A $Q$-polynomial structure for the Attenuated Space poset $\mathcal A_q(N,M)$

The goal of this article is to display a $Q$-polynomial structure for the Attenuated Space poset $\mathcal A_q(N,M)$. The poset $\mathcal A_q(N,M)$ 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 $\mathcal A_q(N,M)$ 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 $A, A^* \in {\rm Mat}_X(\mathbb C)$ with the following entries. For $y, z \in X$ the matrix $A$ has $(y,z)$-entry $1$ (if $y$ covers $z$); $q^{{\rm dim}\,y}$ (if $z$ covers $y$); and 0 (if neither of $y,z$ covers the other). The matrix $A^*$ is diagonal, with $(y,y)$-entry $q^{-{\rm dim}\,y}$ 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 ${\rm Mat}_X(\mathbb C)$ generated by $A, A^*$. We show that $A,A^*$ act on each irreducible $T$-module as a Leonard pair.