On the Existence of Generalized Parking Spaces for Complex Reflection Groups

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 $φ_k$ given by $φ_k(w) = k^{\dim V^w}$ 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 $φ_{n+1}$ is the permutation character on (classical) parking functions. In this paper, we give a complete answer to the question when $φ_k$ (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.

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