The switching element for a Leonard pair

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. Let $v_0,v_1,...,v_d$ (resp. $w_0,w_1,...,w_d$) denote a basis for $V$ referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation $S: V \to V$ that sends $v_0$ to a scalar multiple of $v_d$, fixes $w_0$, and sends $w_i$ to a scalar multiple of $w_i$ for $1 \leq i \leq d$. We call $S$ the {\it switching element}. We describe $S$ from many points of view.