Does anyone know please of some/few references, where people explain and construct explicitly some (finite-dimensional irreducible) complex representations of finite groups of Lie type? I am particularly interested in groups consisting of two by two matrices with entries in some finite field (and similar related finite groups), as well as their complex representations.
I know that there is a whole theory of such representations, but I am not familiar with algebraic groups, only with Lie groups (over $\mathbb{R}$ and over $\mathbb{C}$), and would like for the time being to study some "baby" examples.
Edit 1: as an update, I found online the following notes by Gerhard Hiss http://www.math.rwth-aachen.de/~Gerhard.Hiss/Preprints/StAndrewsBath09.pdf. I think this may be a good start for me.
There is a lot of sophisticated theory on the classification of these representations and their characters (Deligne-Lusztig and many others, the notes mentioned above give a short introduction).
But there seems to be not much published about concrete representations. In "practice" only baby examples occur because the smallest dimensions of non-trivial representations (complex or other non-defining characteristic) grow quickly with the group (e.g., over the complex numbers for $SL_2(q)$ with odd $q$ this is $(q-1)/2$ or for $E_8(q)$ it is $q(q^4-q^2+1)(q^8-q^6+q^4-q^2+1)(q^4+1)(q^8-q^4+1)(q^2+1)^2$).
One reference with explicit representations over the smallest possible field is Böge, S., Realisierung (p−1)-dimensionaler Darstellungen von PSL(2,p), Arch. Math. (Basel) 60 (1993), no. 2, 121–127.