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

Let Γ be a dual polar graph with diameter D⩾3, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field Fq equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. From a pair of a vertex x of Γ and a maximal c...

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Date:	2018
Main Authors:	Lee, J.-H., Tanaka, H.
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Cite this:	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials / J.-H. Lee, H. Tanaka // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-2094552025-11-22T01:04:37Z Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials Lee, J.-H. Tanaka, H. Let Γ be a dual polar graph with diameter D⩾3, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field Fq equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. From a pair of a vertex x of Γ and a maximal clique C containing x, we construct a 2D-dimensional irreducible module for a nil-DAHA of type (C₁∨, C₁), 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 Γ based on the type of the form. The authors thank Daniel Orr for helpful comments on nil-DAHAs of type (C∨n, Cn). They also thank Paul Terwilliger for many valuable discussions, Marta Mazzocco and Alexei Zhedanov for bringing the authors’ attention to [21, 22], and the anonymous referees for carefully reading the paper. Part of this work was done while Jae-Ho Lee was visiting Tohoku University as a JSPS Postdoctoral Fellow. Hajime Tanaka was supported by JSPS KAKENHI Grant Numbers JP25400034 and JP17K05156. An extended abstract of this work appeared in the proceedings of FPSAC’17, London, UK, July 2017; cf. [18]. 2018 Article Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials / J.-H. Lee, H. Tanaka // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E30; 20C08; 33D45; 33D80 arXiv: 1709.07825 https://nasplib.isofts.kiev.ua/handle/123456789/209455 https://doi.org/10.3842/SIGMA.2018.009 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Let Γ be a dual polar graph with diameter D⩾3, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field Fq equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. From a pair of a vertex x of Γ and a maximal clique C containing x, we construct a 2D-dimensional irreducible module for a nil-DAHA of type (C₁∨, C₁), 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 Γ based on the type of the form.
format	Article
author	Lee, J.-H. Tanaka, H.
spellingShingle	Lee, J.-H. Tanaka, H. Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Lee, J.-H. Tanaka, H.
author_sort	Lee, J.-H.
title	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
title_short	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
title_full	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
title_fullStr	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
title_full_unstemmed	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
title_sort	dual polar graphs, a nil-daha of rank one, and non-symmetric dual q-krawtchouk polynomials
publisher	Інститут математики НАН України
publishDate	2018
url	https://nasplib.isofts.kiev.ua/handle/123456789/209455
citation_txt	Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials / J.-H. Lee, H. Tanaka // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT leejh dualpolargraphsanildahaofrankoneandnonsymmetricdualqkrawtchoukpolynomials AT tanakah dualpolargraphsanildahaofrankoneandnonsymmetricdualqkrawtchoukpolynomials
first_indexed	2025-11-27T19:18:00Z
last_indexed	2025-11-27T19:18:00Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 009, 27 pages Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials Jae-Ho LEE † and Hajime TANAKA ‡ † Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA E-mail: jaeho.lee@unf.edu URL: https://sites.google.com/site/jaeholeemath/ ‡ Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan E-mail: htanaka@tohoku.ac.jp URL: http://www.math.is.tohoku.ac.jp/~htanaka/ Received September 25, 2017, in final form January 29, 2018; Published online February 10, 2018 https://doi.org/10.3842/SIGMA.2018.009 Abstract. Let Γ be a dual polar graph with diameter D > 3, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field Fq equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt in- dex D. From a pair of a vertex x of Γ and a maximal clique C containing x, we construct a 2D-dimensional irreducible module for a nil-DAHA of type (C∨1 , C1), and establish its con- nection 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 Γ based on the type of the form. Key words: dual polar graph; nil-DAHA; dual q-Krawtchouk polynomial; Terwilliger algebra; Leonard system 2010 Mathematics Subject Classification: 05E30; 20C08; 33D45; 33D80 1 Introduction Q-polynomial distance-regular graphs are viewed as finite analogues of compact symmetric spaces of rank one, and have been extensively studied; cf. [1, 2, 10, 11]. By a famous theorem of Leonard [19], [1, Section 3.5], the duality property of Q-polynomial distance-regular graphs characterizes the terminating branch of the Askey scheme [14] of (basic) hypergeometric or- thogonal polynomials, at the top (i.e., 4φ3) of which are the q-Racah polynomials. A central tool in studying such a graph is the Terwilliger algebra T = T (x) [31, 32, 33], which is a non- commutative semisimple matrix C-algebra attached to every vertex x of the graph. Cherednik [3, 4, 5, 6] introduced the double affine Hecke algebras (DAHAs) for reduced affine root systems and used them to prove several conjectures for the Macdonald polynomials. Sahi [25] then extended the concept to the non-reduced affine root systems of type (C∨n , Cn) and proved the duality conjecture and other conjectures for the Koornwinder polynomials, which are the Macdonald polynomials attached to the affine root systems of type (C∨n , Cn). For n = 1, these polynomials are the Askey–Wilson polynomials which are of 4φ3, and the q-Racah polynomials are a discretization of the Askey–Wilson polynomials. mailto:jaeho.lee@unf.edu https://sites.google.com/site/jaeholeemath/ mailto:htanaka@tohoku.ac.jp http://www.math.is.tohoku.ac.jp/~htanaka/ https://doi.org/10.3842/SIGMA.2018.009 2 J.-H. Lee and H. Tanaka Recently, the first author [16] found a link between the theories of Q-polynomial distance- regular graphs and the DAHAs. Namely, he considered a Q-polynomial distance-regular graph Γ corresponding to q-Racah polynomials. He further assumed that Γ possesses a clique C with maximal possible size (called a Delsarte clique), and defined a semisimple matrix C-algebra T = T(x,C) attached to C and a vertex x ∈ C, which contains T (x) as a subalgebra. Then he showed that the so-called primary T-module has the structure of an irreducible module for the DAHA of type (C∨1 , C1), and studied how the two module structures are related. In the subse- quent paper [17], he captured in this context what should be called the non-symmetric q-Racah polynomials, which are the finite counterpart of the non-symmetric Askey–Wilson polynomials discussed by Sahi [25], and succeeded in giving an explicit combinatorial description of their orthogonality relations. We note that the expression for the non-symmetric q-Racah polyno- mials in [17] agrees with the one for the non-symmetric Askey–Wilson polynomials given by Koornwinder and Bouzeffour [15]. A big goal in this project is to establish a “non-symmetric version” of Leonard’s theorem mentioned above. As the next attempt towards this goal, we discuss the dual polar graphs in this paper, and specialize the above situation to this case. The dual polar graphs are a classical family of Q-polynomial distance-regular graphs arising naturally as homogeneous spaces of finite classical groups by maximal parabolic subgroups, and correspond to the dual q-Krawtchouk polynomials which are of 3φ2; cf. [26, 27]. In particular, we will obtain the non-symmetric dual q-Krawtchouk polynomials and describe their recurrence and orthogonality relations; cf. Theorems 9.5 and 10.7. There are multiple motivations for the present work. First, for the q-Racah case, there is indeed no known example of a Q-polynomial distance-regular graph with large diameter (say, at least ten) having such a maximal clique, except the ordinary polygons with even order, which we view as rather trivial. Hence we may say that the theory developed in [16, 17] still remains somewhat at the algebraic/parametric level, whereas we will deal with concrete (and non-trivial) combinatorial examples in this paper. Second, there are of course other candidates of examples to be considered, such as the Grassmann graphs corresponding to the dual q-Hahn polynomials which lie in between the q-Racah and the dual q-Krawtchouk polynomials, but we decided to focus on the dual polar graphs, mainly because they exhibit quite a strong regularity of being regular near polygons, so that the computations become far simpler than those in [16, 17]. Though many of our results can also be obtained in principle by taking appropriate limits of the (much involved) results in [16, 17], this fact motivates us to work out the details for this particular case rather independently of [16, 17]. Third, we will encounter a nil- DAHA of type (C∨1 , C1), which is obtained by specializing some of the defining relations of the DAHA of type (C∨1 , C1). The nil-DAHAs were introduced and discussed recently by Cherednik and Orr [7, 8, 9], and our results demonstrate the fundamental importance of the concept in the theory of Q-polynomial distance-regular graphs; cf. Theorems 8.13 and 8.14. We may remark that our results are also relevant to recent work by Mazzocco [21, 22]; cf. Remarks 8.4 and 8.15. The layout of the paper is as follows. In Section 2, we recall some background concerning Q-polynomial distance-regular graphs and their Terwilliger algebras. In Section 3, we recall the dual polar graphs and their properties. We fix a dual polar graph Γ with diameter D > 3, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field Fq equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index D. We fix a maximal clique C of Γ, and discuss the Terwilliger algebra of Γ with respect to C in the sense of Suzuki [28]. In Section 4, we also fix a vertex x ∈ C and define the generalized Terwilliger algebra T = T(x,C) of Γ. We then define from x and C a 2D-dimensional subspace W of the standard module for T, and show that it is an irreducible T-module. In Section 5, we recall the basic theory concerning Leonard systems, a certain linear algebraic framework for Leonard’s theorem introduced by Terwilliger [34]. We focus on the class of Leonard systems corresponding to the dual q-Krawtchouk polynomials and discuss their properties. To Non-Symmetric Dual q-Krawtchouk Polynomials 3 the T-module W we attach four Leonard systems, all of which belong to this class, and we study each of these Leonard systems in detail in Sections 6 and 7. In Section 8, we introduce a nil-DAHA H of type (C∨1 , C1), and define a 2D-dimensional representation H → End(W). Our first main results of this paper describe how the T-action on W is related to the H-action; cf. Theorems 8.13 and 8.14. In Section 9, we define the non-symmetric dual q-Krawtchouk polynomials `±i using the representation H → End(W) and discuss a role of these Laurent polynomials in W. In fact, we will obtain two expressions for the `±i . Recurrence relations involving at most four terms will also be given; cf. Theorem 9.5. The standard (Hermitian) inner product on W gives rise to an inner product on the 2D-dimensional vector space to which the `±i belong, which ultimately leads in Section 10 to our second main result of this paper, i.e., a combinatorial description of the orthogonality relations for the `±i ; cf. Theorem 10.7. 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 e which we assign to Γ based on the type of the non-degenerate form. Throughout this paper, we use the following notation. For a given non-empty finite set X, let MatX(C) be the C-algebra consisting of the complex square matrices indexed by X. Let V = VX be the C-vector space consisting of the complex column vectors indexed by X. We endow V with the inner product 〈u, v〉 = utv for u, v ∈ V , where t denotes transpose and ¯ denotes complex conjugate. We abbreviate ‖u‖2 = 〈u, u〉 for all u ∈ V . For every y ∈ X, let ŷ be the vector in V with a 1 in the y-coordinate and 0 elsewhere. For a subset Y ⊂ X, let Ŷ = ∑ y∈Y ŷ ∈ V denote its (column) characteristic vector. A Laurent polynomial f(η) ∈ C[η, η−1] in the variable η is said to be symmetric if f(η) = f(η−1), and non-symmetric otherwise. Note that the symmetric Laurent polynomials are precisely the polynomials in ξ := η+ η−1. Let q be a prime power. For r ∈ C and an integer n > 0, let (r; q)n = (1− r)(1− rq) · · · ( 1− rqn−1 ) , [ n 1 ] = [ n 1 ] q = qn − 1 q − 1 . For r1, r2, . . . , rk+1, s1, s2, . . . , sk ∈ C, let k+1φk ( r1, r2, . . . , rk+1 s1, s2, . . . , sk ∣∣∣∣ q, η) = ∞∑ n=0 (r1; q)n(r2; q)n · · · (rk+1; q)n (s1; q)n(s2; q)n · · · (sk; q)n ηn (q; q)n . 2 Distance-regular graphs Let Γ be a finite simple connected graph with vertex set X and diameter D. For x ∈ X, let Γi(x) = {y ∈ X : ∂(x, y) = i}, 0 6 i 6 D, where ∂ denotes the path-length distance. We abbreviate Γ(x) := Γ1(x). We call Γ distance- regular if there are non-negative integers ai, bi, ci, 0 6 i 6 D, called the intersection numbers of Γ, such that bD = c0 = 0, bi−1ci 6= 0, 1 6 i 6 D, and ai = \|Γi(x) ∩ Γ(y)\|, bi = \|Γi+1(x) ∩ Γ(y)\|, ci = \|Γi−1(x) ∩ Γ(y)\| for every pair of vertices x, y ∈ X with ∂(x, y) = i, where Γ−1(x) = ΓD+1(x) := ∅. Fig. 1 shows a small example of a distance-regular graph with D = 3. From now on, assume that Γ is distance-regular. The ith distance matrix of Γ is the 0-1 matrix Ai ∈ MatX(C) such that (Ai)x,y = 1 if and only if ∂(x, y) = i. The Bose–Mesner algebra of Γ is the subalgebra M of MatX(C) generated by the Ai. We note that M is semisimple since it is closed under conjugate-transposition. Observe also that A1Ai = bi−1Ai−1 + aiAi + ci+1Ai+1, 0 6 i 6 D, 4 J.-H. Lee and H. Tanaka Figure 1. The complement of the 2× 4-grid. from which it follows that, for 0 6 i 6 D, there is a polynomial vi ∈ C[ξ] with deg(vi) = i such that vi(A1) = Ai. It follows that the adjacency matrix A := A1 of Γ generates M , and that the Ai form a basis for M . In particular, we have dim(M) = D + 1. Since A is real symmetric and generates M , it has D + 1 mutually distinct real eigenvalues θ0, θ1, . . . , θD, which we call the eigenvalues of Γ. We will always set θ0 := b0, the valency (or degree) of Γ. For 0 6 i 6 D, let Ei ∈ MatX(C) be the orthogonal projection onto the eigenspace of θi. Then we have A = D∑ i=0 θiEi, and the Ei form another basis for M consisting of the primitive idempotents, i.e., EiEj = δi,jEi, D∑ i=0 Ei = I (the identity matrix). We note that E0V = CX̂. Observe that M is also closed under entrywise multiplication, denoted ◦. We say that Γ is Q-polynomial with respect to the ordering {Ei}Di=0 (or {θi}Di=0) if there are scalars a∗i , b ∗ i , c ∗ i , 0 6 i 6 D, such that b∗D = c∗0 = 0, b∗i−1c ∗ i 6= 0, 1 6 i 6 D, and E1 ◦ Ei = 1 \|X\| (b∗i−1Ei−1 + a∗iEi + c∗i+1Ei+1), 0 6 i 6 D, where we set b∗−1E−1 = c∗D+1ED+1 := 0. If this is the case, then for 0 6 i 6 D, there is a polynomial v∗i ∈ C[ξ] with deg(v∗i ) = i such that v∗i (\|X\|E1) = \|X\|Ei, where the multiplication is under ◦. In particular, if we write E1 = 1 \|X\| D∑ i=0 θ∗iAi, (2.1) then the θ∗i are (real and) mutually distinct. Note also that θ∗0 = trace(E1) = rank(E1). Assume that Γ is Q-polynomial with respect to the ordering {Ei}Di=0. Fix a ‘base vertex’ x ∈ X. For 0 6 i 6 D, let E∗i = E∗i (x) := diag(Aix̂) ∈ MatX(C). Note that E∗i E ∗ j = δi,jE ∗ i , D∑ i=0 E∗i = I. The E∗i form a basis for the dual Bose–Mesner algebra M∗ = M∗(x) of Γ with respect to x. We call A∗ = A∗(x) := \|X\|diag(E1x̂) ∈ MatX(C) the dual adjacency matrix of Γ with respect to x. We have A∗ = D∑ i=0 θ∗iE ∗ i , so that A∗ generates M∗, and that the θ∗i are the eigenvalues of A∗, which we call the dual eigenvalues of Γ. The Terwilliger (or subconstituent) algebra T = T (x) with respect to x is the subalgebra of MatX(C) generated by M , M∗ [31, 32, 33]. We note that T is again semisimple, is generated by A, A∗, and that any two non-isomorphic irreducible T -modules in V are orthogonal. The following are relations in T (cf. [31, Lemma 3.2]): E∗i AE ∗ j = EiA ∗Ej = 0 if \|i− j\| > 1, 0 6 i, j 6 D. (2.2) Non-Symmetric Dual q-Krawtchouk Polynomials 5 Observe also that E∗i X̂ = Aix̂, 0 6 i 6 D, from which it follows that the (D + 1)-dimensional subspace Mx̂ = M∗X̂ = D⊕ i=0 CAix̂ = D⊕ i=0 CEix̂ (2.3) of V is an irreducible T -module, called the primary T -module. We note that A.Aix̂ = bi−1Ai−1x̂+ aiAix̂+ ci+1Ai+1x̂, A∗.Aix̂ = θ∗iAix̂, 0 6 i 6 D. (2.4) We refer the reader to [1, 2, 10, 11] for more detailed information. 3 Dual polar graphs Let D be a positive integer. Let V be one of the following spaces over the finite field Fq equipped with a non-degenerate form:1 space dimension form e [CD(q)] 2D alternating 1 [BD(q)] 2D + 1 quadratic 1 [DD(q)] 2D quadratic (maximal Witt index D) 0 [2DD+1(q)] 2D + 2 quadratic (non-maximal Witt index D) 2 [2A2D(r)] 2D + 1 Hermitian (q = r2) 3 2 [2A2D−1(r)] 2D Hermitian (q = r2) 1 2 A subspace of V is called (totally) isotropic if the form vanishes completely on it. We note that maximal isotropic subspaces have dimension D. Let X be the set of all maximal isotropic subspaces of V. The dual polar graph (on V) has vertex set X, where two vertices x, y are adjacent if and only if dim(x∩y) = D−1; cf. [2, Section 9.4]. We have ∂(x, y) = D−dim(x∩y) in general, so that the diameter equals D. Assumption 3.1. For the rest of this paper, we will always assume that Γ is a dual polar graph with diameter D > 3. First we summarize some results that we need. The graph Γ is distance-regular with inter- section numbers ai = ( qe − 1 )[i 1 ] , bi = qi+e [ D − i 1 ] , ci = [ i 1 ] , 0 6 i 6 D. (3.1) Note that ai = cia1, 0 6 i 6 D. The eigenvalues of Γ are given by θi = qe [ D − i 1 ] − [ i 1 ] , 0 6 i 6 D, (3.2) and Γ is Q-polynomial with respect to the ordering2 {θi}Di=0, where θ0 > θ1 > · · · > θD. Moreover, the corresponding dual eigenvalues are given by θ∗i = q(1 + qD+e−2) 1− q + q(1 + qD+e−2)(1 + qD+e−1) (q − 1)(1 + qe−1) q−i, 0 6 i 6 D. (3.3) 1The scalar e is from [2, Section 9.4] and is assigned to V to give unified descriptions to various formulas regarding the dual polar spaces. 2The dual polar graph on [2A2D−1(r)] has another Q-polynomial ordering, which is θ0, θD, θ1, θD−1, . . . in terms of the natural ordering θ0 > θ1 > · · · > θD; cf. [1, p. 304]. However, it turns out that the maximal cliques do not behave well with this ordering (cf. [30, Theorem 8.8]), so that we will not pay attention to it in this paper. 6 J.-H. Lee and H. Tanaka See [2, Theorem 9.4.3] and [37, Lemma 16.5]. The dual polar graph Γ is an example of a regular near polygon (cf. [2, Section 6.4]), which means that Γ does not have (i.e., K1,1,2) as an induced subgraph, and that for every y ∈ X and a maximal clique C, there is a unique z ∈ C nearest to y, provided that ∂(y, C) < D. Note that the former condition implies that every edge lies in a unique maximal clique. We also note that Γ is more specifically a regular near 2D-gon,3 i.e., there is in fact no vertex y at distance D from C. Let C be a maximal clique in Γ. By the above comments, we have \|C\| = a1 + 2 = 1 + qe, (3.4) which attains the Hoffman bound 1 − θ0θ−1D (cf. [2, Proposition 4.4.6]). In other words, C is a so-called Delsarte clique. For 0 6 i 6 D − 1, define the ith distance neighbor of C by Ci = {y ∈ X : ∂(y, C) = i}. Then we have AiĈ = Ĉi + qeĈi−1, 0 6 i 6 D, (3.5) where C−1 = CD := ∅, from which it follows that MĈ = D−1⊕ i=0 CĈi. (3.6) In other words, {Ci}D−1i=0 is an equitable partition of X. In particular, there are non-negative integers ãi, b̃i, c̃i, 0 6 i 6 D− 1, called the intersection numbers of C, such that b̃D−1 = c̃0 = 0, b̃i−1c̃i 6= 0, 1 6 i 6 D − 1, and ãi = \|Ci ∩ Γ(y)\|, b̃i = \|Ci+1 ∩ Γ(y)\|, c̃i = \|Ci−1 ∩ Γ(y)\| for every y ∈ Ci. We may remark that a clique in a distance-regular graph satisfies (3.6) precisely when it is a Delsarte clique; cf. [11, Section 13.7]. Lemma 3.2. The following (i), (ii) hold: (i) c̃i = ci, 1 6 i 6 D − 1. (ii) b̃i = bi+1, 0 6 i 6 D − 2. Proof. (i) Let y ∈ Ci. Recall that there is a unique z in C ∩ Γi(y) since Γ is a regular near polygon. Then we have Ci−1 ∩ Γ(y) = Γi−1(z) ∩ Γ(y), and the result follows. (ii) With the above notation, pick any z′ ∈ C with z′ 6= z. Then z′ ∈ Γi+1(y), and we have Ci+1 ∩ Γ(y) = Γi+2(z ′) ∩ Γ(y), as desired. � Using (3.1), Lemma 3.2, and ãi + b̃i + c̃i = θ0, we have ãi = qe [ i+ 1 1 ] − [ i 1 ] , b̃i = qi+1+e [ D − i− 1 1 ] , c̃i = [ i 1 ] , 0 6 i 6 D − 1. (3.7) 3The identity ai = cia1, 0 6 i 6 D, mentioned above is a consequence of this; cf. [2, Theorem 6.4.1]. Non-Symmetric Dual q-Krawtchouk Polynomials 7 We now recall the Terwilliger algebra of Γ with respect to C in the sense of Suzuki [28]. For 0 6 i 6 D − 1, let Ẽ∗i = Ẽ∗i (C) := diag(Ĉi) ∈ MatX(C). Note that Ẽ∗i Ẽ ∗ j = δi,jẼ ∗ i , D−1∑ i=0 Ẽ∗i = I. The Ẽ∗i form a basis for the dual Bose–Mesner algebra M̃∗ = M̃∗(C) of Γ with respect to C. The dual adjacency matrix of Γ with respect to C is defined by (cf. [16, equation (50)]) Ã∗ = Ã∗(C) := \|X\| \|C\| diag(E1Ĉ) = 1 \|C\| ∑ y∈C A∗(y). From (2.1), (3.4), and (3.5), it follows that 1 \|C\| E1Ĉ = 1 \|X\| D−1∑ i=0 θ̃∗i Ĉi, (3.8) where θ̃∗i = 1 1 + qe θ∗i + qe 1 + qe θ∗i+1, 0 6 i 6 D − 1, so that (cf. [16, Lemma 4.11]) Ã∗ = D−1∑ i=0 θ̃∗i Ẽ ∗ i . By (3.3), we have θ̃∗i = q(1 + qD+e−2) 1− q + q(1 + qD+e−2)(1 + qD+e−1) (q − 1)(1 + qe) q−i, 0 6 i 6 D − 1. (3.9) In particular, Ã∗ has D mutually distinct real eigenvalues and hence generates M̃∗. The Ter- williger algebra T̃ = T̃ (C) with respect to C is the subalgebra of MatX(C) generated by M , M̃∗. We note that T̃ is semisimple and is generated by A, Ã∗. By virtue of (2.2), the following are relations in T̃ : Ẽ∗i AẼ ∗ j = EiÃ ∗Ej = 0 if \|i− j\| > 1, 0 6 i, j 6 D, (3.10) where we set Ẽ∗D := 0 for convenience. The subspace (3.6) of V is an irreducible T̃ -module with dimension D, called the primary T̃ -module. We note that A.Ĉi = b̃i−1Ĉi−1 + ãiĈi + c̃i+1Ĉi+1, Ã∗.Ĉi = θ̃∗i Ĉi, 0 6 i 6 D − 1. (3.11) Remark 3.3. The Bose–Mesner algebra M coincides in this case with the commutant of the corresponding classical group acting on X, whereas the Terwilliger algebras T , T̃ are subalgebras of those of maximal parabolic subgroups. See also [26, 27]. 4 The algebra T We continue to discuss the dual polar graph Γ. Assumption 4.1. For the rest of this paper, we will fix a vertex x ∈ X and a maximal clique C containing x. 8 J.-H. Lee and H. Tanaka C−0 C+ 0 C−1 C+ 1 C−2 C+ 2 C−3 C+ 3 C0 C1 C2 C3 Γ0 Γ1 Γ2 Γ3 Γ4 Figure 2. The partition {C±i } D−1 i=0 when D = 4. Definition 4.2 ([16, Definition 5.20]). The generalized Terwilliger algebra of Γ with respect to x, C is the subalgebra T = T(x,C) of MatX(C) generated by T = T (x), T̃ = T̃ (C). We note that T is semisimple and is generated by A, A∗, and Ã∗, where A∗Ã∗ = Ã∗A∗. We now refine the equitable partition {Ci}D−1i=0 of X as follows C−i = Γi(x) ∩ Ci, C+ i = Γi+1(x) ∩ Ci, 0 6 i 6 D − 1. See Fig. 2. For notational convenience, we set C−−1 = C+ −1 = C−D = C+ D := ∅. (4.1) Observe that Ci = C−i ∪ C + i , 0 6 i 6 D − 1, and that Γi(x) = C+ i−1 ∪ C − i , 0 6 i 6 D. Lemma 4.3. The following (i), (ii) hold: (i) For 0 6 i 6 D − 1 and z ∈ C−i , we have Y C−i−1 C+ i−1 C−i C+ i C−i+1 \|Γ(z) ∩ Y \| ci 0 ai bi − bi+1 bi+1 . (ii) For 0 6 i 6 D − 1 and z ∈ C+ i , we have Y C+ i−1 C−i C+ i C−i+1 C+ i+1 \|Γ(z) ∩ Y \| ci ci+1 − ci ai+1 0 bi+1 . In particular, the partition {C±i } D−1 i=0 is again equitable.4 Proof. Use Lemma 3.2. � Lemma 4.4. We have \|C−i \| = qie i∏ n=1 qD − qn qn − 1 , \|C+ i \| = q(i+1)e i∏ n=1 qD − qn qn − 1 , 0 6 i 6 D − 1. In particular, the C±i are non-empty. 4See [12, Proposition 1.3] for a more general result. Non-Symmetric Dual q-Krawtchouk Polynomials 9 Proof. It follows from Lemma 4.3 that ci\|C−i \| = bi\|C−i−1\|, ci\|C+ i \| = bi\|C+ i−1\|. The result follows from this and (3.1), together with \|C−0 \| = 1 and \|C+ 0 \| = qe (cf. (3.4)). � Let W be the linear span of the Ĉ±i . Thus, W has the following ordered orthogonal basis: C = { Ĉ−0 , Ĉ + 0 , Ĉ − 1 , Ĉ + 1 , . . . , Ĉ − D−1, Ĉ + D−1 } . (4.2) Proposition 4.5. The subspace W is an irreducible T-module. Proof. It is clear that W is closed under the actions of the E∗i and the Ẽ∗i . Moreover, since {C±i } D−1 i=0 is an equitable partition by Lemma 4.3, W is also A-invariant. It follows that W is a T-module. We now show the irreducibility. Since T is semisimple, W is an orthogonal direct sum of irreducible T-modules. Among these T-modules, there is one which is not orthogonal to CX̂ = E0W, denoted by W0. Then we have 0 6= E0W0 ⊂ E0W, so that X̂ ∈ E0W0 ⊂ W0, from which it follows that W = M∗M̃∗X̂ ⊂W0. Hence W0 = W, and the result follows. � We call W the primary T-module. Note that W contains both the primary T -module Mx̂ and the primary T̃ -module MĈ. Let Mx̂⊥ (resp. MĈ⊥) be the orthogonal complement of Mx̂ (resp. MĈ) in W. In Sections 6 and 7, we will show that Mx̂⊥ (resp. MĈ⊥) is also an irreducible T -module (resp. T̃ -module). Thus, W decomposes in two ways: W = Mx̂⊕Mx̂⊥ (orthogonal direct sum of irreducible T -modules) (4.3) = MĈ ⊕MĈ⊥ (orthogonal direct sum of irreducible T̃ -modules). (4.4) We end this section by describing the actions of A, A∗, and Ã∗ on W in terms of the basis C. Lemma 4.6. For 0 6 i 6 D − 1, we have A.Ĉ−i = qD+e − qi+e q − 1 Ĉ−i−1 + ( qe − 1 )qi − 1 q − 1 Ĉ−i + qiĈ+ i + qi+1 − 1 q − 1 Ĉ−i+1, A.Ĉ+ i = qD+e − qi+e q − 1 Ĉ+ i−1 + qe+iĈ−i + ( qe − 1 )qi+1 − 1 q − 1 Ĉ+ i + qi+1 − 1 q − 1 Ĉ+ i+1. Proof. From Lemma 4.3 it follows that A.Ĉ−i = biĈ − i−1 + aiĈ − i + (ci+1 − ci)Ĉ+ i + ci+1Ĉ − i+1, A.Ĉ+ i = biĈ + i−1 + (bi − bi+1)Ĉ − i + ai+1Ĉ + i + ci+1Ĉ + i+1 for 0 6 i 6 D − 1. Evaluate the two identities using (3.1). � Lemma 4.7. For 0 6 i 6 D − 1, we have A∗.Ĉ−i = θ∗i Ĉ − i , A∗.Ĉ+ i = θ∗i+1Ĉ + i , Ã∗.Ĉ±i = θ̃∗i Ĉ ± i . Proof. Clear. � 10 J.-H. Lee and H. Tanaka 5 Leonard systems of dual q-Krawtchouk type Let d be a positive integer, and let W be a vector space over C with dim(W) = d + 1. An element A ∈ End(W) is called multiplicity-free if it has d + 1 mutually distinct eigenvalues. Suppose that A is multiplicity-free, and let {θi}di=0 be an ordering of the eigenvalues of A. Then there is a sequence of elements {Ei}di=0 in End(W) such that AEi = θiEi, EiEj = δi,jEi, d∑ i=0 Ei = I, where I is the identity of End(W). We call Ei the primitive idempotent of A associated with θi. Definition 5.1. A Leonard system on W is a sequence Φ = ( A; A∗; {Ei}di=0; {E∗i }di=0 ) that satisfies the following axioms (LS1)–(LS4): (LS1) Each of A,A∗ is a multiplicity-free element in End(W). (LS2) {Ei}di=0 (resp. {E∗i }di=0) is an ordering of the primitive idempotents of A (resp. A∗). (LS3) E∗iAE∗j = EiA ∗Ej = 0 if \|i− j\| > 1, 0 6 i, j 6 d. (LS4) There is no proper subspace of W which is both A- and A∗-invariant. We call d the diameter of Φ. We note that the above definition is taken from [13, Definition 2.1] (and the paragraph following it), and is easily shown to be equivalent to the original definition in [34, Definition 1.4]. Let Φ = (A; A∗; {Ei}di=0; {E∗i }di=0) be a Leonard system on W. Note that each of the following is also a Leonard system on W: Φ∗ := ( A∗; A; {E∗i }di=0; {Ei}di=0 ) , Φ⇓ := ( A; A∗; {Ed−i}di=0; {E∗i }di=0 ) . (5.1) A Leonard system Ψ on a vector space W′ is isomorphic to Φ if there is a C-algebra isomorphism σ : End(W)→ End(W′) such that Ψ = Φσ := (Aσ; A∗σ; {Eσi }di=0; {E∗σi }di=0). For 0 6 i 6 d, let θi (resp. θ∗i ) be the eigenvalue of A (resp. A∗) associated with Ei (resp. E∗i ). Then there are non-zero scalars {φi}di=1 in C and a C-algebra isomorphism \ : End(W) → Matd+1(C) (the full matrix algebra) such that [34, Theorem 3.2] A\ =  θ0 0 1 θ1 1 θ2 · · · · 0 1 θd  , A∗\ =  θ∗0 φ1 0 θ∗1 φ2 θ∗2 · · · · φd 0 θ∗d  . We call {φi}di=1 the first split sequence of Φ. Let {φi}di=1 be the first split sequence of Φ⇓ and call this the second split sequence of Φ. The parameter array of Φ is the sequence( {θi}di=0; {θ∗i }di=0; {φi}di=1; {φi}di=1 ) . It is clear that the parameter array is a complete invariant for the isomorphism classes of Leonard systems. Terwilliger [36, Section 5] displayed all the parameter arrays of Leonard systems in parametric form. We now recall the dual q-Krawtchouk family of Leonard systems on which we will focus. Non-Symmetric Dual q-Krawtchouk Polynomials 11 Definition 5.2 ([36, Example 5.9]). The Leonard system Φ is said to have dual q-Krawtchouk type if there are scalars α, α∗, β, β∗, γ with ββ∗γ 6= 0 such that θi = α + βq−i + γqi, θ∗i = α∗ + β∗q−i for 0 6 i 6 d, and φi = ββ∗q1−2i ( 1− qi )( 1− qi−d−1 ) , φi = γβ∗qd+1−2i(1− qi)(1− qi−d−1) for 1 6 i 6 d. We call (α, α∗, β, β∗, γ ; q, d) the parameter sequence of Φ. Take a non-zero vector u ∈ E0W. By [35, Lemma 10.2], the vectors {E∗i u}di=0 form a basis for W, called a Φ-standard basis. In view of (LS3), (LS4), there are scalars ai, bi, ci, 0 6 i 6 d such that bd = c0 = 0, bi−1ci 6= 0, 1 6 i 6 d, and A.E∗i u = bi−1E∗i−1u + aiE ∗ i u + ci+1E∗i+1u, 0 6 i 6 d, (5.2) where b−1E∗−1u = cd+1E∗d+1u := 0. We call ai, bi, ci the intersection numbers of Φ. Observe that ai + bi + ci = θ0, 0 6 i 6 d. (5.3) By [35, Theorem 17.7], we have bi = φi+1 (θ∗i − θ∗0)(θ∗i − θ∗1) · · · (θ∗i − θ∗i−1) (θ∗i+1 − θ∗0)(θ∗i+1 − θ∗1) · · · (θ∗i+1 − θ∗i ) , 0 6 i 6 d− 1, (5.4) ci = φi (θ∗i − θ∗i+1)(θ∗i − θ∗i+2) · · · (θ∗i − θ∗d) (θ∗i−1 − θ∗i )(θ∗i−1 − θ∗i+1) · · · (θ∗i−1 − θ∗d) , 1 6 i 6 d. (5.5) Define a finite sequence of polynomials {vi}di=0 in C[ξ] with deg(vi) = i by v0 := 1 and ξvi = bi−1vi−1 + aivi + ci+1vi+1, 0 6 i 6 d− 1, (5.6) where b−1v−1 := 0. From (5.2) and (5.6) it follows that (cf. [35, Theorem 13.4]) vi(A).E∗0u = E∗i u, 0 6 i 6 d. Consider the following normalization: fi := vi vi(θ0) , 0 6 i 6 d, where using (5.3), (5.6), we have by induction (cf. [35, Lemma 13.2]) vi(θ0) = b0b1 · · · bi−1 c1c2 · · · ci ( 6= 0), 0 6 i 6 d. (5.7) By [35, Theorem 17.4], for 0 6 i 6 d we have fi = i∑ n=0 (θ∗i − θ∗0)(θ∗i − θ∗1) · · · (θ∗i − θ∗n−1)(ξ − θ0)(ξ − θ1) · · · (ξ − θn−1) φ1φ2 · · · φn . Define the scalars mi, 0 6 i 6 d, by mi = trace(EiE ∗ 0), 0 6 i 6 d, (5.8) 12 J.-H. Lee and H. Tanaka so that E∗0EiE ∗ 0 = miE ∗ 0, 0 6 i 6 d; cf. [35, Lemma 9.2]. By [35, Theorem 17.12], we have mi = φ1φ2 · · · φiφ1φ2 · · · φd−i (θ∗0 − θ∗1) · · · (θ∗0 − θ∗d)(θi − θ0) · · · (θi − θi−1)(θi − θi+1) · · · (θi − θd) . Assume now that Φ has dual q-Krawtchouk type as in Definition 5.2. Then we have5 bi = β ( 1− qi−d ) , ci = γ ( 1− qi ) , 0 6 i 6 d. (5.9) The values of the fi at ξ = θj are given by (cf. [36, Example 5.9]) fi(θj) = 3φ2 ( q−i, q−j , β−1γqj 0, q−d ∣∣∣∣ q, q) , 0 6 i, j 6 d. (5.10) It follows that the fi are the dual q-Krawtchouk polynomials [14, Section 14.17] in the variable ξ = ξ(x) := α + βq−x + γqx. We also have mi = βi γi (q−d; q)i(1− βγ−1q−2i) qi(i−1)(q; q)i(βγ−1q−d−i; q)d+1 , 0 6 i 6 d. (5.11) Following [15], we fix a non-zero scalar τ such that τ2 = β−1γ, (5.12) and renormalize the fi so that they are monic6 as symmetric Laurent polynomials in η: hi(η) := (q−d; q)i τi 3φ2 ( q−i, τη−1, τη 0, q−d ∣∣∣∣ q, q) , 0 6 i 6 d. (5.13) The hi depend on the parameters q, d, and τ, and we will write hi(η) = hi(η; τ, d; q), 0 6 i 6 d. (5.14) We note that hi has highest degree i and lowest degree −i in η, and that hi(η) = (q−d; q)i τi fi ( ξ ( logq τ−1η )) , 0 6 i 6 d. (5.15) Let W be a vector space over C containing W as a subspace, and let X ∈ End(W) be invertible such that (X + X−1)W ⊂W and that A = αI + βτ ( X + X−1 ) = αI + β ( τ−1X )−1 + γ ( τ−1X ) holds on W. Then it follows from (5.15) that hi(X)\|W = (q−d; q)i τi fi(A) = τi(q; q)ivi(A), where we have also used (5.7) and (5.9). In particular, we have hi(X).E∗0u = τi(q; q)iE∗i u, 0 6 i 6 d. (5.16) 5See also [31, Theorem 2.1]. In the notation of [31, Theorem 2.1], the Leonard systems of dual q-Krawtchouk type correspond to Case (I) with s∗ = r1 = r2 = 0. 6A Laurent polynomial in η is called monic if the coefficient of its highest degree term in η equals 1. Non-Symmetric Dual q-Krawtchouk Polynomials 13 6 Irreducible T -submodules of W Recall the primary T-module W. In this section, we focus on the decomposition (4.3) as T - modules. We first discuss the primary T -module Mx̂. Consider the sequence Φ := ( A;A∗; {Ei}Di=0; {E∗i }Di=0 )∣∣ Mx̂ , where \|Mx̂ means that each of the elements in the sequence is restricted to Mx̂. Proposition 6.1. The sequence Φ is a Leonard system on Mx̂ that has dual q-Krawtchouk type with parameter sequence (α, α∗, β, β∗, γ ; q,D), where7 α = qe − 1 1− q , β = qD+e q − 1 , γ = 1 1− q , α∗ = q(1 + qD+e−2) 1− q , β∗ = q(1 + qD+e−2)(1 + qD+e−1) (q − 1)(1 + qe−1) . Moreover, {Aix̂}Di=0 is a Φ-standard basis for Mx̂. Proof. It follows from (2.2) and (2.3) that Φ is a Leonard system on Mx̂. Moreover, {Aix̂}Di=0 is a Φ-standard basis for Mx̂, since Aix̂ = E∗i X̂, 0 6 i 6 D, and X̂ ∈ E0Mx̂. By virtue of (2.4) and (5.2), we routinely obtain the parameter array of Φ from (3.1), (3.2), (3.3) using (5.4), (5.5) (or (5.9)), and find that Φ has dual q-Krawtchouk type as given above. � We note that the intersection numbers of Φ coincide with those of Γ. We next consider the T -module Mx̂⊥, the orthogonal complement of Mx̂ in W. Observe that E0Mx̂⊥ = 0 since E0W = CX̂ ⊂Mx̂. From (2.1) and (3.4) it follows that \|\|E1x̂\|\|2 = θ∗0 \|X\| , 〈E1x̂, E1Ĉ〉 = 1 \|X\| (θ∗0 + qeθ∗1), \|\|E1Ĉ\|\|2 = 1 + qe \|X\| (θ∗0 + qeθ∗1), (6.1) from which it follows that w := c · ( E1Ĉ − θ∗0 + qeθ∗1 θ∗0 E1x̂ ) ∈ E1Mx̂⊥, where we take c := \|X\|(α∗ + β∗) α∗β∗qe−1(1− q) for convenience. By (2.1), (3.3), (3.8), and (3.9), we routinely obtain w = D−1∑ i=0 (( q−i − 1 ) Ĉ−i + ( qD−i−1 − 1 ) Ĉ+ i ) . Note that E∗0w = E∗Dw = 0. Let u⊥i := E∗i+1w = ( qD−i−1 − 1 ) Ĉ+ i + ( q−i−1 − 1 ) Ĉ−i+1, 0 6 i 6 D − 2. (6.2) Since dim(Mx̂⊥) = D − 1, the vectors {u⊥i } D−2 i=0 form an orthogonal basis for Mx̂⊥. Consider the sequence Φ⊥ := ( A;A∗; {Ei}D−1i=1 ; {E∗i }D−1i=1 )∣∣ Mx̂⊥ . 7See also [33, Example 6.1(6)]. 14 J.-H. Lee and H. Tanaka Proposition 6.2. The T -module Mx̂⊥ is irreducible, and Φ⊥ is a Leonard system on Mx̂⊥ that has dual q-Krawtchouk type with parameter sequence (α⊥, α∗⊥, β⊥, β∗⊥, γ⊥ ; q,D − 2), where8( α⊥, α∗⊥, β⊥, β∗⊥, γ⊥ ) = ( α, α∗, βq−1, β∗q−1, γq ) . Moreover, {u⊥i } D−2 i=0 is a Φ⊥-standard basis for Mx̂⊥. Proof. By Lemma 4.6, we have A.u⊥i = qD+e−1 − qi+e q − 1 u⊥i−1 + ( qe − 1 )qi+1 − 1 q − 1 u⊥i + qi+2 − q q − 1 u⊥i+1, 0 6 i 6 D − 2, (6.3) where u⊥−1 = u⊥D−1 := 0. Note that the coefficients of u⊥i−1, u ⊥ i+1 are non-zero. Every T - submodule of Mx̂⊥ is spanned by some of the u⊥i since it is M∗-invariant, and (6.3) then shows that it must be either Mx̂⊥ or 0. Hence Mx̂⊥ is an irreducible T -module. It also follows from (6.3) that Mx̂⊥ = Mu⊥0 . In particular, Mx̂⊥ is spanned by the vectors Eiu ⊥ 0 , 0 6 i 6 D. Note that E0u ⊥ 0 = 0 and E1u ⊥ 0 6= 0 since E0Mx̂⊥ = 0 and w ∈ E1Mx̂⊥ = CE1u ⊥ 0 . Hence it follows from (2.2) and the irreducibility of Mx̂⊥ that Eiu ⊥ 0 6= 0, 2 6 i 6 D − 1 and EDu ⊥ 0 = 0. From these comments and (2.2), it follows that Φ⊥ is a Leonard system on Mx̂⊥, and that {u⊥i } D−2 i=0 is a Φ⊥-standard basis for Mx̂⊥. Note that A∗.u⊥i = θ∗i+1u ⊥ i , 0 6 i 6 D − 2. The parameter array of Φ⊥ can now be computed as in the proof of Proposition 6.1. � We may remark that every irreducible T -module indeed affords a Leonard system of dual q-Krawtchouk type; cf. [37, Theorem 23.1]. Let π be the orthogonal projection from W onto Mx̂, i.e., (1− π)Mx̂ = 0, πMx̂⊥ = 0. Then we have Aπ = πA and A∗π = πA∗ on W. Observe that Ĉ+ i−1 = qi − 1 qD − 1 Aix̂+ qi qD − 1 u⊥i−1, Ĉ−i = qD − qi qD − 1 Aix̂+ qi 1− qD u⊥i−1 (6.4) for 1 6 i 6 D − 1, from which it follows that π.Ĉ+ i−1 = qi − 1 qD − 1 ( Ĉ+ i−1 + Ĉ−i ) , π.Ĉ−i = qD − qi qD − 1 ( Ĉ+ i−1 + Ĉ−i ) (6.5) for 1 6 i 6 D − 1. Moreover, we have π.Ĉ−0 = Ĉ−0 and π.Ĉ+ D−1 = Ĉ+ D−1. 7 Irreducible T̃ -submodules of W In this section, we focus on the decomposition (4.4) as T̃ -modules. We first discuss the primary T̃ -module MĈ. Consider the sequence Φ̃ := ( A; Ã∗; {Ei}D−1i=0 ; {Ẽ∗i }D−1i=0 )∣∣ MĈ . 8See also [29, Theorem 6.9], [32, Theorem 4.6]. Non-Symmetric Dual q-Krawtchouk Polynomials 15 Proposition 7.1. The sequence Φ̃ is a Leonard system on MĈ that has dual q-Krawtchouk type with parameter sequence (α̃, α̃∗, β̃, β̃∗, γ̃ ; q,D − 1), where9 (cf. Proposition 6.1)( α̃, α̃∗, β̃, β̃∗, γ̃ ) = ( α, α∗, β, 1 + qe−1 1 + qe β∗, γ ) . Moreover, {Ĉi}D−1i=0 is a Φ̃-standard basis for MĈ. Proof. We first remark that EDĈ = 0. Indeed, the Hoffman bound 1 − θ0θ −1 D on the size of a clique follows from the fact that ED is positive semidefinite, and the bound is attained precisely when the characteristic vector of the clique vanishes on EDV . From this comment, (3.6), and (3.10), it follows that Φ̃ is a Leonard system on MĈ. Moreover, {Ĉi}D−1i=0 is a Φ̃- standard basis for MĈ, since Ĉi = Ẽ∗i X̂, 0 6 i 6 D − 1, and X̂ ∈ E0MĈ. In view of (3.11), the parameter array of Φ̃ can be computed from (3.2), (3.7), and (3.9), as in the proof of Proposition 6.1. � We note that the intersection numbers of Φ̃ coincide with those of C. We next consider the T̃ -module MĈ⊥, the orthogonal complement of MĈ in W. From (6.1) it follows that w̃ := c̃ · ( E1x̂− 1 1 + qe E1Ĉ ) ∈ E1MĈ⊥, where we take c̃ := \|X\|q(1 + qe) β∗(1− q) for convenience. By (2.1), (3.3), (3.8), and (3.9), we obtain w̃ = D−1∑ i=0 ( −qe−iĈ−i + q−iĈ+ i ) . Let ũ⊥i := Ẽ∗i w̃ = −qe−iĈ−i + q−iĈ+ i , 0 6 i 6 D − 1. (7.1) Since dim(MĈ⊥) = D, the vectors {ũ⊥i } D−1 i=0 form an orthogonal basis for MĈ⊥. Consider the sequence Φ̃⊥ := ( A; Ã∗; {Ei}Di=1; {Ẽ∗i }D−1i=0 )∣∣ MĈ⊥ . Proposition 7.2. The T̃ -module MĈ⊥ is irreducible, and Φ̃⊥ is a Leonard system on MĈ⊥ that has dual q-Krawtchouk type with parameter sequence (α̃⊥, α̃∗⊥, β̃⊥, β̃∗⊥, γ̃⊥ ; q,D−1), where( α̃⊥, α̃∗⊥, β̃⊥, β̃∗⊥, γ̃⊥ ) = ( α, α∗, βq−1, 1 + qe−1 1 + qe β∗, γq ) . Moreover, {ũ⊥i } D−1 i=0 is a Φ̃⊥-standard basis for MĈ⊥. Proof. By Lemma 4.6, we have A.ũ⊥i = qD+e−1 − qe+i−1 q − 1 ũ⊥i−1 + qe+i − qe − qi+1 + 1 q − 1 ũ⊥i + qi+2 − q q − 1 ũ⊥i+1, 0 6 i 6 D − 1, where ũ⊥−1 = ũ⊥D := 0. The result is proved using this identity as in the proof of Proposi- tion 6.2. � 9See [30, Proposition 4.6] for a more general result. 16 J.-H. Lee and H. Tanaka Let π̃ be the orthogonal projection from W onto MĈ, i.e., (1− π̃)MĈ = 0, π̃MĈ⊥ = 0. Then we have Aπ̃ = π̃A and Ã∗π̃ = π̃Ã∗ on W. Observe that Ĉ−i = 1 1 + qe Ĉi − qi 1 + qe ũ⊥i , Ĉ+ i = qe 1 + qe Ĉi + qi 1 + qe ũ⊥i (7.2) for 0 6 i 6 D − 1, from which it follows that π̃.Ĉ−i = 1 1 + qe ( Ĉ−i + Ĉ+ i ) , π̃.Ĉ+ i = qe 1 + qe ( Ĉ−i + Ĉ+ i ) (7.3) for 0 6 i 6 D − 1. 8 A nil-DAHA of type (C∨ 1 , C1) In this section, we introduce a nil-DAHA of type (C∨1 , C1) and show that the primary T-modu- le W also has a module structure for this algebra. Let κ0, κ1, κ ′ 0, κ ′ 1 ∈ C be non-zero scalars. Recall that the DAHA H = H(κ0, κ1, κ ′ 0, κ ′ 1; q) of type (C∨1 , C1) is generated by T±10 , T±11 , and X±1, subject to the relations ([20, Section 6.4], [25, Section 3]) (Ti − κi) ( Ti + κ−1i ) = 0, (T′i − κ′i) ( T′i + κ′−1i ) = 0, i = 0, 1, where T′0 := q−1/2XT−10 , T′1 := X−1T−11 . Cherednik and Orr [9] (cf. [7, 8]) introduced the concept of nil-DAHAs for reduced affine root systems. The procedure for obtaining nil-DAHAs from ordinary DAHAs discussed in [9, Section 2.5] works for the non-reduced affine root systems of type (C∨n , Cn) as well, with a bit of extra flexibility in the specialization.10 It will turn out that the following specialization for type (C∨1 , C1) is the one which is well-suited to our situation: Definition 8.1. Let κ, κ′ ∈ C be non-zero scalars. Let H = H(κ, κ′) be the C-algebra generated by T±1, U, and X±1, subject to the relations11 (T − κ) ( T + κ−1 ) = 0, (T′ − κ′) ( T′ + κ′−1 ) = 0, U(U + 1) = 0, U′2 = 0, where T′ = XT−1, U′ = X−1(U + 1). We call H a nil-DAHA of type (C∨1 , C1). Remark 8.2. Our nil-DAHA H is obtained from H as follows. Let T̈1 := κ1T1, T̈ ′ 1 := κ1T ′ 1. Then T̈′1 = κ1X −1(T1 − κ1 + κ−11 ) = X−1 ( T̈1 − κ21 + 1 ) , (8.1) 10We learned this procedure for type (C∨n , Cn) from Daniel Orr. 11The definition of a nil-DAHA of type (C∨1 , C1) given here is different from the one in [18, Definition 5.1]. In fact, the former is a homomorphic image of the latter. Non-Symmetric Dual q-Krawtchouk Polynomials 17 and we have( T̈1 − κ21 )( T̈1 + 1 ) = 0, ( T̈′1 − κ1κ′1 )( T̈′1 + κ1κ ′−1 1 ) = 0. (8.2) Take T±10 ,X±1, and T̈1 as new generators for H. We now set κ′1 = 1 and let κ1 → 0. Then (8.1) and (8.2) become T̈′1 = X−1 ( T̈1 + 1 ) , T̈1 ( T̈1 + 1 ) = 0, ( T̈′1 )2 = 0, and we obtain the presentation of H in Definition 8.1 by the replacement( T0, q −1/2X, T̈1, κ0, κ ′ 0 ) → (T,X,U, κ, κ′). We note that, if we instead set κ′1 = κ1 and let κ1 → 0, then the second identity in (8.2) becomes T̈′1 ( T̈′1 + 1 ) = 0. This gives another version of a nil-DAHA of type (C∨1 , C1), which we expect would be suitable for the Grassmann graphs [2, Section 9.3] corresponding to the dual q-Hahn polynomials [14, Section 14.7]. We may also apply the above procedure to T0 as well to get more variations. Remark 8.3. By either or both of the replacements T → −T, X→ −X, it is immediate to see that the four nil-DAHAs H(±κ,±κ′) are all isomorphic. Remark 8.4. Mazzocco [22] defined seven new algebras as confluences of the DAHA H, and established a new link between the theory of the Painlevé equations and part of the left side of the q-Askey scheme. One of these algebras, called HIII, turns out to be isomorphic to the C- algebra having the same generators and relations as the nil-DAHA H, plus one further relation U′ = qUX; cf. [22, equations (3.86)–(3.91)]. For the rest of this paper, we set κ = q−e/2, κ′ = √ −1q−D/2 (8.3) in Definition 8.1, where we recall that q is assumed to be a prime power in our context. Our first goal is to define a 2D-dimensional representation of H = H(κ, κ′). To this end, we consider the following matrices: • for 0 6 i 6 D − 1, let t(i) = ( q−e/2 − qe/2 qe/2 q−e/2 0 ) , u′(i) = ( 0 0 − √ −1 q(D−e)/2−i 0 ) ; • for 1 6 i 6 D − 1, let t′(i) = √ −1 ( q−D/2(qD − qi + 1) qD/2(qi−D − 1) q−D/2(1− qi) qi−D/2 ) , u(i) = ( −1 1− qD−i 0 0 ) , and let t′(0) = (√ −1 q−D/2 ) , t′(D) = (√ −1 q−D/2 ) , u(0) = ( 0 ) , and u(D) = ( −1 ) . Lemma 8.5. The following (i), (ii) hold: (i) (t(i)− κ)(t(i) + κ−1) = 0 for 0 6 i 6 D − 1. (ii) (t′(i)− κ′)(t′(i) + κ′−1) = 0 for 0 6 i 6 D. 18 J.-H. Lee and H. Tanaka Proof. (i) Immediate from det(t(i)) = −1 and trace(t(i)) = κ− κ−1. (ii) Similar to the proof of (i) above. � Lemma 8.6. The following (i), (ii) hold: (i) u(i)(u(i) + 1) = 0 for 0 6 i 6 D. (ii) (u′(i))2 = 0 for 0 6 i 6 D − 1. Proof. Routine. � Define the 2D × 2D block diagonal matrices t, t′, u, and u′ by t = blockdiag ( t(0), t(1), . . . , t(D − 1) ) , t′ = blockdiag ( t′(0), t′(1), . . . , t′(D − 1), t′(D) ) , u = blockdiag ( u(0), u(1), . . . , u(D − 1), u(D) ) , u′ = blockdiag ( u′(0), u′(1), . . . , u′(D − 1) ) . Moreover, let x = t′t. Proposition 8.7. There is a representation H→ Mat2D(C) such that T 7→ t, T′ 7→ t′, U 7→ u, U′ 7→ u′, X 7→ x. Proof. From Lemmas 8.5 and 8.6 it follows that (t− κ) ( t + κ−1 ) = (t′ − κ′) ( t′ + κ′−1 ) = u(u + 1) = u′2 = 0. In particular, t, t′, and x are invertible. We have t′ = xt−1 by definition, and it is a straightforward matter to show that xu′ = u + 1. It follows that t±1, u, and x±1 satisfy the defining relations for H, and the result follows. � Corollary 8.8. There is an H-module structure on W such that t, t′, u, u′, and x are respectively the matrices representing the actions of T, T′, U, U′, and X in the ordered basis C from (4.2). By Corollary 8.8, W is now a module for both T and H = H(κ, κ′), where κ, κ′ are given as in (8.3). We next discuss how the two module structures are related. Let A = X + X−1, A∗ = √ −1 q−D/2(TU′ + UT′), Ã∗ = √ −1 q−D/2(TU′ + qUT′). For the rest of this paper, we also let τ = √ −1 q−(D+e)/2. (8.4) Note that τ2 = β−1γ, where β, γ are from Proposition 6.1; cf. (5.12). The following four lemmas are checked by straightforward calculations. Recall (4.1). Lemma 8.9. For 0 6 i 6 D − 1, the actions of X±1 on Ĉ−i are given as linear combinations with the following terms and coefficients: X.Ĉ−i : term coefficient Ĉ+ i−1 (τqD + τ−1)(qi−D − 1) Ĉ−i (τqD + τ−1)qi−D Ĉ+ i τ(qD − qi+1 + 1) Ĉ−i+1 τ(1− qi+1) X−1.Ĉ−i : term coefficient Ĉ−i−1 τ−1(1− qi−D) Ĉ+ i−1 (τqD + τ−1)(1− qi−D) Ĉ−i 0 Ĉ+ i −τ(qD − qi + 1) Non-Symmetric Dual q-Krawtchouk Polynomials 19 Lemma 8.10. For 0 6 i 6 D − 1, the actions of X±1 on Ĉ+ i are given as linear combinations with the following terms and coefficients: X.Ĉ+ i : term coefficient Ĉ+ i−1 τ−1(1− qi−D) Ĉ−i −τ−1qi−D Ĉ+ i 0 Ĉ−i+1 0 X−1.Ĉ+ i : term coefficient Ĉ−i τ−1qi−D+1 Ĉ+ i (τqD + τ−1)qi−D+1 Ĉ−i+1 0 Ĉ+ i+1 τ(1− qi+1) Lemma 8.11. For 0 6 i 6 D − 1, the actions of A on Ĉ±i are given as linear combinations with the following terms and coefficients: A.Ĉ−i : term coefficient Ĉ−i−1 τ−1(1− qi−D) Ĉ+ i−1 0 Ĉ−i (τqD + τ−1)qi−D Ĉ+ i τqi(1− q) Ĉ−i+1 τ(1− qi+1) A.Ĉ+ i : term coefficient Ĉ+ i−1 τ−1(1− qi−D) Ĉ−i τ−1qi−D(q − 1) Ĉ+ i (τqD + τ−1)qi−D+1 Ĉ−i+1 0 Ĉ+ i+1 τ(1− qi+1) Lemma 8.12. For 0 6 i 6 D − 1, the actions of A∗, Ã∗ on Ĉ±i are given by A∗.Ĉ−i = q−iĈ−i , A∗.Ĉ+ i = q−i−1Ĉ+ i , Ã∗.Ĉ−i = q−iĈ−i , Ã∗.Ĉ+ i = q−iĈ+ i . Recall the generators A, A∗, and Ã∗ of T. We now present our first main result. Theorem 8.13. On W, we have A = α+ βτA, A∗ = α∗ + β∗A∗, Ã∗ = α̃∗ + β̃∗Ã∗, where α, α∗, α̃∗, β, β∗, β̃∗ are from Propositions 6.1 and 7.1. Moreover, W is an irreducible H-module. Proof. The three identities follow from Lemmas 4.6, 4.7, 8.11, 8.12, and Propositions 6.1, 7.1. Since W is an irreducible T-module by Proposition 4.5 and since A, A∗, and Ã∗ generate T, it follows from these identities that W is irreducible as an H-module. � Recall the orthogonal projection π (resp. π̃) from W onto Mx̂ (resp. MĈ). The following result illustrates (to some extent) how we arrived at the H-module structure on W given above: Theorem 8.14. On W, we have π = T′ + κ′−1 κ′ + κ′−1 , π̃ = T + κ−1 κ+ κ−1 . Proof. Use (6.5), (7.3). � Remark 8.15. A direct computation shows that we also have u′ = qux in Proposition 8.7, so that, by Theorem 8.13, W is indeed an irreducible module for the algebra HIII mentioned in Remark 8.4. The finite-dimensional irreducible modules for H have been classified by Oblomkov 20 J.-H. Lee and H. Tanaka and Stoica [24] in the general case where the scalar q is not a root of unity. See also [23]. It would be an interesting problem to specialize the classification to H and/or HIII. We may remark that Mazzocco [21, Section 3] obtained (among other results) a faithful representation of HIII on C[η, η−1]. She introduced the non-symmetric Al-Salam–Chihara polynomials in proving the faithfulness of this representation, and the non-symmetric dual q-Krawtchouk polynomials `±i which we will discuss in Sections 9 and 10 are a discretization of these Laurent polynomials. 9 Non-symmetric dual q-Krawtchouk polynomials Recall the four Leonard systems Φ, Φ⊥, Φ̃, and Φ̃⊥, and their standard bases {Aix̂}Di=0, {u⊥i } D−2 i=0 , {Ĉi}D−1i=0 , and {ũ⊥i } D−1 i=0 from Propositions 6.1, 6.2, 7.1, and 7.2, respectively. We consider the monic dual q-Krawtchouk (Laurent) polynomials from (5.13) attached to these Leonard systems. More specifically, with the notation (5.14) we let hi(η) = hi(η; τ,D; q), 0 6 i 6 D, h⊥i (η) = hi(η; τq,D − 2; q), 0 6 i 6 D − 2, h̃i(η) = hi(η; τ,D − 1; q), 0 6 i 6 D − 1, h̃⊥i (η) = hi(η; τq,D − 1; q), 0 6 i 6 D − 1 for Φ, Φ⊥, Φ̃, and Φ̃⊥, respectively, where the scalar τ is from (8.4). Indeed, we have β−1γ = β̃−1γ̃ = τ2 and (β⊥)−1γ⊥ = (β̃⊥)−1γ̃⊥ = (τq)2. We moreover set p⊥(η) = η−1(η − τ) ( η − τ−1q−D ) , p̃(η) = η−1 ( η − τ−1q−D ) , p̃⊥(η) = η−1(η − τ). Note that p⊥, p̃, and p̃⊥ are monic. Using these Laurent polynomials, we now define `+i (η) = hi+1 − p⊥h⊥i τ i+1(1− qD)(q; q)i , `−i (η) = qD − qi τ i(qD − 1)(q; q)i ( hi − 1− qi qD − qi p⊥h⊥i−1 ) , ˜̀+ i (η) = qe ( p̃ h̃i − p̃⊥h̃⊥i ) τ i(1 + qe)(q; q)i , ˜̀− i (η) = p̃ h̃i + qep̃⊥h̃⊥i τ i(1 + qe)(q; q)i for 0 6 i 6 D − 1, where h⊥−1 := 0 and h⊥D−1(η) := D−1∏ n=1 ( η + η−1 − τqn − τ−1q−n ) = η1−D D−1∏ n=1 ( η − τqn )( η − τ−1q−n ) . Let L = D−1∑ i=−D C ηi ⊂ C [ η, η−1 ] . (9.1) Observe that dim(L) = 2D, and that the `±i and the ˜̀±i all belong to L. Our definition of these Laurent polynomials is explained by the following result (and its proof): Proposition 9.1. For 0 6 i 6 D − 1, we have Ĉ+ i = `+i (X).x̂ = ˜̀+ i (X).x̂, Ĉ−i = `−i (X).x̂ = ˜̀− i (X).x̂. (9.2) In particular, we have `+i = ˜̀+ i , `−i = ˜̀− i for 0 6 i 6 D−1. Moreover, the `±i form a basis for L. Non-Symmetric Dual q-Krawtchouk Polynomials 21 Proof. Setting i = 0 in Lemma 8.9, we have X.x̂ = ( τ + τ−1q−D ) x̂+ τ ( qD − q + 1 ) Ĉ+ 0 + τ(1− q)Ĉ−1 , X−1.x̂ = −τqDĈ+ 0 . By these identities, (6.2), and (7.1), we have u⊥0 = τ−1q−1p⊥(X).x̂, Ĉ0 = p̃(X).x̂, ũ⊥0 = −qe p̃⊥(X).x̂. (9.3) We first show that Ĉ+ i = `+i (X).x̂. By virtue of Propositions 6.1, 6.2, and Theorem 8.13, we may apply the discussions in the last paragraph of Section 5 to the Leonard systems Φ, Φ⊥. Assume that 0 6 i 6 D − 2. Then, from (5.16) and (6.4) it follows that Ĉ+ i = qi+1 − 1 qD − 1 hi+1(X) τ i+1(q; q)i+1 .x̂+ qi+1 qD − 1 h⊥i (X) τ iqi(q; q)i .u⊥0 , which, together with (9.3), gives Ĉ+ i = `+i (X).x̂. When i = D − 1, by (5.16) we have Ĉ+ D−1 = ADx̂ = hD(X) τD(q; q)D .x̂. On the other hand, from Proposition 6.1 and Theorem 8.13, it follows that h⊥D−1(X) = β1−Dτ1−D D−1∏ n=1 (A− θn) on W. Hence it follows from Proposition 6.2 that h⊥D−1(X) vanishes on Mx̂⊥, so that by (9.3) we have in particular h⊥D−1(X)p⊥(X).x̂ = τq h⊥D−1(X).u⊥0 = 0. (9.4) Combining these comments, we obtain Ĉ+ D−1 = `+D−1(X).x̂. The other identities in (9.2) are similarly (and more easily) proved, using also Propositions 7.1, 7.2, and (7.2). Finally, from (9.2) it follows that the `±i and the ˜̀±i form two bases for L, but then (9.2) again shows that we must have `+i = ˜̀+ i , `−i = ˜̀− i for 0 6 i 6 D − 1. This completes the proof. � Definition 9.2. We call the Laurent polynomials `±i = ˜̀± i , 0 6 i 6 D − 1, the non-symmetric dual q-Krawtchouk polynomials. Remark 9.3. We have given two expressions for the non-symmetric dual q-Krawtchouk poly- nomials. The first one, i.e., the `±i , is a specialization of the expressions for the non-symmetric q-Racah polynomials12 in [17, Proposition 7.8] and for the non-symmetric Askey–Wilson poly- nomials in [15, Section 4] (but with a different normalization; cf. [17, Section 11]). It is a natural guess that the polynomials studied in [15, 17] also have expressions akin to the second one, i.e., the ˜̀±i . We next show that the `±i satisfy recurrence relations with at most four terms. As a by- product of the proof of Proposition 9.1 it follows that Lemma 9.4. On W, we have p⊥(X)h⊥D−1(X) = 0. 12Strictly speaking, this is true except for `+D−1, in which case we adjusted it so that it has no term of degree D. A similar adjustment should also be possible for the non-symmetric q-Racah polynomials. 22 J.-H. Lee and H. Tanaka Proof. From Proposition 9.1 and (9.4) it follows that p⊥(X)h⊥D−1(X).Ĉ±i = `±i (X)h⊥D−1(X)p⊥(X).x̂ = 0 for 0 6 i 6 D − 1. � For convenience, let `±−1(η) := 0, `+D(η) := − η−1p⊥h⊥D−1 τD+1(q; q)D , `−D(η) := p⊥h⊥D−1 τD(q; q)D . Observe that η−1p⊥h⊥D−1 ≡ q−Dη−D−1 (modL), p⊥h⊥D−1 ≡ ηD (modL). (9.5) Theorem 9.5. The following (i), (ii) hold: (i) For 0 6 i 6 D−1, η±1`−i are linear combinations with the following terms and coefficients: η`−i : term coefficient `+i−1 (τqD + τ−1)(qi−D − 1) `−i (τqD + τ−1)qi−D `+i τ(qD − qi+1 + 1) `−i+1 τ(1− qi+1) η−1`−i : term coefficient `−i−1 τ−1(1− qi−D) `+i−1 (τqD + τ−1)(1− qi−D) `−i 0 `+i −τ(qD − qi + 1) (ii) For 0 6 i 6 D−1, η±1`+i are linear combinations with the following terms and coefficients: η`+i : term coefficient `+i−1 τ−1(1− qi−D) `−i −τ−1qi−D `+i 0 `−i+1 0 η−1`+i : term coefficient `−i τ−1qi−D+1 `+i (τqD + τ−1)qi−D+1 `−i+1 0 `+i+1 τ(1− qi+1) Proof. Looking at the two expressions `±i = ˜̀± i it follows that `+i has highest degree (at most) i − 1 and lowest degree −i − 1, whereas `−i has highest degree i and lowest degree −i. Hence, except for η−1`+D−1, η` − D−1 we have η±1`±i ∈ L, and the result follows from Lemmas 8.9, 8.10 and Proposition 9.1. For the remaining two cases, using (9.5) we routinely find that η−1`+D−1 ≡ τ ( 1− qD ) `+D (modL), η`−D−1 ≡ τ ( 1− qD ) `−D (modL). Since `±D(X) vanish on W by virtue of Lemma 9.4, the result again follows from Lemmas 8.9, 8.10 and Proposition 9.1. � 10 Orthogonality relations Recall the subspace L of C[η, η−1] from (9.1). In this section, we define a Hermitian inner product on L and show that the non-symmetric dual q-Krawtchouk polynomials `±i are orthogonal with respect to that inner product. Non-Symmetric Dual q-Krawtchouk Polynomials 23 From Proposition 9.1 and Lemma 9.4 it follows that the minimal polynomial of X on W has degree 2D and is given by ηDp⊥h⊥D−1, which has the following 2D simple zeros: λi := τqi, 0 6 i 6 D − 1, λ−i := τ−1q−i, 1 6 i 6 D. (10.1) In particular, X is multiplicity-free on W with the above eigenvalues, and is therefore diagona- lizable on W. Our aim is to explicitly describe eigenvectors of X on W. To this end, recall the Leonard systems Φ, Φ⊥ on Mx̂, Mx̂⊥ from Propositions 6.1 and 6.2, respectively, and observe that {Eix̂}Di=0, {Eiu⊥0 } D−1 i=1 form Φ∗- and (Φ⊥)∗-standard bases for Mx̂, Mx̂⊥, respectively; cf. (5.1). We will work with the following ordered orthogonal basis for W: B = { E0x̂, E1x̂, E1u ⊥ 0 , E2x̂, E2u ⊥ 0 , . . . , ED−1x̂, ED−1u ⊥ 0 , EDx̂ } . Recall the orthogonal projection π (resp. π̃) from W onto Mx̂ (resp. MĈ). Lemma 10.1. The following (i), (ii) hold: (i) The matrix representing π in B is blockdiag ( π(0), π(1), . . . , π(D − 1), π(D) ) , where π(0) = π(D) = (1), and π(i) = diag(1, 0) for 1 6 i 6 D − 1. (ii) The matrix representing π̃ in B is blockdiag ( π̃(0), π̃(1), . . . , π̃(D − 1), π̃(D) ) , where π̃(0) = (1), π̃(D) = (0), and for 1 6 i 6 D − 1, π̃(i) = 1 (qD − 1)(1 + qe) × ( (qe + qi)(qD−i − 1) q−1(1 + qD+e−i)(qi − 1)(qe + qi)(qD−i − 1) q (1 + qD+e−i)(qi − 1) ) . Proof. (i) Immediate from πEix̂ = Eix̂, 0 6 i 6 D, and πEiu ⊥ 0 = 0, 1 6 i 6 D − 1. (ii) We also recall the Leonard systems Φ̃, Φ̃⊥ on MĈ, MĈ⊥ from Propositions 7.1 and 7.2, respectively, and consider the Φ̃∗- and (Φ̃⊥)∗-standard bases {EiĈ0}D−1i=0 , {Eiũ⊥0 }Di=1 for MĈ, MĈ⊥, respectively. First, we obtain π̃(0) = (1) and π̃(D) = (0) since E0x̂ ∈ MĈ and EDx̂ ∈MĈ⊥. Next, let 1 6 i 6 D − 1. By (6.4), (7.1), and since Ĉ0 = Ĉ−0 + Ĉ+ 0 , we have Ĉ0 = x̂+ q − 1 qD − 1 Ax̂+ q qD − 1 u⊥0 , ũ⊥0 = −qex̂+ q − 1 qD − 1 Ax̂+ q qD − 1 u⊥0 . Then, using (3.2) we have EiĈ0 = ( 1 + q − 1 qD − 1 θi ) Eix̂+ q qD − 1 Eiu ⊥ 0 = (qe + qi)(qD−i − 1) qD − 1 Eix̂+ q qD − 1 Eiu ⊥ 0 , and likewise we have Eiũ ⊥ 0 = (1 + qD+e−i)(1− qi) qD − 1 Eix̂+ q qD − 1 Eiu ⊥ 0 . Solving these equations for Eix̂, Eiu ⊥ 0 , it follows that Eix̂ = 1 1 + qe EiĈ0 − 1 1 + qe Eiũ ⊥ 0 , Eiu ⊥ 0 = (1 + qD+e−i)(qi − 1) q(1 + qe) EiĈ0 + (qe + qi)(qD−i − 1) q(1 + qe) Eiũ ⊥ 0 . By these comments and since π̃EiĈ0 = EiĈ0, π̃Eiũ ⊥ 0 = 0, we obtain π̃(i) given above. � 24 J.-H. Lee and H. Tanaka Lemma 10.2. The matrix representing X on W in B is blockdiag ( x(0), x(1), . . . , x(D − 1), x(D) ) , where x(0) = (τ), x(D) = (τ−1q−D), and for 1 6 i 6 D − 1, x(i) = τ qD − 1 ( qD+e−i − qD+e + qD − qi q−1(1 + qD+e−i)(qi − 1)(qe + qi)(qD−i − 1) qD+1 −q2D+e−i + qD+i + qD+e − qD ) . Proof. Since X = T′T, the result routinely follows from Theorem 8.14 and Lemma 10.1. � Define the vectors yi, −D 6 i 6 D − 1, by yi = (qD−i − 1)(1 + qD+e−i) (qD − 1)(1 + qD+e−2i) Eix̂+ qD−i+1 (qD − 1)(1 + qD+e−2i) Eiu ⊥ 0 , y−i = qD−2i(qi − 1)(qe + qi) (qD − 1)(1 + qD+e−2i) Eix̂− qD−i+1 (qD − 1)(1 + qD+e−2i) Eiu ⊥ 0 for 1 6 i 6 D − 1, and y0 = E0x̂, y−D = EDx̂. Observe that the yi are real vectors. We normalized the yi so that yi + y−i = Eix̂, 1 6 i 6 D − 1. (10.2) Proposition 10.3. For −D 6 i 6 D − 1, yi is an eigenvector of X on W with eigenvalue λi. Moreover, we have D−1∑ i=−D yi = x̂. Proof. The first assertion follows from direct computations using Lemma 10.2. By (10.2), we have D−1∑ i=−D yi = y0 + D−1∑ i=1 (yi + y−i) + y−D = D∑ i=0 Eix̂ = x̂, as desired. � By Proposition 10.3, the yi form an eigenbasis of X on W. Since B is an orthogonal basis for W, it follows that 〈yi,yj〉 6= 0 implies j ∈ {i,−i}. We now compute these inner products. Lemma 10.4. The following (i), (ii) hold: (i) For 0 6 i 6 D, we have \|\|Eix̂\|\|2 = (−1)iqi(D+e−i+1) (q−D; q)i(1 + qD+e−2i) (q; q)i(−qe−i; q)D+1 . (ii) For 1 6 i 6 D − 1, we have \|\|Eiu⊥0 \|\|2 = (−1)i−1qi(D+e−i+1)+D−2 (q−D; q)i+1(1 + qD+e−2i) (q; q)i−1(−qe−i+1; q)D−1 . Non-Symmetric Dual q-Krawtchouk Polynomials 25 Proof. Let mi (resp. m⊥i ) denote the scalar mi from (5.8) for the Leonard system Φ (resp. Φ⊥). (i) Since \|\|Eix̂\|\|2 = 〈x̂, E∗0EiE∗0 x̂〉 = mi, the result follows from Proposition 6.1 and (5.11). (ii) By (6.2) and Lemma 4.4, we have \|\|u⊥0 \|\|2 = qe−1 ( qD−1 − 1 )( qD − 1 ) . Since \|\|Eiu⊥0 \|\|2 = m⊥i−1\|\|u⊥0 \|\|2 in this case, the result follows from Proposition 6.2 and (5.11). � Lemma 10.5. For 1 6 i 6 D − 1, we have \|\|yi\|\|2 = (−1)iqi(D+e−i) (q1−D; q)i(1 + qD + qD+e−i − qD−i) (q; q)i(−qe−i; q)D(1 + qD+e−2i) , \|\|y−i\|\|2 = (−1)i−1qi(D+e−i−2)+D+e (q1−D; q)i−1(1 + qD + qi−e − qi) (q; q)i−1(−qe−i+1; q)D(1 + qD+e−2i) , 〈yi,y−i〉 = (−1)iqi(D+e−i−1)+D (q1−D; q)i (q; q)i−1(−qe−i+1; q)D−1(1 + qD+e−2i) . Moreover, \|\|y0\|\|2 = 1 (−qe; q)D , \|\|y−D\|\|2 = 1 (−q−e; q)D . Proof. The above values are routinely computed using Lemma 10.4. � Proposition 10.6. For f, g ∈ L, we have 〈f(X).x̂, g(X).x̂〉 = D−1∑ i=−D f(λi)g(λi) \|\|yi\|\|2 + D−1∑ i=1 ( f(λi)g(λ−i) + f(λ−i)g(λi) ) 〈yi,y−i〉. Proof. From Lemma 10.3 it follows that f(X).x̂ = f(X). D−1∑ i=−D yi = D−1∑ i=−D f(λi)yi, and similarly for g(X).x̂. Since the yi are real, the result is immediate from this comment. � Define the Hermitian form 〈·, ·〉L on the subspace L from (9.1) by13 〈f, g〉L = D−1∑ i=−D f(λi)g(λi)\|\|yi\|\|2 + D−1∑ i=1 ( f(λi)g(λ−i) + f(λ−i)g(λi) ) 〈yi,y−i〉 (10.3) for f, g ∈ L, where the λi are from (10.1), and the \|\|yi\|\|2 and the 〈yi,y−i〉 are given in Lemma 10.5. Since 〈f, f〉L = 0 implies f ∈ ( p⊥h⊥D−1C[η, η−1] ) ∩L = 0, it follows that 〈·, ·〉L is an inner product on L. From (10.2) it also follows that, when restricted to symmetric polynomials, (10.3) gives an inner product for which the (symmetric) dual q-Krawtchouk polynomials (5.10) for the Leonard system Φ are orthogonal; cf. [14, Section 14.17], [35, Section 16]. We now present the orthogonality relation for the non-symmetric dual q-Krawtchouk poly- nomials `±i from Definition 9.2: Theorem 10.7. With reference to the inner product (10.3), we have 〈`µi , ` ν j 〉L = δi,jδµ,ν \|Cµi \| for 0 6 i, j 6 D − 1 and µ, ν ∈ {+,−}, where the \|C±i \| are given in Lemma 4.4. Proof. Immediate from Propositions 9.1 and 10.6. � 13We would like to point out here that the terms involving the 〈yi,y−i〉 are missing in the definition of the corresponding Hermitian form for the non-symmetric q-Racah polynomials given in [17, equation (77)]. A corrected statement will be given elsewhere. 26 J.-H. Lee and H. Tanaka Acknowledgements The authors thank Daniel Orr for helpful comments on nil-DAHAs of type (C∨n , Cn). They also thank Paul Terwilliger for many valuable discussions, Marta Mazzocco and Alexei Zhedanov for bringing the authors’ attention to [21, 22], and the anonymous referees for carefully reading the paper. Part of this work was done while Jae-Ho Lee was visiting Tohoku University as a JSPS Postdoctoral Fellow. 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Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials

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