The Erd\H{o}s-Ko-Rado basis for a Leonard system

We introduce and discuss an Erd\H{o}s-Ko-Rado basis for the underlying vector space of a Leonard system $\Phi = (A; A^; {E_i}_{i=0}^d ; {E_i^ }_{i=0}^d)$thatsatisfiesamildconditionontheeigenvaluesof$A$and$A^*$. We describe the transition matrices to/from other known bases, as well as the matrices representing $A$ and $A^{\ast }$ with respect to the new basis. We also discuss how these results can be viewed as a generalization of the linear programming method used previously in the proofs of the “Erd\H{o}s-Ko-Rado theorems” for several classical families of $Q$-polynomial distance-regular graphs, including the original 1961 theorem of Erd\H{o}s, Ko, and Rado.