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Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having asvertices the maximal isotropic subspaces of a finite-dimensional vector spaceover the finite field $\mathbb{F}_q$ equipped with a non-degenerate form(alternating, quadratic, or Hermitian) with Witt index $D$. From a pair of avertex $x$ 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 withrespect to $x$, $C$. Using this module, we then define the non-symmetric dual$q$-Krawtchouk polynomials and derive their recurrence and orthogonalityrelations from the combinatorial points of view. We note that our results donot depend essentially on the particular choice of the pair $x$, $C$, and thatall the formulas are described in terms of $q$, $D$, and one other scalar whichwe assign to $\Gamma$ based on the type of the form.

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