I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the character tables of the irreps.
I heard of those Mackey theory for a while (which should be helpful for the Weyl group over classical simple Lie Groups), but I failed to find some nice references for it. It will be very helpful if there is a nice account on the theory.
Best Answer
Let me add a useful reference book, probably no longer in print but found in many libraries: R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley-Interscience, 1985. The book includes a lot of information about the irreducible representations of Weyl groups (though not with complete proofs) in the context of characters of finite groups of Lie type, unipotent classes in simple algebraic groups, Springer correspondence, etc. Chapters 11 and 13 involve the Weyl groups most heavily. (Though the book by Chriss and Ginzburg goes much deeper into the geometry of the Springer correspondence, the treatment there is mostly limited to the case of symmetric groups which is often more straightforward than the general case; for this Carter gives a helpful overview.)
For example, the $25 \times 25$ character table of the Weyl group of type $F_4$ originally worked out by T. Kondo in a 1965 journal article is displayed on page 413 of the book, while the role of these characters in the Springer correspondence is summarized on page 428 (involving the previous study of unipotent classes for the simple algebraic group of type $F_4$). Like all character tables of Weyl groups, this one has entries in $\mathbb{Z}$. It's not at all easy in a case like $F_4$ to write down explicit integral matrices affording the irreducible representations, but fortunately the characters alone are sufficient for some applications like those developed by Carter. One caveat is that notation for conjugacy classes and characters differs in various sources.