$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$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 $F_q$.

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one of two people whose names differ only by spaces. @mathseeker, if you are the same as the author of this post, then you should flag the post and ask the moderators to merge your account. If you arenotthe same, then your edit (which I approved thinking you were the same!) was excessive and should not have been made. $\endgroup$