Parking spaces

Let W be a Weyl group with root lattice Q and Coxeter number h  . The elements of the finite torus Q/(h+1)QQ/(h+1)Q are called the W-parking functions, and we call the permutation representation of W on the set of W-parking functions the (standard) W-parking space. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new W-parking spaces, called the noncrossing parking space and the algebraic parking space, with the following features:
• They are defined more generally for real reflection groups.
• They carry not just W  -actions, but W×CW×C-actions, where C is the cyclic subgroup of W generated by a Coxeter element.
• In the crystallographic case, both are isomorphic to the standard W-parking space. Our Main Conjecture is that the two new parking spaces are isomorphic to each other as permutation representations of W×CW×C. This conjecture ties together several threads in the Catalan combinatorics of finite reflection groups. Even the weakest form of the Main Conjecture has interesting combinatorial consequences, and this weak form is proven in all types except E7E7 and E8E8. We provide evidence for the stronger forms of the conjecture, including proofs in some cases, and suggest further directions for the theory.