would vanish and what is the phenomenon behind it that works for $n \geq 4$ also.

I have explored the same scenario for $\GL(3,q)$ and here for a cuspidal character $y_1$ of $\GL(2,q)$ and a linear character $y_2$ of $\GL(1,q)$, the induced character $\Ind(y_1\otimes y_2)$ vanishes on the conjugacy classes whose representatives have characteristic polynomial either equal to $\prod_{i=1}^{3}(x-\alpha_i),$ for distinct $\alpha_i \in F_q^{*}$; or it is an irreducible polynomial of degree 3 over $F_q$.

Answer 1

Your conjugacy classes in $\operatorname{GL}(3, q)$ are called regular split and regular elliptic, respectively. An induced character will always vanish on conjugacy classes that do not intersect the support of the inducing character; so every properly parabolically induced character vanishes on all regular elliptic conjugacy classes, and the vanishing on regular split elements is explained by the fact that, though they are conjugate into every Levi, they do not intersect the support of the character of a regular cuspidal representation. (I mean "regular" here in the Deligne–Lusztig sense, which includes all cuspidals for general linear groups.)