QQ-polynomial distance-regular graphs with a1=0a1=0 and a2≠0a2≠0

Let ΓΓ denote a QQ-polynomial distance-regular graph with diameter D≥3D≥3 and intersection numbers a1=0a1=0, a2≠0a2≠0. Let XX denote the vertex set of ΓΓ and let A∈MatX(C) denote the adjacency matrix of ΓΓ. Fix x∈Xx∈X and let A∗∈MatX(C) denote the corresponding dual adjacency matrix. Let TT denote the subalgebra of MatX(C) generated by A,A∗A,A∗. We call TT the Terwilliger algebra   of ΓΓ with respect to xx. We show that up to isomorphism there exists a unique irreducible TT-module WW with endpoint 11. We show that WW has dimension 2D−22D−2. We display a basis for WW which consists of eigenvectors for A∗A∗. We display the action of AA on this basis. We show that WW appears in the standard module of ΓΓ with multiplicity k−1k−1, where kk is the valency of ΓΓ.