Grassmann graphs, degenerate DAHA, and non-symmetric dual q-Hahn polynomials

Abstract

We discuss the Grassmann graph Jq(N,D) with N≥2D, having as vertices the D-dimensional subspaces of an N-dimensional vector space over the finite field Fq. This graph is distance-regular with diameter D; to avoid trivialities we assume D≥3. Fix a pair of a Delsarte clique C of Jq(N,D) and a vertex x in C. We construct a 2D-dimensional irreducible module W for the Terwilliger algebra T of Jq(N,D) associated with the pair x, C. We show that W is an irreducible module for the confluent Cherednik algebra HV and describe how the T-action on W is related to the HV-action on W. Using the HV-module W, we define non-symmetric dual q-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint.