The Terwilliger Algebra of a Distance-Regular Graph of Negative Type

Let $\Gamma$ denote a distance-regular graph with diameter $D \ge 3$. Assume $\Gamma$ has classical parameters $(D,b,\alpha,\beta)$ with $b < -1$. Let $X$ denote the vertex set of $\Gamma$ and let $A \in MX$ denote the adjacency matrix of $\Gamma$. Fix $x \in X$ and let $A^* \in MX$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A, A^*$. We call $T$ the {\em Terwilliger algebra} of $\Gamma$ with respect to $x$. We show that up to isomorphism there exist exactly two irreducible $T$-modules with endpoint 1; their dimensions are $D$ and $2D-2$. For these $T$-modules we display a basis consisting of eigenvectors for $A^*$, and for each basis we give the action of $A$