  Let $W$ be an irreducible finite complex reflection group acting on a complex
vector space $V$. For a positive integer $k$, we consider a class function
$\varphi_k$ given by $\varphi_k(w) = k^{\dim V^w}$ for $w \in W$, where $V^w$
is the fixed-point subspace of $w$. If $W$ is the symmetric group of $n$
letters and $k=n+1$, then $\varphi_{n+1}$ is the permutation character on
(classical) parking functions. In this paper, we give a complete answer to the
question when $\varphi_k$ (resp. its $q$-analogue) is the character of a
representation (resp. the graded character of a graded representation) of $W$.
As a key to the proof in the symmetric group case, we find the greatest common
divisors of specialized Schur functions. And we propose a unimodality
conjecture of the coefficients of certain quotients of principally specialized
Schur functions.
