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Values of unipotent characters on non-unipotent elements of $GL_n$\operatorname{GL}_n(q)$

I$\DeclareMathOperator\GL{GL}$I had posted this question on stackexchange, but couldn't get any answer; So I am posting it here.

Let $GL_n(q)$$\GL_n(q)$ denote the finite general linear group over a field of size $q$, and $\chi$ be a unipotent character corresponding to a partition $\lambda$ of $n$. I am currently reading the paper "A Geometric Approach to the Representations of the Full Linear Group Over a Galois Field""A geometric approach to the representations of the full linear group over a Galois field" by R. Steinberg, and I am assuming that the characters defined there are indeed the '(twisted) unipotent ' characters. Please correct me if I am wrong.

Now, if we choose the partition $(1^n)$, we end up having a special character which is known as Steinberg character and the character values of this are well known; for any non-semisimple element it is zero, and for semisimple element $g$ it is (up to sign) size of the Sylow $p$-subgroup of the centralizer of $g$ in $GL_n(q)$$\GL_n(q)$. Furthermore, the sign can be derived from Theorem 9.2 of the paper "Spherical Buildings and the Character of the Steinberg Representation""Spherical buildings and the character of the Steinberg representation" by C.W. Curtis, G.I. Lehrer, and J. Tits.

I do not understand this fully, but vaguely, for other unipotent characters the values at unipotent elements are given by Green functions for the case of $GL_n(q)$$\GL_n(q)$, and in general some cohomological description applies for other finite reductive groups.

Here is my question: what are the character values of unipotent characters (other than the Steinberg character) on non-unipotent elements of $GL_n(q)$$\GL_n(q)$, at least on the semisimple elements? A reference will be really helpful. I appreciate any help you can provide.

Values of unipotent characters on non-unipotent elements of $GL_n(q)$

I had posted this question on stackexchange, but couldn't get any answer; So I am posting it here.

Let $GL_n(q)$ denote the finite general linear group over a field of size $q$, and $\chi$ be a unipotent character corresponding to a partition $\lambda$ of $n$. I am currently reading the paper "A Geometric Approach to the Representations of the Full Linear Group Over a Galois Field" by R. Steinberg, and I am assuming that the characters defined there are indeed the '(twisted) unipotent ' characters. Please correct me if I am wrong.

Now, if we choose the partition $(1^n)$, we end up having a special character which is known as Steinberg character and the character values of this are well known; for any non-semisimple element it is zero, and for semisimple element $g$ it is (up to sign) size of the Sylow $p$-subgroup of the centralizer of $g$ in $GL_n(q)$. Furthermore, the sign can be derived from Theorem 9.2 of the paper "Spherical Buildings and the Character of the Steinberg Representation" by C.W. Curtis, G.I. Lehrer, and J. Tits.

I do not understand this fully, but vaguely, for other unipotent characters the values at unipotent elements are given by Green functions for the case of $GL_n(q)$, and in general some cohomological description applies for other finite reductive groups.

Here is my question: what are the character values of unipotent characters (other than the Steinberg character) on non-unipotent elements of $GL_n(q)$, at least on the semisimple elements? A reference will be really helpful. I appreciate any help you can provide.

Values of unipotent characters on non-unipotent elements of $\operatorname{GL}_n(q)$

$\DeclareMathOperator\GL{GL}$I had posted this question on stackexchange, but couldn't get any answer; So I am posting it here.

Let $\GL_n(q)$ denote the finite general linear group over a field of size $q$, and $\chi$ be a unipotent character corresponding to a partition $\lambda$ of $n$. I am currently reading the paper "A geometric approach to the representations of the full linear group over a Galois field" by R. Steinberg, and I am assuming that the characters defined there are indeed the '(twisted) unipotent ' characters. Please correct me if I am wrong.

Now, if we choose the partition $(1^n)$, we end up having a special character which is known as Steinberg character and the character values of this are well known; for any non-semisimple element it is zero, and for semisimple element $g$ it is (up to sign) size of the Sylow $p$-subgroup of the centralizer of $g$ in $\GL_n(q)$. Furthermore, the sign can be derived from Theorem 9.2 of the paper "Spherical buildings and the character of the Steinberg representation" by C.W. Curtis, G.I. Lehrer, and J. Tits.

I do not understand this fully, but vaguely, for other unipotent characters the values at unipotent elements are given by Green functions for the case of $\GL_n(q)$, and in general some cohomological description applies for other finite reductive groups.

Here is my question: what are the character values of unipotent characters (other than the Steinberg character) on non-unipotent elements of $\GL_n(q)$, at least on the semisimple elements? A reference will be really helpful. I appreciate any help you can provide.

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Values of unipotent characters on non-unipotent elements of $GL_n(q)$

I had posted this question on stackexchange, but couldn't get any answer; So I am posting it here.

Let $GL_n(q)$ denote the finite general linear group over a field of size $q$, and $\chi$ be a unipotent character corresponding to a partition $\lambda$ of $n$. I am currently reading the paper "A Geometric Approach to the Representations of the Full Linear Group Over a Galois Field" by R. Steinberg, and I am assuming that the characters defined there are indeed the '(twisted) unipotent ' characters. Please correct me if I am wrong.

Now, if we choose the partition $(1^n)$, we end up having a special character which is known as Steinberg character and the character values of this are well known; for any non-semisimple element it is zero, and for semisimple element $g$ it is (up to sign) size of the Sylow $p$-subgroup of the centralizer of $g$ in $GL_n(q)$. Furthermore, the sign can be derived from Theorem 9.2 of the paper "Spherical Buildings and the Character of the Steinberg Representation" by C.W. Curtis, G.I. Lehrer, and J. Tits.

I do not understand this fully, but vaguely, for other unipotent characters the values at unipotent elements are given by Green functions for the case of $GL_n(q)$, and in general some cohomological description applies for other finite reductive groups.

Here is my question: what are the character values of unipotent characters (other than the Steinberg character) on non-unipotent elements of $GL_n(q)$, at least on the semisimple elements? A reference will be really helpful. I appreciate any help you can provide.