The nucleus of a Q-polynomial distance-regular graph

Let Γ denote a Q-polynomial distance-regular graph with diameter D≥ 1. For a vertex x of Γ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix A of Γ and the dual adjacency matrix A^*=A^*(x) of Γ with respect to x. We introduce a T-module N = N(x) called the nucleus of Γ with respect to x. We describe N from various points of view. We show that all the irreducible T-submodules of N are thin. Under the assumption that Γ is a nonbipartite dual polar graph, we give an explicit basis for N and the action of A, A^* on this basis. The basis is in bijection with the set of elements for the projective geometry LD(q), where $GF(q)$ is the finite field used to define Γ.