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The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the Green functions. I also have a very basic understanding of perverse sheaves, but not tons of experience with geometric representation theory. I am struggling to read Lusztig's Orange book and I was wondering if there was a more accessible/modern introduction to the theory?

For those interested, here are my main goals in understanding this: I am working in theoretical computer science, and am trying to find combinatorial methods to express these characters. For example, J.A. Green's paper on the charcters of $GL_n(\mathbb{F}_q)$ gives such a formula. The end goal is to be able to do this for all finite groups of lie type. From what I understand, Shoji-Lusztig give an algorithm based on the Springer correspondence, but I am still struggling on how to compute the so-called "Y" functions. In the case we can find a split unipotent element, from what I have read there is a method of computing these functions, but I am unsure of what to do in the non-split case. Honestly an overview article that summarizes most of this work would be quite helpful.

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    $\begingroup$ For large $p$, the Green functions are trigonometric sums: see Kazhdan - Proof of Springer's hypothesis (MSN) and, for the application to characters, Springer - Trigonometric sums … (MSN). $\endgroup$
    – LSpice
    Commented May 16, 2019 at 19:52
  • $\begingroup$ This is useful, and I have seen Springer's work before. However, I believe that this approach still forces me to compute some cohomology groups at one point, which is too expensive complexity theory wise. $\endgroup$
    – Sheel Stueber
    Commented May 16, 2019 at 21:21

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I'm not quite sure what you are looking for, but Green's work (though combinatorial and influential) was only one of the inputs for the Deligne-Lusztig paper of 1976.

It might help for example to look at the paper of Chang-Ree on $G_2 (q)$ here or the more complicated case of Sp(4,q) for q odd treated by Green's student Bhama Srinivasan here. Both of rthese are attempts to implement what became Ian G. Macdonald's conjectural formulation leading to the Deligne-Lusztig paper.

While the Chang-Ree paper is less accessible, the 1968 paper by Srinivasan is freely available now from the AMS Transactions. Perhaps you'd also find her Lecture Notes in Mathematics 764 (Springer, 1979) helpful. In any case, the implementation of the Deligne-Lusztig procedure for constructing virtual characters of finite groups of Lie type is rather intricate in general and goes well beyond Green's early work.

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  • $\begingroup$ Thank you for the reply Prof Humphreys, I have seen the Srinivasan paper before and it was definitely helpful in understanding the theory. . Specifically I suppose what I am looking for is a method of computing the characters in an algorithmic sense without having to actually compute any cohomology groups, as the known algorithms for this are very expensive from the complexity theory perspective. Do you know if such a computation is possible? $\endgroup$
    – Sheel Stueber
    Commented May 16, 2019 at 21:26
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    $\begingroup$ I'm skeptical about combinatorial formulas for Deligne-Lusztig characters, since the subject involves a range of techniques from combinatorics to geometric representation theory. It might be worthwhile to look at the most recent preprint on arXiv by Meinolf Geck, where he looks for Green functions when $p$ is small. $\endgroup$
    – Jim Humphreys
    Commented May 20, 2019 at 20:35

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