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.


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 orthogonal polynomials, at the top (i.e., 4 φ 3 ) 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 noncommutative 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 , C n ) 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 , C n ). For n = 1, these polynomials are the Askey-Wilson polynomials which are of 4 φ 3 , and the q-Racah polynomials are a discretization of the Askey-Wilson polynomials. arXiv:1709.07825v2 [math.CO] 10 Feb 2018 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 , C 1 ), 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 Mat X (C) be the C-algebra consisting of the complex square matrices indexed by X. Let V = V X be the C-vector space consisting of the complex column vectors indexed by X. We endow V with the inner product u, v = u t v 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, letŷ be the vector in V with a 1 in the y-coordinate and 0 elsewhere. For a subset Y ⊂ X, letŶ = y∈Yŷ ∈ 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 − rq n−1 , n 1 = n 1 q = q n − 1 q − 1 .

Distance-regular graphs
Let Γ be a finite simple connected graph with vertex set X and diameter D. For x ∈ X, let where ∂ denotes the path-length distance. We abbreviate Γ(x) := Γ 1 (x). We call Γ distanceregular if there are non-negative integers a i , b i , c i , 0 i D, called the intersection numbers of Γ, such that b D = c 0 = 0, b i−1 c i = 0, 1 i D, and Fig. 1 shows a small example of a distance-regular graph with D = 3. From now on, assume that Γ is distance-regular. The i th distance matrix of Γ is the 0-1 matrix A i ∈ Mat X (C) such that (A i ) x,y = 1 if and only if ∂(x, y) = i. The Bose-Mesner algebra of Γ is the subalgebra M of Mat X (C) generated by the A i . We note that M is semisimple since it is closed under conjugate-transposition. Observe also that from which it follows that, It follows that the adjacency matrix A := A 1 of Γ generates M , and that the A i 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 θ 0 , θ 1 , . . . , θ D , which we call the eigenvalues of Γ. We will always set θ 0 := b 0 , the valency (or degree) of Γ. For 0 i D, let E i ∈ Mat X (C) be the orthogonal projection onto the eigenspace of θ i . Then we have and the E i form another basis for M consisting of the primitive idempotents, i.e., Observe that M is also closed under entrywise multiplication, denoted •. We say that Γ is Q-polynomial with respect to the ordering where the multiplication is under •. In particular, if we write then the θ * i are (real and) mutually distinct. Note also that θ * 0 = trace(E 1 ) = rank(E 1 ). Assume that Γ is Q-polynomial with respect to the ordering 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 Mat X (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]): Observe also that E * iX = A ix , 0 i D, from which it follows that the (D + 1)-dimensional subspace of V is an irreducible T -module, called the primary T -module. We note that We refer the reader to [1,2,10,11] for more detailed information.

Dual polar graphs
Let D be a positive integer. Let V be one of the following spaces over the finite field F q equipped with a non-degenerate form: 1 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 intersection numbers Note that a i = c i a 1 , 0 i D. The eigenvalues of Γ are given by and Γ is Q-polynomial with respect to the ordering 2 {θ i } D i=0 , where θ 0 > θ 1 > · · · > θ D . Moreover, the corresponding dual eigenvalues are given by 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., K 1,1,2 ) 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 which attains the Hoffman bound 1 − θ 0 θ −1 D (cf. [2,Proposition 4.4.6]). In other words, C is a so-called Delsarte clique. For 0 i D − 1, define the i th distance neighbor of C by

Then we have
where C −1 = C D := ∅, from which it follows that for every y ∈ C i . 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: Proof . (i) Let y ∈ C i . Recall that there is a unique z in C ∩ Γ i (y) since Γ is a regular near polygon. Then we have C i−1 ∩ Γ(y) = Γ i−1 (z) ∩ Γ(y), and the result follows.
Using (3.1), Lemma 3.2, and a i + b i + c i = θ 0 , we have We now recall the Terwilliger algebra of Γ with respect to C in the sense of Suzuki [28]. For The E * i form a basis for the dual Bose-Mesner algebra M * = M * (C) of Γ with respect to C. The dual adjacency matrix of Γ with respect to C is defined by (cf. [16, equation (50)]) From (2.1), (3.4), and (3.5), it follows that where so that (cf. [16,Lemma 4.11]) In particular, A * has D mutually distinct real eigenvalues and hence generates M * . The Terwilliger algebra T = T (C) with respect to C is the subalgebra of Mat X (C) generated by M , M * . We note that T is semisimple and is generated by A, A * . By virtue of (2.2), the following are relations in T : where we set E * D := 0 for convenience. The subspace (3.6) of V is an irreducible T -module with dimension D, called the primary T -module. We note that 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 , T are subalgebras of those of maximal parabolic subgroups. See also [26,27].

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.
We note that T is semisimple and is generated by See Fig. 2. For notational convenience, we set The following (i), (ii) hold: In particular, the partition Lemma 4.4. We have In particular, the C ± i are non-empty.
Let W be the linear span of theĈ ± i . Thus, W has the following ordered orthogonal basis: Proof . It is clear that W is closed under the actions of the E * i and the E * i . Moreover, since We call W the primary T-module. Note that W contains both the primary T -module Mx and the primary T -module MĈ. Let Mx ⊥ (resp. MĈ ⊥ ) be the orthogonal complement of Mx (resp. MĈ) in W. In Sections 6 and 7, we will show that Mx ⊥ (resp. MĈ ⊥ ) is also an irreducible T -module (resp. T -module). Thus, W decomposes in two ways: We end this section by describing the actions of A, A * , and A * on W in terms of the basis C.

Proof . From Lemma 4.3 it follows that
for 0 i D − 1. Evaluate the two identities using (3.1).

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 {θ i } d i=0 be an ordering of the eigenvalues of A. Then there is a sequence of elements where I is the identity of End(W). We call E i the primitive idempotent of A associated with θ i .
that satisfies the following axioms (LS1)-(LS4): 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].
) be a Leonard system on W. Note that each of the following is also a Leonard system on W: be the first split sequence of Φ ⇓ and call this the second split sequence of Φ. The parameter array of Φ is the sequence 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.
Take a non-zero vector u ∈ E 0 W. By [35,Lemma 10.2], the vectors {E * i u} d i=0 form a basis for W, called a Φ-standard basis. In view of (LS3), (LS4), there are scalars , 1 i d.
Consider the following normalization: By [35,Theorem 17.4], for 0 i d we have Define the scalars m i , 0 i d, by .
Assume now that Φ has dual q-Krawtchouk type as in Definition 5.2. Then we have 5 The values of the f i at ξ = θ j are given by (cf. [36,Example 5.9]) It follows that the f i are the dual q-Krawtchouk polynomials [14,Section 14.17] in the variable We also have Following [15], we fix a non-zero scalar τ such that τ 2 = β −1 γ, (5.12) and renormalize the f i so that they are monic 6 as symmetric Laurent polynomials in η: The h i depend on the parameters q, d, and τ, and we will write We note that h i has highest degree i and lowest degree −i in η, and that 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 holds on W. Then it follows from (5.15) that where we have also used (5.7) and (5.9). In particular, we have where | Mx means that each of the elements in the sequence is restricted to Mx.
We note that the intersection numbers of Φ coincide with those of Γ. We next consider the T -module Mx ⊥ , the orthogonal complement of Mx in W. Observe
Proof . By Lemma 4.6, we have Every Tsubmodule of Mx ⊥ is spanned by some of the u ⊥ i since it is M * -invariant, and (6.3) then shows that it must be either Mx ⊥ or 0. Hence Mx ⊥ is an irreducible T -module. It also follows from (6.3) that Mx ⊥ = M u ⊥ 0 . In particular, Mx ⊥ is spanned by the vectors Hence it follows from (2.2) and the irreducibility of Mx ⊥ that E i u ⊥ 0 = 0, 2 i D − 1 and E D u ⊥ 0 = 0. From these comments and (2.2), it follows that Φ ⊥ is a Leonard system on Mx ⊥ , and that 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].
Then we have Aπ = πA and A * π = πA * on W. Observe that
Moreover, {Ĉ i } D−1 i=0 is a Φ-standard basis for MĈ. Proof . We first remark that E DĈ = 0. Indeed, the Hoffman bound 1 − θ 0 θ −1 D on the size of a clique follows from the fact that E D is positive semidefinite, and the bound is attained precisely when the characteristic vector of the clique vanishes on E D V . From this comment, (3.6), and (3.10), it follows thatΦ is a Leonard system on MĈ. Moreover, In view of (3.11), the parameter array of Φ 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 Φ coincide with those of C. We next consider the T -module MĈ ⊥ , the orthogonal complement of MĈ in W. From (6.1) it follows that where we take for convenience. By (2.1), (3.3), (3.8), and (3.9), we obtain form an orthogonal basis for MĈ ⊥ . Consider the sequence where u ⊥ −1 = u ⊥ D := 0. The result is proved using this identity as in the proof of Proposition 6.2.
Then we have A π = πA and A * π = π A * on W. Observe that In this section, we introduce a nil-DAHA of type (C ∨ 1 , C 1 ) and show that the primary T-module W also has a module structure for this algebra. Let κ 0 , κ 1 , κ 0 , κ 1 ∈ C be non-zero scalars. Recall that the DAHA H = H(κ 0 , κ 1 , κ 0 , κ 1 ; q) of type (C ∨ 1 , C 1 ) is generated by T ±1 0 , T ±1 1 , and X ±1 , subject to the relations ([20, Section 6.4], [25, Section 3]) 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 , C n ) as well, with a bit of extra flexibility in the specialization. 10 It will turn out that the following specialization for type (C ∨ 1 , C 1 ) 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 relations 11 We call H a nil-DAHA of type (C ∨ 1 , C 1 ).
Remark 8.2. Our nil-DAHA H is obtained from H as follows. LetT 1 := κ 1 T 1 ,T 1 := κ 1 T 1 . ThenT 1) 10 We learned this procedure for type (C ∨ n , Cn) from Daniel Orr. 11 The 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. and we have Take T ±1 0 , X ±1 , andT 1 as new generators for H. We now set κ 1 = 1 and let κ 1 → 0. Then (8.1) and (8.2) becomë and we obtain the presentation of H in Definition 8.1 by the replacement We note that, if we instead set κ 1 = κ 1 and let κ 1 → 0, then the second identity in (8.2) becomes This gives another version of a nil-DAHA of type (C ∨ 1 , C 1 ), 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 T 0 as well to get more variations. For the rest of this paper, we set 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: Lemma 8.5. The following (i), (ii) hold: Proof . (i) Immediate from det(t(i)) = −1 and trace(t(i)) = κ − κ −1 .
(ii) Similar to the proof of (i) above.
Define the 2D × 2D block diagonal matrices t, t , u, and u by

Moreover, let
x = t t.
Proposition 8.7. There is a representation H → Mat 2D (C) such that Proof . From Lemmas 8.5 and 8.6 it follows that 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 For the rest of this paper, we also let Note that τ 2 = β −1 γ, 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 i D − 1, the actions of X ±1 onĈ − i are given as linear combinations with the following terms and coefficients: Lemma 8.10. For 0 i D − 1, the actions of X ±1 onĈ + i are given as linear combinations with the following terms and coefficients: Lemma 8.11. For 0 i D − 1, the actions of A onĈ ± i are given as linear combinations with the following terms and coefficients: Recall the generators A, A * , and A * of T. We now present our first main result.
Since W is an irreducible T-module by Proposition 4.5 and since A, A * , and A * generate T, it follows from these identities that W is irreducible as an H-module.
Recall the orthogonal projection π (resp. π) from W onto Mx (resp. MĈ). The following result illustrates (to some extent) how we arrived at the H-module structure on W given above: Proof . Use (6.5), (7.3).  [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 H III . We may remark that Mazzocco [21,Section 3] obtained (among other results) a faithful representation of H III 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 ⊥ , Φ, and Φ ⊥ , and their standard bases 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 for Φ, Φ ⊥ , Φ, and Φ ⊥ , respectively, where the scalar τ is from (8.4). Indeed, we have Note that p ⊥ , p, and p ⊥ are monic. Using these Laurent polynomials, we now define 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 i D − 1, we havê In particular, we have Moreover, the ± i form a basis for L.
Proof . Setting i = 0 in Lemma 8.9, we have By these identities, (6.2), and (7.1), we have We first show thatĈ + i = + i (X).x. 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 i D − 2. Then, from (5.16) and (6.4) it follows that which, together with (9.3), givesĈ + i = + i (X).x. When i = D − 1, by (5.16) we havê On the other hand, from Proposition 6.1 and Theorem 8.13, it follows that Hence it follows from Proposition 6.2 that h ⊥ D−1 (X) vanishes on Mx ⊥ , so that by (9.3) we have in particular Combining these comments, we obtainĈ + D−1 = + D−1 (X).x. 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 This completes the proof.
Definition 9.2. We call the Laurent polynomials ± i = ± i , 0 i D − 1, the non-symmetric dual q-Krawtchouk polynomials. Remark 9.3. We have given two expressions for the non-symmetric dual q-Krawtchouk polynomials. The first one, i.e., the ± i , is a specialization of the expressions for the non-symmetric q-Racah polynomials 12 in [17, Proposition 7.8] and for the non-symmetric Askey-Wilson polynomials 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 byproduct of the proof of Proposition 9.1 it follows that Lemma 9.4. On W, we have 12 Strictly 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.
Proof . From Proposition 9.1 and (9.4) it follows that For convenience, let ± −1 (η) := 0, Observe that Theorem 9.5. The following (i), (ii) hold: (i) For 0 i D −1, η ±1 − i are linear combinations with the following terms and coefficients: (ii) For 0 i D −1, η ±1 + i are linear combinations with the following terms and coefficients: 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 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.

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. 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 η D p ⊥ h ⊥ D−1 , which has the following 2D simple zeros: In particular, X is multiplicity-free on W with the above eigenvalues, and is therefore diagonalizable on W. Our aim is to explicitly describe eigenvectors of X on W. To this end, recall the Leonard systems Φ, Φ ⊥ on Mx, Mx ⊥ from Propositions 6.1 and 6.2, respectively, and observe that form Φ * -and (Φ ⊥ ) * -standard bases for Mx, Mx ⊥ , respectively; cf. (5.1). We will work with the following ordered orthogonal basis for W: Recall the orthogonal projection π (resp. π) from W onto Mx (resp. MĈ).
Proof . Since X = T T, the result routinely follows from Theorem 8.14 and Lemma 10.1.
Define the vectors y i , −D i D − 1, by for 1 i D − 1, and Observe that the y i are real vectors. We normalized the y i so that By Proposition 10.3, the y i form an eigenbasis of X on W. Since B is an orthogonal basis for W, it follows that y i , y j = 0 implies j ∈ {i, −i}. We now compute these inner products.