An A-invariant subspace for bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint 2, both thin

Let (Gamma) denote a bipartite distance-regular graph with vertex set X, diameter D >= 4, and valency k >= 3. Let C-X denote the vector space over C consisting of column vectors with entries in C and rows indexed by X. For z is an element of X, let (z) over cap denote the vector in C-X with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of Gamma and let T = T (x) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T -modules with endpoint 2, and they both are thin. Fix y is an element of X such that partial derivative(x, y) = 2, where partial derivative denotes path-length distance. For 0 <= i, j <= D define w(ij) = Sigma(z) over cap, where the sum is over all z is an element of X such that partial derivative(x, z) = i and partial derivative(y, z) = j. We define W = span{w(ij) vertical bar 0 <= i, j <= D}. In this paper we consider the space MW = span{mw vertical bar m is an element of M, w is an element of W}, where M is the Bose-Mesner algebra of Gamma. We observe that MW is the minimal A-invariant subspace of C-X which contains W, where A is the adjacency matrix of Gamma. We show that 4D - 6 <= dim(MW) <= 4D - 2. We display a basis for MW for each of these five cases, and we give the action of A on these bases.