Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual $q$-Krawtchouk Polynomials

Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index $D$. From a pair of a vertex $x$ of $\Gamma$ and a maximal clique $C$ containing $x$, we construct a $2D$-dimensional irreducible module for a nil-DAHA of type $(C^{\vee}_1, C_1)$, and establish its connection to the generalized Terwilliger algebra with respect to $x$, $C$. Using this module, we then define the non-symmetric dual $q$-Krawtchouk polynomials and derive their recurrence and orthogonality relations from the combinatorial points of view. We note that our results do not depend essentially on the particular choice of the pair $x$, $C$, and that all the formulas are described in terms of $q$, $D$, and one other scalar which we assign to $\Gamma$ based on the type of the form.