Leonard pairs arising from dual polar spaces

Abstract

A Leonard pair is an ordered pair of diagonalizable linear transformations on a finite-dimensional vector space such that each of the two transformations acts on an eigenbasis for the other one in an irreducible tridiagonal form. In this paper, we consider Leonard pairs arising from a class of graded posets called dual polar spaces. Let P be a dual polar space with rank N and let T be its incident algebra (over the complex field) generated by the raising matrix R, the lowering matrix L and the projection matrices $F_i(0\le i\le N)$. Define two elements $A,A^{⁎}$ of T: $A={\sum }_{i=0}^{N-1}{\alpha }_{i}R{F}_i+{\sum }_{i=0}^N{\theta }_i{F}_i$ and $A^{⁎}={\sum }_{i=1}^N{\alpha }_i^{⁎}L{F}_i+{\sum }_{i=0}^N{\theta }_i^{⁎}{F}_i$. Assume $N\ge 8$. We first give a necessary and sufficient condition for A and $A^{⁎}$ to satisfy the so-called tridiagonal relations. Then these results allow us to further display a necessary and sufficient condition for A and $A^{⁎}$ acting on any irreducible T-module as a Leonard pair.