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$\DeclareMathOperator\GL{GL}$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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For character values at semisimple elements there is a neat way to do it.

Step-1.

A classical theorem of Lusztig and Srinivasan expresses any irreducible character of $\mathrm{GL}_n(\mathbb{F}_q)$ as a linear combination of Deligne--Lusztig characters. But note that it may take some works to write out the coefficients explicitly (I think you need the information of rational semisimple conjugacy classes and some representation theory of symmetric groups). For details, see e.g. 11.7.3 in Digne--Michel's book Representations of finite groups of Lie type (2nd edition).

Step-2.

Apply the character formula of Deligne--Lusztig characters at semisimple elements. See 7.5.3 in Carter's book Finite groups of Lie type.

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  • $\begingroup$ Thank you so much for your answer. The reference in Carter's book is a little bit understandable to me; however, I couldn't spot 11.7.3 in Digne-Michel's book. I mean there is 11.7, but it is a theorem on something else. Could you please rectify this reference, if needed, otherwise please let me know another reference. $\endgroup$
    – user300
    Commented Aug 13 at 13:36
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    $\begingroup$ @user300 It is Thm-11.7.3, not 11.7 (note that the citation is for the 2nd edition of the book, not the 1st edition; these two editions have very different numberings). Alternatively, you can take a look at Corollary~2.4.19 in Geck--Malle's book "Character theory of finite groups of Lie type", or Lusztig--Srinivasan's original paper (it should be a short paper with similar title). $\endgroup$
    – user148212
    Commented Aug 13 at 13:56

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