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 nonrational characters of the generic ring \(A\) defined over \(R=\mathbb{C}[q]\). 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 give a simple uniform proof of the fact that all the central idempotents of \(A^{\overline {K}}\) are elements of \(A^{\widetilde {K}}\), where \(\widetilde{K}=\mathbb{C}[\sqrt{q_ \alpha}]\).
scientific article; zbMATH DE number 784011
scientific article