Leonard triples and hypercubes

Let ▫$V$▫ denote a vector space over ▫$mathbb{C}$▫ with finite positive dimension. By a Leonard triple on ▫$V$▫ we mean an ordered triple of linear operators on ▫$V$▫ such that for each of these operators there exists a basis of ▫$V$▫ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let ▫$D$▫ denote a positive integer and let ▫${mathcal{Q}}_D$▫ denote the graph of the ▫$D$▫-dimensional hypercube. Let ▫$X$ denote the vertex set of ▫${mathcal{Q}}_D$▫ and let ▫$A in {mathrm{Mat}}_X ({mathbb{C}})$▫ denote the adjacency matrix of ▫${mathcal{Q}}_D$▫. Fix ▫$x in X$▫ and let ▫$A^ast in {mathrm{Mat}}_X({mathbb{C}})$▫ denote the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫${mathrm{Mat}}_X({mathbb{C}})$ generated by ▫$A,A^ast$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫${mathcal{Q}}_D$▫ with respect to ▫$x$▫. The matrices ▫$A$▫ and ▫$A^ast$▫ are related by the fact that ▫$2iA = A^ast A^varepsilon - A^varepsilon A^ast$▫ and ▫$2iA^ast = A^varepsilon A - AA^varepsilon$▫, where ▫$2iA^varepsilon = AA^ast - A^ast A$▫ and ▫$i^2 = -1$▫. We show that the triple ▫$A$▫, ▫$A^ast$▫, ▫$A^varepsilon$▫ acts on each irreducible ▫$T$▫-module as a Leonard triple. We give a detailed description of these Leonard triples