The switching element for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → 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 Leonard pair on V . Let {vi} d i=0 (resp. {wi} d i=0 ) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S : V → V that sends v0 to a scalar multiple of vd, fixes w0, and sends wi to a scalar multiple of wi for 1 ≤ i ≤ d. We call S the switching element. We describe S from many points of view.