A REMARK ON THE IRREDUCIBLE CHARACTERS AND FAKE DEGREES OF FINITE REAL REFLECTION GROUPS

Beynon and Lusztig have shown that the fake degrees of almost all irreducible characters of finite real reflection groups are palindromes, and that the exceptions to this rule correspond to the non rational characters of the generic ring A defined over R = C[q]. Their proof consists of a case-by-case check. In this note we give an explanation for this phenomenon and some related facts about fake degrees. Moreover, in the situation where we allow for distinct parameters q(alpha) in the definition of A, we shall give a simple uniform proof of the fact that all the central idempotents of A(($) over bar K) are elements of A(($) over tilde K) where ($) over tilde K = C[root q(alpha)].