The Terwilliger algebra of a distance-regular graph of negative type

Let Γ denote a distance-regular graph with diameter D⩾3. Assume Γ has classical parameters (D,b,α,β) with b<-1. Let X denote the vertex set of Γ and let A∈MatX(C) denote the adjacency matrix of Γ. Fix x∈X and let A∗∈MatX(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of MatX(C) generated by A,A∗. We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions are D and 2D-2. For these T-modules we display a basis consisting of eigenvectors for A∗, and for each basis we give the action of A.