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Séminaire Lotharingien de Combinatoire, 78B.42 (2017), 12 pp.

# Jae-Ho Lee and Hajime Tanaka

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

**Abstract.**
Let Γ be a dual polar graph with diameter *D* >= 3. From every
pair of a vertex of Γ and a maximal clique containing it, we
construct a 2*D*-dimensional irreducible module for a nil-DAHA of type
(*C*^{v}_{1},*C*_{1}).
Using this module, we define non-symmetric dual
*q*-Krawtchouk polynomials and describe their orthogonality
relations.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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