Relating Deligne-Lusztig virtual representation characters to Green functions

Question

I have 2 questions - the first is what the title refers to, and the second is something I want a reference on (I thought I'd include them in one post since they are very strongly related). Sorry this post is a bit long, I tried to put as much as detail as I could ..

$1$-st question: I'm interested only in the group $GL_n(F_q)$. In Carter's book "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters", in Chapter 7 "The generalized characters of Deligne-Lusztig", the construction of the virtual representations $R_{T, \theta}$ as alternating sums of $l$-adic cohomology of Deligne-Lusztig varieties is given in some details, and a series of formulae are proved in the chapter about these ($T$ a torus, and $\theta$ a character of $T^{F}$). It says that if $\theta \in \widehat{T^{F}}$ is in general position, then $\pm R_{T, \theta}$ is irreducible. The following formula is given (also in http://en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory), where $g=su=us$, $s,u$ being the semisimple and unipotent parts, and $Q_{T}(u) = R_{T, 1}(u)$, $C^{0}(s)$ being the identity connected component of the centralizer of $s$, and $F$ the Frobenius endomorphism.

$ R_{T, \theta}(g) = \frac{1}{ | C^{0}(s)^{F} |} \sum_{ x \in G^{F}, x^{-1}sx \in T^{F} } \theta ( x^{-1} s x) Q_{x T x^{-1}}^{C^{0}(s)} (u) $

The book then says that $Q_{T}(u)$ is a Green function, depends only on the torus (I understand it will not change if we conjugate the torus in $G^F$ either so essentially corresponds to an element of $S_n$ for the group general linear group of size $n$, which is what I'm most curious about; unless I'm mistaken). The book does not give an explicit formulae for these $Q_{T}(u)$, but it does give orthogonality relations and such - explicit formulae is what I"m looking for:

Question: What's an explicit formulae for these $Q_{T}(u)$? How does this relate to the Green function that I've been studying from in Macdonald's book "Symmetric Functions and Hall polynomials", in the chapter "Characters of $GL_n$ over a finite field" -i.e., how do I express the character $ \pm R_{T, \theta}$ as a sum of the irreducible characters described by Green functions in Macdonald's book (or a single irreducible character in the case where $\theta$ is in general position)?

In that book, I've learnt that the polynomials correspond to symmetric functions $S_{\lambda}$, via a correspondence that maps $A$, the sums of the representation ring of for all $n$, to $B$, an algebra generated by elementary symmetric functions in independent variables $X_{i,f}$ ($f$ ranges over all irreducible polynomials in $\mathbb{F}_{q}[t]$). I'm sorry I'm being a bit vague right here - it would take pages to define precisely all the notation that Macdonald uses in his book; feel free to work with any alternative explicit definitions of these Green functions (but please include a reference so I know where to look it up).

$2$-nd question: I have looked through Carter's book and Digne&Michel's book on the same topic, but I have been unable to find a reference which gives the representing matrices for these virtual representations $\pm R_{T, \theta}$ of these finite Lie type groups (the fact that they are defined with alternating sum complicates matters somewhat). I'm not so interested in the entries of the representing matrices as such, just a construction for the module which enables you to find the representing matrices. Can anyone suggest a good reference for this? The closest I can find is Lusztig's original book "Characters of reductive groups over finite fields", where it mentions that $l$-adic intersection homology can be used as a substitute (this was from what I can see in googlebooks preview); but I hear this book is horrible to learn from, and I'm not entirely certain if what's given there is what I'm looking for (I don't have a copy of the book at present).

Answer 1

The Green function $Q_T(u)$ is the value of the Deligne-Lusztig character $R_T^\theta$ at $u$ (a unipotent element), which turns out not to depend on $\theta$, hence the notation. Conjugacy classes of rational tori in $GL_n$ are parametrized by conjugacy classes in the symmetric group, so this means you have one Green's function for each partition of $n$. For $GL_n$ they are (I assume) what Macdonald calls Green functions (I don't have a copy of the book to hand), but regardless they can be made combinatorially explicit in various ways.

To relate to Macdonald's book/Green's paper, you can use the orthogonality formulas for the $R_T^1$s to show that taking appropriate linear combinations of them according to character values of the symmetric group, you get an orthonormal set of class functions. You can then check that these are irreducible characters, and in fact they are just the representations of $GL_n(\mathbb F_q)$ which you get as constituents of $\text{Ind}_B^G(1)$, where $B$ is the set of upper triangular matrices, the so-called unipotent representations. Thus the change of basis matrix for class functions on the unipotents between Green functions and irreducible unipotent characters is just given by the character table of $S_n$. I think this is worked out in the book by Digne and Michel in one of the later chapters on examples, if you need a reference.