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