A Q-polynomial structure for the Attenuated Space poset Aq(N,M)

The goal of this article is to display a Q-polynomial structure for the Attenuated Space poset $A_q(N,M)$. The poset $A_q(N,M)$ is briefly described as follows. Start with an $(N+M)$-dimensional vector space H over a finite field with q elements. Fix an M-dimensional subspace h of H. The vertex set X of $A_q(N,M)$ consists of the subspaces of H that have zero intersection with h. The partial order on X is the inclusion relation. The Q-polynomial structure involves two matrices $A,A^{⁎}\in {\mathrm{Mat}}_X\left(C\right)$ with the following entries. For $y,z\in X$ the matrix A has $(y,z)$-entry 1 (if y covers z); $q^{\dim y}$ (if z covers y); and 0 (if neither of $y,z$ covers the other). The matrix $A^{⁎}$ is diagonal, with $(y,y)$-entry $q^{-\dim y}$ for all $y\in X$. By construction, $A^{⁎}$ has $N+1$ eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with $2N+1$ eigenspaces. We show that $A^{⁎}$ acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q-polynomial. We show that $A,A^{⁎}$ satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of ${\mathrm{Mat}}_X\left(C\right)$ generated by $A,A^{⁎}$. We show that $A,A^{⁎}$ act on each irreducible T-module as a Leonard pair.