## Generalized exponents and forms

##### J. Algebr. Comb. 22(1), 115-132 (2005)
DOI: 10.1007/s10801-005-6908-y

### Summary

Summary: We consider generalized exponents of a finite reflection group acting on a real or complex vector space $V$. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations ( $V$ tensor a linear character) using the theory of semi-invariant differential forms. Springer's theory of regular numbers gives a formula when the group is generated by dim $V$ reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.

### Keywords/Phrases

keywords reflection group, invariant theory, generalized exponents, Coxeter group, fake degree, hyperplane arrangement, derivations