The nucleus of a Q-polynomial distance-regular graph

Let denote a Q-polynomial distance-regular graph with diameter D 1. For a vertex x of the corresponding subconstituent algebra T=T (x) is generated by the adjacency matrix A of and the dual adjacency matrix A^*=A^* (x) of with respect to x. We introduce a T-module N = N (x) called the nucleus of 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 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.