Extending the parking space

The action of the symmetric group SnSn on the set ParknParkn of parking functions of size n   has received a great deal of attention in algebraic combinatorics. We prove that the action of SnSn on ParknParkn extends to an action of Sn+1Sn+1. More precisely, we construct a graded Sn+1Sn+1-module VnVn such that the restriction of VnVn to SnSn is isomorphic to ParknParkn. We describe the SnSn-Frobenius characters of the module VnVn in all degrees and describe the Sn+1Sn+1-Frobenius characters of VnVn in extreme degrees. We give a bivariate generalization Vn(ℓ,m) of our module VnVn whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.