In Carter's book (Finite groups of Lie type), he reviews the truncated induction procedure (called j-operation in the text) of Macdonald-Lusztig-Spaltenstein in great detail for the classical Weyl groups. But, as far as I can see, he doesn't say much about how this operation works in the exceptional cases. My questions are : 


1) What is the map between $W\_s$ (a subgroup of an exceptional Weyl group, together with a choice of rep if other than sign) and the irrep obtained as $Irr(W) = j^W\_{W\_S}$ ? 


2) Which subset of $Irr(W)$ arise as Macdonald representations in the exceptional cases ? (For Macdonald representations, the $W\_s$ is the Weyl group attached to a subsystem and the induction is from the sign rep of $W\_s$). 



Truncated induction for exceptional cases
Maybe I can partially answer your second question by refocusing it somewhat. Carter's chapters 11-13 cover a lot of ground and were hard to organize in a straight line fashion, but the main theme is the study of *unipotent* characters of a finite group of Lie type (consisting of fixed points in a suitable algebraic group under a Frobenius map). This turns out to involve the finitely many unipotent classes of the algebraic groups along with a close study of Weyl group representations using Springer theory. When Carter wrote his book not all details had been completed, but since then the entire theory has been refined by Lusztig and others.


Now fix a Weyl group $W$ (or more generally a finite Coxeter group), which can be assumed to have a connected Coxeter-Dynkin diagram. In his 1971 note in *Bull. London Math. Soc.* Macdonald developed an elementary method for constructing most (not necessarily all) irreducible representations of $W$, using its standard realization as a finite reuclidean refletion group. Here there is a truncated induction process from a known irreducible representaiton of a proper reflection subgroup. 


Later in the 1970s there were important papers by Deligne-Lusztig along with one in 1979 by Lusztig-Spaltenstein (on which Carter bases part of his Chapter 11). A two-part paper in *Indag. Math.* (1979, 1982) by Lusztig dealt with the construction of Weyl group representations, going beyond Macdonald's elementary method by incorporating Springer's deep 1976 study of Weyl group representations in the context of unipotent classes (or nilpotent orbits in the Lie algebra). Much of this theory is at least outlined in Carter's last chapters.


Along the way Springer representations tend to supersede Macdonald representations of $W$. Briefly, all irreducible representations of $W$ occur in the top cohomology of Springer fibers (tensored with possibly nontrivial characters of component groups) for a desingularization of the nilpotent variety. Those representations involving only the trivial character of a component group are *Springer representations* forming a collection $\overline{\mathcal{S}}\_W$, which in turn contains a subset $\mathcal{S}\_W$ of *special representations*. It turns out that the Springer representations are in natural bijection with nilpotent orbits, while special ones correspond to special orbits (not easy to characterize intrinsically) and there is a natural duality on the latter orbits generalizating the usual one for partitions in type $A$. Lusztig showed that at least the Springer representations can be constructed using the Lusztig-Spaltenstein method.


In Carter's last chapters you can find detailed information about the Springer representations along with the special ones. For instance, in type $G\_2$, there are six irreducible representations of the dihedral group $W$ of order 12, of which five are Springer and only three special. In type $E\_8$, there are 112 irreducible representations of $W$, of which 70 are Springer and in turn 46 are special. 


ADDED: A more up-to-date book than Carter's is *Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras* (London Mathematical Society Monographs, Oxford, 2000) by Geck and Pfeiffer. They call Lusztig-Spaltenstein truncated induction "$j$-induction". 


[EDIT] Software developed for the *Atlas of Lie Groups* [here](http://liegroups.org/web/) shows that Macdonald's original approach constructs all irreducible representations of $W$ in type $G\_2$ but
for other exceptional types is not closely related to Springer theory. (In classical types other than $D\_n$, Macdonald showed that his method gives all irreducibles.)


