A Linear Map Acts as a Leonard Pair with Each of the Generators of Usl2

Let ℱ denote an algebraically closed field with a characteristic not two. Fix an integer d≥3; let x, y, and z be the equitable basis of sl2 over ℱ. Let V denote an irreducible sl2-module with dimension d+1; let A∈EndV. In this paper, we show that if each of the pairs A,x, A,y, and A,z acts on V as a Leonard pair, then these pairs are of Krawtchouk type. Moreover, A is a linear combination of 1, x, y, and z.

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−1is 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.

ON THE WITT INDEX OF THE BILINEAR FORM DETERMINED BY A LEONARD PAIR

Let a,b,d be nonnegative integers such that a + b = d + 1. Let ℝ be the field of real numbers. We prove that there is always a Leonard pair A, A⋆ of d + 1 by d + 1 matrices over ℝ such that the associated bilinear form of P. Terwilliger on ℝd + 1 has the Witt index a - b.