### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 009, 27 pages      arXiv:1709.07825      https://doi.org/10.3842/SIGMA.2018.009

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

Jae-Ho Lee a and Hajime Tanaka b
a) Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA
b) Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

Received September 25, 2017, in final form January 29, 2018; Published online February 10, 2018

Abstract
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.

Key words: dual polar graph; nil-DAHA; dual $q$-Krawtchouk polynomial; Terwilliger algebra; Leonard system.

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