Generalized Q-functions for GKM

Recently we explained that the classical Q Schur functions stand behind various well known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with monomial potential Xn+1. We suggest to use the Hall-Littlewood polynomials at the parameter equal to the n-th root of unity as a generalization of the Q Schur functions from n=2 to arbitrary n>2. They are associated with n-strict Young diagrams and are independent of time-variables pkn with numbers divisible by n. These are exactly the properties possessed by the generalized Kontsevich model (GKM), thus its partition function can be expanded in such functions Q(n). However, the coefficients of this expansion remain to be properly identified. At this moment, we have not found any “superintegrability” property ∼character, which expressed these coefficients through the values of Q at delta-loci in the n=2 case. This is not a big surprise, because for n>2 our suggested Q functions are not looking associated with characters.