Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$ and Bose-Mesner algebra $M$. For $\theta\in C\cup \infty$ we define a 1 dimensional subspace of $M$ which we call $M(\theta)$. If $\theta\in C$ then $M(\theta)$ consists of those $Y$ in $M$ such that $(A-\theta I)Y\in C A_D$, where $A$ (resp. $A_D$) is the adjacency matrix (resp. $D$th distance matrix) of $\Gamma.$ If $\theta = \infty$ then $M(\theta)= C A_D$. By a {\it pseudo primitive idempotent} for $\theta$ we mean a nonzero element of $M(\theta)$. We use pseudo primitive idempotents to describe the irreducible modules for the Terwilliger algebra, that are thin with endpoint one.