The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups

In this paper, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod $q$ hyperplane arrangements. We prove that the permutation character is a quasi-polynomial in $q$, and show that it can be expressed by the sum of the induced characters of an equivariant version of the Ehrhart quasi-polynomials. Furthermore, we consider the case of the Coxeter arrangements, and compute in detail for type $A_{\mathrm{\ell }}$.