Self-dual Leonard pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider a pair A, A* of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A*. Such an automorphism is unique, and called the duality A ↔ A*. In the present paper we give a comprehensive description of this duality. In particular,we display an invertible F-linearmap T on V such that the map X → TXT −1 is the duality A ↔ A*. We express T as a polynomial in A and A*. We describe how T acts on 4 flags, 12 decompositions, and 24 bases for V.