Abelianization of Matrix Orthogonal Polynomials

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1 Introduction

The paper is mainly concerned with the geometry of matrix non-hermitian biorthogonal polynomials (BMOPs) for a (possibly complex–valued) weight matrix W (z). These are two sequences of polynomials Pn(z) n N, Pn∨(z) n N of degrees at most n such that { } ∈ { } ∈

Pn(z)W (z)Pm∨(z)dz = δnmHn, (1.1) Zγ for some matrices Hn, where γ is some chosen contour in C along which W (z) is defined and such that all the moments are defined and finite. The notion of MOPs and BMOPs is a non-commutative version of the standard notion of (bi)orthogonality of polynomials and the literature is vast and dates back to Krein [23]. It has found several applications beyond approximation theory in several areas and is a staple of both approximation theory and mathematical physics see for example [26, 11, 21, 22, 9, 7, 8]. There have been a few recent publications [13, 10, 17] where a connection was found between the matrix biorthogonality and “scalar” biorthogonality on the “spectral curve” of W (z), namely, the curve defined by det(y1 W (z)) = 0. In the quoted literature the matrix W (z) is a rational function of z so that the spectral curve− is necessarily an algebraic Riemann surface. It is interesting to point out the the spectral curve is not really an invariant object associated with the class of biorthogonal polynomials in the following sense: in the scalar case multiplying the weight by a nonzero constant does not affect the resulting orthogonal polynomials. On the contrary, in the matrix case we can multiply W on the two sides by arbitrary invertible constant matrices W (z) W (z) = LW (z)R, 7→ L, R GLn(C). This transformation does affect the biorthogonal matrix polynomials, albeit in a simple way: ∈ 1 1 Pn PnL− and Pn∨ R− Pn∨. However this same transformation does not preserve the spectfral curve. 7→This point notwithstanding,7→ the idea of associating scalar biorthogonality on a covering curve is interesting and potentially useful in studying asymptotic properties. The general philosophy of associating a “scalar” set of data to a “matrix” is at the roots of much of the theory of integrable systems in several guises; for example (see [3] Ch. 3,5 for a recent review, and see also [1, 2, 6]) the isospectral deformations of a rational Lax matrix L can be realized as linear evolution on the Jacobian of its spectral curve of a line bundle L , which encodes the whole matrix L up to conjugation by constant invertible matrices. The same general idea is behind Hitchin’s abelianization of Higgs bundles [16], namely the idea of associating to a pair of vector bundle V and Higgs field Φ H0(End(V ) ) a line bundle, L ,. over the spectral curve det(η1 Φ)=0 embedded in . This should sufficiently∈ motivate the⊗K wording of the title. − K Summary of results. There are two main sets of results in the paper, the first necessary for the second. The first is contained in Section 2 and describes the generalization to higher genus of the notion of biorthogonality. The notion that we generalize is that of multi-point Padé approximation as in [24] and corresponding notion of biorthogonality along a contour γ with respect to a weight function Y : γ C. This falls within the same general area as the recent [4], where a suitable→ notion of Padéapproximation on Riemann surfaces was defined and shown how it is linked with a (scalar) orthogonality of sections in a sequence of line bundles of increasing degree. While the general idea is fruitful, we need to modify it to be applicable to the present situation. Here instead we consider not functions, but sections of appropriate twists of the semicanonical line bundle on a Riemann surface of higher genus. Since these notionsC are less common, especially in the literature on approximation theory, we cursorily illustrate below how they apply in the familiar context of ordinary (scalar) orthogonal polynomials.

Half-diﬀerentials. In genus zero, (non-hermitian) orthogonal polynomials for a weight function Y (z) on a contour γ are polynomials pn(z) of degree at most n such that

pn(z)pm(z)Y (z)dz = δnmhn, (1.2) Zγ

2 for some constants hn. The (formalistic) point of view we take is that we have actually sections of the semicanonical bundle, √ , ψn(z)= pn(z)√dz and the orthogonality (1.2) reads instead K

ψn(z)ψm(z)Y (z)= δnmhn. (1.3) Zγ While this is almost entirely an aesthetic modification in genus zero, it is this the notion that we want to generalize to higher genus, where the modification is of substance. Another effect of this point of view is in the counting of degrees; while a polynomial pn can be thought of as a meromorphic function with a pole of order n at infinity, the half-differential pn√dz has a pole of order n + 1. Thus the space of polynomials n of degree n is identified with P the space, Pn, of meromorphic sections of √ with a single pole of order at most n +1 at . The orthogonal polynomials are also the denominatorsK of the Padé approximation problem∞ for the Weyl–Stiltjes function Y (w)dw W (z) := . (1.4) z w Zγ − In keeping with the above shift of point of view we can interpret the above as a section half-differential:

(z) := S(z, w)ψ (w)Y (w), (1.5) W 0 Zγ √dz√dw where we have introduced the Szegö kernel S(z, w) = z w . In [4] the Cauchy kernel was used instead to generalize the notion of Weyl-Stiltjes transform; that pivotal− role is here played by the Szegökernel. The definition of this kernel is classical [15, 18] and it involves an additional g = genus( ) complex parameters. In higher genus these extra g complex parameters (rather, 2g real ones)C are encoded in the choice of a representation : π1( ) U(1), or, which is the same, a flat line bundle. The notion of polynomials is thus replaced by the notionX of holomorphicC → sections of a line bundle obtained by tensoring the flat bundle with the bundle √ X K of half-differentials, and allowing poles at an increasing sequence Dn < Dn+1 of divisors of degree n. The “dual” space is constructed similarly, but starting from the dual flat bundle ∨ (i.e. the representation of π1 given by the reciprocal multipliers). X The notion of Szegökernel subordinated to the choice of is described in Section 2.2, where we also describe a notion of biorthogonality with respect to a pairing (2.14) ofX the form

ϕ, ϕ∨ := Y ϕϕ∨, h i Zγ where γ is a contour, Y : γ C a (sufficiently smooth on γ) weight function and ϕ, ϕ∨ two sections of √ , → X ⊗ K ∨ √ respectively. X The⊗ K notion of biorthogonal section is given in Def. 2.2 and we show how the resulting biorthogonal sections can be expressed in familiar determinantal form (2.17) and with a Heine–like formula (2.25). We then show how these sections can be characterized in terms of a Riemann–Hilbert problem 2.5 in a parallel way as it was recently done in [4] and as it was done historically in the context of orthogonal polynomials in [19]. The issue of existence and uniqueness of the solution is brought into focus in Prop. 2.6. In Sec. 2.2.1 we describe the Padéproblem associated to these biorthogonal sections as sketched above. In Sec. 2.3 we describe the Christoffel–Darboux–Szegökernel and its properties, which generalizes the familiar notion of Christoffel–Darboux kernel for orthogonal polynomials. The biorthogonality that we describe here is the higher genus version of the multi-point Padéapproximation, see for example [24]. In Section 2.1 we introduce the notions by the way of a guide example in the familiar setting of genus 0.

The second part is contained in Sec. 3. We add to the data a meromorphic function Z : CP1 of degree r C → and choose the sequence of divisors Dn so that its support is contained in the pole divisor, Ξ, of Z. The hinges of the construction are contained in Thm. 3.1 and Prop. 3.2: the main idea here is that any meromorphic section of √ with poles only at the poles of Z can be written as a linear combination of any basis of r sections of H0X ⊗ √KK(Ξ) with polynomial coefficients in z = Z(p). These polynomial coefficients are the entries of the Matrix BiorthogonalX ⊗ Polynomials, see Corollary 3.4 and the main Theorem 3.5. Section 3.3 shows the inverse construction to that of Section 3; namely we show how to associate to matrix orthogonal polynomials for a rational weight matrix W (z) the set of data (flat bundle etc.) described in Sec. 3. This construction is essentially a reformulation of [10] and uses the theory of the spectralX problem for rational Lax

3 matrices [1, 2, 3, 6] in an essential way.

Finally we conclude with an extensive list of examples in Section 4: we offer two examples where the Riemann surface is of genus zero. These two examples are outside (but not by much) of the scope of [10] because the weight matrix isC not a rational function. They are the “projection” of the ordinary Laguerre and Hermite polynomials by a rational map of degree 2 to give a matrix orthogonality. See Sec. 4.1.1, 4.1.2. We also consider the example of being an elliptic curve (genus one) in Sec. 4.2. In this case we can write explicitly all the main objects (Szegökernel,C etc.) in terms of classical functions (Jacobi theta functions, Weierstraß functions). We consider an example of “finite” orthogonality in Sec. 4.3. This means that the weight function Y is meromorphic and the contour γ is a closed contractible contour on the elliptic curve, containing all poles in the simply connected part. In this case the (scalar) biorthogonality is easily seen to be of finite rank. This section is inspired by [17, 13] where the matrix biorthogonal polynomials were used to express the statistical properties of the random tilings of the aztec diamond. The examples contained in Sec. 4.3 and Sec. 4.4 provide a generalization of the setup in loc. cit.. The precise relationship is explained in Remark 4.10.

Notations. - : a Riemann surface of genus g with canonical bundle ; C K - J( ) the Jacobian of ; C C - A : J( ) the Abel map; C → C -Θ, Θ[α,β] the Riemann theta function (with characteristic [α, β], respectively); - if p is a point or more generally for a divisor D, then Θ(p) is a shorthand for Θ(A(p)), or Θ(D) stands for Θ(∈A C(D)), respectively;

- E(p, q): Klein’s prime form [[14], pag 16]. - Given a line bundle on and a divisor D the vector space of holomorphic sections of of (D) is denoted by H0( (D)) and h0( L(D))C denotes its dimension. Holomorphic sections of (D) are the sameL as meromorphic sections,L ϕ, of satisfyingL div(ϕ) D. L L ≥− - the sheaf of holomorphic functions; ( p) the sheaf of meromorphic functions with at most a pole at p (at leastO a zero, for the case with ). O ± − - for ϕ H0(L ) its divisor is denoted by div(ϕ). ∈ 2 Biorthogonality on a Riemann surface 2.1 Guide example: (multi-point) Padéapproximations in genus 0 The standard example of orthogonal polynomials is reviewed in the next paragraph. In the subsequent one we review the case of multi–point Padéapproximations. The two examples depend on a choice of strictly increasing sequence Dn < Dn+1 of positive divisors of the indicated degree. For such a choice we define

0 Pn := H (√ (Dn )) = P∨, dim Pn = n +1. (2.1) K +1 n The Szegökernel is simply √dz√dw S(z, w)= . (2.2) z w − In genus zero any flat bundle are trivial because CP1 is simply connected; thus there is only one Szegökernel. Given a contour γ and Y : γ X C a sufficiently smooth function (we tacitly assume the necessary conditions so that all integrals we write are convergent)→ we define the Weyl-Stiltjes section of √ as K Y (w)dw (z)= √dz (2.3) W z w Zγ −

4 The standard (multi point) Padéapproximation problem can be formulated as follows. Consider a fixed spanning section ϕ of H0(√ (D )). Note that in genus 0 these have no zeros (e.g. ϕ = √dz if D = ). Let us denote by 0 K 1 0 1 ∞ R the divisor of ϕ0 (which is of the form R = p0 = D1 where p0 is the single pole of ϕ0, typically ). − − 0 ∞ Then the Padéproblem can be formulated as that of finding a section Qn 1 H (√ (Dn+1 D1)) and a 0 − ∈ K − section ψn H (√ (Dn )) such that ∈ K +1

ϕ0(p)Qn 1(p) ψn(p) (p)= ( Dn R). (2.4) − − W O − −

Example: single-point case. Let Dn = n . Then Pn = P∨ is identified with the space of polynomials of ∞ n degree n by ϕ = p(z)√dz (note that √dz has a simple pole at ). Since ϕ0(z) = √dz it follows that (z) is essentially≤ the usual Weyl-Stiltjes transforms of a weight function ∞Y (z) on γ (times √dz). The PadéproblemW 2.7 is the usual Padéapproximation whose denominators notoriously give orthogonal polynomials [25].

Example: multi-point case. A slightly less standard case is when Dn consists of n points (not all the same). D n 1 To simplify suppose n = + j=1− zj with all points pairwise distinct. we can choose as (ordered) basis of 0 ∞ Pn := H (√ (Dn )) the sections K +1 P √dz ϕ0 = √dz, ϕℓ = , ℓ 1. (2.5) z zℓ ≥ − Noticing that R = (ϕ0 has only a simple pole and no zeros), the Pad´eproblem now reduces to the multi-point Pad´eapproximation−∞ [24] where Pn(z) √ ψn(z)= n 1 dz (2.6) − (z zℓ) ℓ=1 − and Pn a monic polynomial of degree n. The sectionQQn is of the form

Qn 1(z) Q − √ n(z)= n 1 dz (2.7) − (z zℓ) ℓ=1 − with Qn 1 a polynomial of degree n 1. The Pad´eproblemQ then becomes − ≤ − n 1 Qn 1(z) Y (w)dw − − = 2 2 zℓ . (2.8) Pn(z) − z w O − ∞− γ ℓ ! Z − X=1 Then one obtains (Pn(z) Pn(w))Y (w)dw Qn 1(z)= − (2.9) − z w Zγ − and Pn is obtained by the requirement that

n 1 Pn(w)Y (w)dw − √dz n = zℓ (2.10) (z w) (w zℓ) O −∞− γ ℓ=1 ℓ ! Z − − X=1 namely the linear system of n equations that embodyQ the so-called multiple orthogonality

Pn(w)Y (w)dw Pn(w)Y (w)dw n =0, n =0, j =1,...,n 1. (2.11) (w zℓ) (zj w) (w zℓ) − Zγ ℓ=1 − Zγ − ℓ=1 − which precisely implies thatQ ψn is orthogonal to ϕ0,...,ϕnQ 1. − 2.2 Higher genus Riemann surfaces and biorthogonality Let be a smooth Riemann surface of genus g and Z : CP1 a meromorphic function (the “projection”) of degreeC r. This allows us to think of as a branched cover ofCC →with r sheets. We will denote by lowercase z the value of Z(p) for p but otherwise tryC to keep the function Z and the coordinate z of the target space distinguished from each other.∈ C We recall that the ramiﬁcation points are the points p of the curve where dZ(p) = 0; viceversa, the branch-points are the same as the critical values of Z, i.e. the values of Z(p) at theC ramiﬁcation points.

5 (j) The points above z = will be denoted with multiplicity kj so that kj = r and the corresponding divisor is denoted by Ξ: ∞ ∞ (j) P Ξ= kj . (2.12) ∞ We fix a square root √ of the canonical line bundle ofXwith h0(√ )=0. Let be a flat bundle given by a U(1) representation (with theK same symbol) of the fundamentalC group ofK . For an arbitraryX (generic) positive divisor D an application of the Serre-Riemann-Roch theorem guarantees thatC h0( √ (D)) = deg D: sections of this bundle can be thought of as half-differentials ϕ with multiplicative monodromyX ⊗ K : π ( ) U(1) and with pole X 1 C → divisor not exceeding D: div(ϕ) D. The dual line bundle ∨ is the bundle constructed similarly to the above ≥− 1 X but with multiplier system ∨(γ)= (γ)− , γ π ( ). X X ∈ 1 C

General picture: biorthogonal “polynomials”. Suppose now that Dn is an increasing sequence of positive divisors of degree n: Dn < Dn+1. Thus the corresponding sequences of vector spaces

0 Pn := H ( √ (Dn )) = ψ meromorphic section of √ : div(ψ) Dn (2.13) X ⊗ K +1 X ⊗ K ≥− +1 n o 0 and P∨ := H ( ∨ √ (Dn )) are in the same inclusion: n X ⊗ K +1

Pn Pn , P∨ P∨ , n 0. ⊆ +1 n ⊆ n+1 ≥

The reason for the shift in the indexing is that we want to think of Pn as “polynomials” of degree n and hence P0 should have dimension 1.

1 Example 2.1 For = CP and Dn = n we have Pn = n√dz where ℓ denotes the vector space of polynomials of degree n. C ∞ P P ≤ We use the symbols P, P∨ to mean the direct sum of all the respective sequences. Let γ be a contour ⊂ C that does not intersect any of the divisors Dn, and Y : γ C a smooth function (typically we will take Y to be a meromorphic function on restricted to γ). Then we can→ deﬁne a bilinear pairing: C b b , :P P∨ C (2.14) h bi × →

(ϕ, ϕ∨) ϕ, ϕ∨ := Y ϕϕ∨. (2.15) 7→ h i Zγ

Note that the product ϕϕ∨ is an ordinary single–valued meromorphic diﬀerential. Let us ﬁx two sequences of P P sections ϕn n N, ϕn∨ n N with ϕn spanning n mod n 1 (and respectively for the “dual” space). { } ∈ { } ∈ − The Gram-Schmidt orthogonalization process produces biorthogonal section ψn, ψn∨. We can write them explicitly as follows. Denote the “bimoments” by

n 1 − µa,b := Y ϕaϕb∨,Dn := det µa,b a,b=0. (2.16) Zγ Then

µ00 µ01 ... µ0n µ00 µ01 ... µ0,n 1 ϕ0 − µ10 µ11 ... µ1n µ10 µ11 ... µ1,n 1 ϕ1    −  . . ψn := det . . , ψn∨ := det . . (2.17) . .      µn 1,0 µn 1,1 ... µn 1,n   µn1 µn2 ... µn,n 1 ϕn   − − −   −   ϕ ϕ ... ϕ     0∨ 1∨ n∨    and they satisfy

Y ψnψm∨ = δnmhn. (2.18) Zγ A simple exercise shows that b hn = Dn+1Dn. (2.19) P P P P If Dn = 0 then ψn (ψn∨) also span n mod n 1, ( n mod n 1, respectively). 6 − −

6 Deﬁnition 2.2 A section ψn Pn is a biorthogonal section if ψn, ϕ∨ = 0 for all j = 0,...,n 1. Similarly a ∈ h j i − section ψ∨ P∨ is a biorthogonal section if ϕj , ψ∨ =0, j =0,...,n 1. n ∈ n h n i −

Exercise 2.3 Show that the notion of biorthogonality does not depend on the choices of the sequences ϕn∨ n N, ϕn n N { } ∈ { } ∈ provided that ϕn spans Pn mod Pn 1. Hence the notion depends only on the choice of the sequence Dn, n N. This is familiar in the genus zero context− and for the “Laurent”orthogonal polynomials [5, 20]. In that context∈ Dn have support at the two points z = 0, and the choice of sequence describes diﬀerent families of orthogonal sections. ∞

Note that biorthogonal sections are deﬁned up to multiplicative nonzero scalar.

Szeg¨okernel. We need to introduce the notion of Szeg¨okernel following2 [18, 15]. A convenient way to think of it is as the reproducing kernel for the pair of spaces ℘,℘∨ in the following sense:

ϕ(p)= res S(p, q)ϕ(q), ϕ∨(q)= res ϕ∨(p)S(p, q). (2.20) q D − p D ∈ ∈ It is conceptually the equivalent of the Cauchy kernel for functions. More formally it is a kernel such that: 1. with respect to p (q) is a section of √ (q) ( √ (p), respectively), namely has a simple pole only on the diagonal p = q and, under analyticX ⊗ continuationK X ⊗ alongK closed loops γ π ( ,p), σ π ( , q) satisﬁes3 ∈ 1 C ∈ 1 C γ σ 1 S(p , q)= (γ)S(p, q), S(p, q )= (σ)− S(p, q). (2.21) X X 2. in any local coordinate ζ it is normalized as follows along the diagonal:

dζ(p) dζ(q) S(p, q)= 1+ ζ(p) ζ(q) (2.22) ζ(p) ζ(q) O − p −p Formulas in terms of Riemann theta functions as well as important determinantal identities (“Fay’s identities”) are presented in Appendix A. The kernel deﬁned here exists and is unique if and only if the ﬂat bundle does not lie on the Theta divisor in the Picard variety of degree zero bundles. What these words mean concretelyX is the g following: choose a basis aj,bj for the homology (the ”a,b” cycles) and denote { }j=1

2iπαj 2iπβj e = (aj ), e = (bj ). (2.23) X X

The numbers αj ,βj are defined modulo integers. Define the following point in the Jacobian J( ) of the curve (we use the same symbol with slight abuse of notation): C X t t g := β τ α J( ), α = (α ,...,αg) , β = (β ,...,βg) R (2.24) X − ∈ C 1 1 ∈ Then we must have Θ( ) = 0, with Θ the Riemann theta function (see App. A). The fact that this is a sufficient condition is clear from theX formula6 (A.2). The necessity is more complicated but it follows from the general Riemann vanishing theorem for Theta functions [14].

Heine formula. A simple exercise using Andreief formula for alternants shows that we can write the orthogonal sections ψn, ψn∨ above in the multiple–integral form:

n 1 n n ψn(p)= det [ϕa(pb)]a,b=0 det [ϕa∨(pb)]a,b=1 Y (pj ) n! n γ j=1 Z Y n 1 n n ψn∨(p)= det [ϕa(pb)]a,b=1 det [ϕa∨(pb)]a,b=0 Y (pj ) (2.25) n! n γ j=1 Z Y where p0 = p (note that one of the two alternants in each integral is of size n + 1).

2The two references above differ in their definition by a sign. We follow the first of them. 3 The symbol π1(C,p) denotes the group of homotopy classes of closed contours on C starting and ending at p.

7 Riemann–Hilbert problem. For any ϕ P, ϕ∨ P∨ we deﬁne ∈ ∈ 1 1 Y [ϕ](p) := S(p, q)ϕ(q)Y (q), Y [ϕ∨](p) := ϕ∨(q)S(q,p)Y (q). (2.26) S −2iπ S −2iπ Zγ Zγ Note that, by the Sokhotskii-Plemeljj formula,

Y [ϕ](p+) Y [ϕ](p )= ϕ(p)Y (p), Y [ϕ∨](p+) Y [ϕ∨](p )= ϕ∨(p)Y (p), S − S − S − S − − p γ. (2.27) ∈

Lemma 2.4 Let ψn be the n-th biorthogonal section for the pairing (2.14). Then ρn := Y [ψn] has the property S

div(ρn) Dn. (2.28) ≥

Viceversa, if ψn Pn is such that ρn = Y [ψn] vanishes at Dn then ψn is a biorthogonal section. Similar statement ∈ S holds for ψn∨.

Proof. Recall that div(ψn) Dn and that Dn > Dn. We proceed by induction on n. The divisor D = p +p ≥− +1 +1 2 1 2 consists of only two points (possibly repeated) and hence ψ1 has at most two simple poles or a double pole at D2. P D Let ϕ0∨ be a spanning section of 0∨; namely, it has a necessarily simple pole at 1 = p1 since by assumption 0 h ( ∨ √ ) = 0. Then, by the Cauchy theorem X ⊗ K

res ϕ0∨ρ1 = ϕ0∨ ((ρ1)+ (ρ1) )= ϕ0∨ψ1Y, (2.29) − − Zγ Zγ where in the last step we have used the Sokhotskii-Plemeljj formula and res denotes the sum over all residues

(outside γ) of the expression. In this case it is just the residue at p1 where ϕ0∨ has a pole. Now ψ1 is a biorthogonal section if and only if (2.29) vanishes. But this means that ρ1 must vanish at p1 = D1. This concludes the initial step of induction. Induction step. Similar considerations show that, for j =0,...n 1 −

res ϕ∨ρn = ϕ∨ψnY. (2.30) D j j n Zγ

By induction, div(ρn) Dn 1 so there is only one residue to consider at Dn Dn 1 = p⋆. If this point appear with D ≥ − − − multiplicity 1 in n then necessarily ϕn∨ 1 has a simple pole at p⋆ and then the vanishing of (2.30) implies that ρn must vanish as well. − If p⋆ has multiplicity k in Dn (and hence multiplicity k + 1 in Dn+1) then we can ﬁnd k + 1 sections in Pn 1 − with poles of orders 1, 2,...,k +1 at p⋆. The induction hypothesis implies that ρn already vanishes of order k at p⋆; then the pairing with a section with a pole of order k +1 at p⋆ promptly implies the vanishing of order k + 1.

Consider the two matrices

ψn Y [ψn] Y [ψn∨] Y [ψn∨ 1] Υn(p) := S , Υn∨(p) := S S − . (2.31) ψn 1 Y [ψn 1] ψn∨ ψn∨ 1 − S − − The Lemma 2.4 shows that their divisor properties (outside of γ) are

(Dn+1) ( Dn) ( Dn) ( Dn 1) Υn = O O − , Υn∨ = O − O − − . (2.32) (Dn) ( Dn 1) (Dn+1) (Dn) O O − − O O Moreover they satisfy the boundary value conditions (jump condition)

1 Y (p) 1 Y (p) Υn(p+)=Υn(p ) , Υn∨(p+)= − Υn∨(p ), p γ, (2.33) − 0 1 0 1 − ∈ where the subscript p denotes the boundary values at p γ from the left (+) or the right ( ). Generically the ± ∈ − (1, 1) entry of Υn is a section of Pn but not in Pn 1 (i.e., it really has an “extra pole”). Similarly, Y [ψn 1] − S − vanishes at Dn 1 but not at Dn. These two conditions specify a Riemann–Hilbert problem on − C

8 Riemann–Hilbert Problem 2.5 Find a 2 2 matrix Υn (Υ∨) of local meromorphic sections of √ ( ∨ √ , × n X⊗ K X ⊗ K respectively) satisfying the two conditions (2.32) and (2.33). Moreover we require that the (1, 1) entry is (Dn ) O +1 but not (Dn) and the (2, 2) entry is ( Dn) but not ( Dn ) O O − O − +1 Note that if Υn solves the RHP (2.5) we are still free to multiply it on the left by a diagonal constant invertible matrix. In general we can dispose of this freedom if we choose some convenient normalization. If we have chosen the bases ϕℓ, ϕℓ∨ then this normalization can be ﬁxed by requiring that

n 1 − (Υn)11(p)= ϕn(p)+ cℓϕℓ(p), res ϕn∨(Υn)22 =1. (2.34) Dn+1 ℓ X=0

We also observe that the product ΥnΥn∨ is a matrix of diﬀerentials without discontinuity across γ, with the (1, 1), (2, 2) entries being holomorphic diﬀerential and the (1, 2), (2, 1) entries having poles at Dn+1 Dn 1 = − − pn + pn+1. Note also that for n = 0 we have ϕ [ϕ ] Υ = 0 Y 0 (2.35) 0 0S 0 We now discuss uniqueness.

Proposition 2.6 The solution Υn of the RHP 2.5 (and similarly for Υn∨) with the normalization (2.34) exists and is unique if and only if the determinant

n 1 − Dn = det µa,b , µa,b = ϕa∨ϕbY (2.36) a,b=0 Zγ ψ∨ does not vanish. In that case the (1, 1) entry is the biorthogonal section ψn ( n , respectively). Dn Dn

Proof. Denote provisionally ψ = (Υn)11 and ρ = (Υn)22. The jump condition (2.33) implies that ψ extends analytically across γ to a meromorphic section of √ . The growth conditions (2.32) imply that div(ψ) Dn X⊗ K ≥− +1 and hence ψ Pn by deﬁnition. The normalization (2.34) implies that we can write ∈ n 1 − ψ(p)= ϕn(p)+ cℓϕℓ(p). (2.37) ℓ X=0

Now the condition that Y [ψ] (in the (1, 2) entry of Υn) is ( n ) means that it vanishes at Dn ; but this S O −D +1 +1 is equivalent to the orthogonality of ψ to all ϕ0,...,ϕn 1 by Lemma 2.4. This condition in turn translates to the linear system; − µ00 ... µ0,n 1 c0 µ0n . . − . .  . .   .  =  .  (2.38) µn 1,0 ... µn 1,n 1 cn 1 µn 1,n  − − −   −   −        which is then uniquely solvable if and only if the determinant Dn = 0. Now consider the (2, 2) entry ρ and let ψ 6 denote the (2, 1) entry of Υn. The jump condition implies that ρ = Y [ψ] and the growth condition implies that S ψ Pn 1. Thus we can write e ∈ − n 1 − e e ψ(p)= βℓϕℓ(p). (2.39) ℓ X=0 e The vanishing condition of ρ at Dn implies that ψ is an orthogonal section. The normalization condition reads

e δℓn = res ϕℓ∨ρ = ψϕℓ∨Y, 0 ℓ n. (2.40) Dn ≤ ≤ Zγ e

9 This is equivalent to the following system:

µ00 ... µ0,n 1 β0 0 . . − . .  . .   .  =  .  (2.41) µn 1,0 ... µn 1,n 1 βn 1 1  − − −   −          Then Dn = 0 if and only if this system (2.41) is uniquely solvable. 6

As noted in [4], in genus zero the matrix of bimoments µab is a Hankel matrix and then the vanishing Dn =0 precludes the compatibility of the two systems (2.38), (2.41). In higher genus one may have Dn = 0 and yet still have compatibility of the two systems (2.38), (2.41) in some non-generic situations. The RHPs 2.5 characterize (normalized) biorthogonal sections by taking the (1, 1) entry of the solutions. The existence of the Riemann–Hilbert characterization of the (normalized) biorthogonal sections is a potentially useful tool to investigate asymptotic issues; this would require to develop a higher genus analog of the Deift–Zhou steepest descent method already used successfully in the ordinary (genus zero) orthogonality [12].

2.2.1 Pad´eapproximation

In a similar vein as [4] we show that the biorthogonal sections ψn, ψn∨ are the denominators of a suitable Pad´e–like pair of approximation problems for the two Weyl “functions”

(p)= S(p, q)ϕ∨(q)Y (q). ∨(p)= ϕ∨(q)S(q,p)Y (q), (2.42) W 0 W 0 Zγ Zγ They are holomorphic sections of √ and ∨ √ on γ, with a jump discontinuity across γ. We will X ⊗ K X ⊗ K C\ consider the Padéproblem for , the one for ∨ being formulated by simply swapping the roles of , ∨. W W X X 0 D 0 D PadéProblem 2.7 Given fixed spanning sections ϕ0, ϕ0∨ of H ( √ ( 1)), H ( ∨ √ ( 1)), respectively, R R X ⊗ K X ⊗ K let = div(ϕ0), ∨ = div(ϕ0∨) be their divisors (of degree g 1). 0 − 0 Find a section Qn 1 H ( ∨ √ (Dn+1 D1) and a section ψn H ( ∨ √ (Dn+1) such that − ∈ X ⊗ K − ∈ X ⊗ K ϕ∨(p)Qn(p) ψn(p) ∨(p)= ( Dn R∨). (2.43) 0 − W O − − Since the degree of Dn + R∨ is n + g, we have no choice but set

ϕ∨(p)Qn(p)= ψn(p)ϕ∨(q)S(q,p) ϕ∨(p)S(p, q)ψn(q) Y (q), (2.44) 0 0 − 0 Zγ and observe that the expression is a meromorphic diﬀerential without discontinuity on γ. Now (2.43) becomes a constraint on the remainder

ϕ∨(p) ψn(q)Y (q)S(p, q)= ( Dn R∨) (2.45) 0 O − − Zγ We can then divide by ϕ0 and obtain

ρn(p) := ψn(q)Y (q)S(p, q)= ( Dn) (2.46) O − Zγ As we have seen in Lemma 2.4 this is the requirement that ψn is a biorthogonal section.

Orthogonality. The spaces P, P∨ are diﬀerent spaces simply because = ∨. However, if takes values in X 6 X X Z = 1, 1 then ∨ = and the pairing is actually a non-hermitian inner product. In this case one promptly 2 { − } X X sees that the matrix of bimoments µa,b (2.16) is symmetric and ψn = ψn∨.

2.3 Christoﬀel-Darboux-Szeg¨oreproducing kernels

Definition 2.8 The Christoffel–Darboux-Szegö(CDS) kernel is the expression

n 1 − 1 Kn(p, q) := ψj (p)ψj∨(q). (2.47) hj j=0 X

10 P P It is immediate to verify that this kernel is the projection on the subspaces n 1, n∨ 1 in the following sense − −

ψj (p) j n 1 ψj∨(q) j n 1 Kn(p, q)Y (q)ψj (q)= ≤ − , Kn(p, q)Y (p)ψ∨(p)= ≤ − (2.48) 0 j n j 0 j n Zγ ≥ Zγ ≥ and the kernel is reproducing:

Kn(p, q)Y (q)Kn(q,t)= Kn(p,t) , Kn(p,p)Y (p)= n. (2.49) q γ Z ∈ Z In order to discuss a “Christoﬀel–Darboux” type of theorem one needs more structure; the nested sequence of divisors Dn will be taken with support within the pole divisor Ξ of the projection map Z of degree r. More precisely,

Assumption 2.9 We consider the sequence of divisors Dn such that Dkr+ℓ = kΞ+ Rℓ with Rℓ positive divisor of degree j r 1, Rj Ξ. ≤ − ≤ Example 2.10 If Ξ= r (j) is a simple divisor, then we may choose j=1 ∞ P ℓ r (j) (j) Dkr ℓ = (k + 1) + k , (2.50) + ∞ ∞ j=1 j ℓ X =X+1 namely we increase the multiplicities of the points at inﬁnity sequentially and periodically. Another example is Ξ = (r 1) (1) + (2); then we may choose − ∞ ∞ (1) Dkr ℓ = kΞ+ ℓ ℓ r 1. (2.51) + ∞ ≤ − A third example is simply Ξ= r , namely, the map Z has a single pole of order r; here the only choice we have is ∞ to take Dn = n . ∞

Let the Assumption 2.9 prevail; then clearly Z(p)ψn(p) Pn r and similarly Z(p)ψ∨(p) P∨ just by ∈ + n ∈ n+r inspection of the degree of the poles. Let us denote by Ψ~ the inﬁnite column vector (ψ0, ψ1,... ) and Ψ~ ∨ the inﬁnite row-vector (ψ0∨, ψ1∨,... ). Thanks to the fact that the multiplication by Z is symmetric

Zϕ, ϕ∨ = ϕ, Zϕ∨ , ϕ P, ϕ∨ P∨, (2.52) h i h i ∀ ∈ ∀ ∈ it follows that the there is an N N matrix Z such that ×

Z(p)Ψ(~ p)= ZΨ(~ p),Z(p)Ψ~ ∨(p)= Ψ~ ∨(p)Z. (2.53)

The lemma hereafter follows immediately from (2.52) and from the degree counting mentioned above.

Lemma 2.11 The N N matrix Z is a ﬁnite-band matrix of size 2r +1 with Za,b =0 if a b > r. × | − | It is convenient to view Z as a tridiagonal block matrix with blocks of size r and similarly to segment the vectors Ψ~ , Ψ~ ∨ into pieces of length r: ~ t ~ Ψk = [ψkr,...,ψkr+r 1] , Ψk∨ = [ψkr∨ ,...,ψkr∨ +r 1], k =0, 1,.... (2.54) − − Then (2.53) can be written

Z(p)Ψ~ k(p)= Ak+1Ψ~ k+1(p)+ BkΨ~ k(p)+ Ck 1Ψ~ k 1(p), (2.55) − − ~ ~ ~ ~ Z(p)Ψk∨(p)= Ψk∨ 1(p)Ak + Ψk∨(p)Bk + Ψk∨+1(p)Ck, (2.56) − where Ak, Bk, Ck are r r matrices. Then we have the simple Corollary (whose proof is immediate) × Corollary 2.12 (“CD identity”) The CDS kernels for n = ℓ r can be written as follows

1 1 Ψ~ ∨(q)H− Aℓ+1Ψ~ ℓ+1(p) Ψ∨ (q)CℓH− Ψ~ ℓ(p) K (p, q)= ℓ ℓ − ℓ+1 ℓ (2.57) ℓr Z(p) Z(q) − where Hℓ = diag(hℓr,...,hℓr+r 1). −

11 3 Matrix (bi)orthogonality

We consider a sequence of divisors Dn chosen according to Assumption 2.9. The first important realization, contained in the next theorem, is that then the sections of Pn Pr 1 can be written as linear combinations of sections in \ − Pr 1 with polynomial coefficients in Z. −We introduce the following notation; we choose a dissection, C˙ = C Σ, of the z–plane so that we can represent the Riemann surface by r copies of the z–plane appropriately identified\ along the various cuts of the dissection. C (a) 1 For z C˙ we denote by z , a = 1,...,r the point in Z− (z) belonging to the corresponding sheet: two points z(a∈),z(b), a = b coincide at z = c if and only if the value c ⊂is aC critical value of Z, namely dZ vanishes at 6 1 some of the points in Z− (c).

r Theorem 3.1 Let ϕj be any basis of holomorphic sections of √ (Ξ), with Ξ the divisor of poles of the { }j=1 X ⊗ K map Z : CP1. Let C → (1) (r) ϕ1(z ) ... ϕ1(z ) . . Φ(z)=  . .  (3.1) ϕ (z(1)) ... ϕ (z(r))  r r    Then the determinant det Φ(z) vanishes only at the critical points of minimal order in the following sense; for any collection ψ1,...,ψr of meromorphic sections of √ (and poles away from the ramiﬁcation points of the map Z) the ratio X ⊗ K r (a) det ψj (z ) F (z) := j,a=1 (3.2) det Φ(z) is a rational function of z without poles at the critical points (i.e. the determinant in the numerator is “divisible” by detΦ)

We could not find an elementary proof that does not use the theory of Theta functions and thus, inevitably, the proof we provide in Appendix A.1 is somewhat technical. However, the reader may find the genus 1 example in Section 4.2 sufficiently illuminating.

0 Proposition 3.2 Let n r and ϕ Pn Pr 1. Let ϕj 1 j r be any basis of H ( √ (Ξ)). Then the following two statements≥ hold. ∈ \ − { } ≤ ≤ X ⊗ K [1] We have the expansion r ϕ(p)= fj(z)ϕj (p), z = Z(p) (3.3) j=1 X n n where fj are polynomials of degree and at least one of them is of degree exactly . ≤⌊ r ⌋ ⌊ r ⌋ [2] Let ψℓ ℓ 0 be an arbitrary sequence such that ψℓ spans Pℓ mod Pℓ 1 and arrange the coeﬃcients of the expansion{ (3.3} ≥) in matrix form as follows −

ψkr(p) ψ0(p) . (k) .  .  = F (z)  .  , z = Z(p). (3.4) ψkr+r 1(p) ψr 1(p)  −   −      Then F (k)(z) is a polynomial matrix of degree k in the indeterminate z and the leading coeﬃcient matrix is invertible.

Proof. [1] It is clear that any linear combination (3.3) belongs to Pn so we need to show the converse. We give here a rather pedestrian proof of this fact. For z C˙ let ~ϕ(z) := ϕ(z(1)),...,ϕ(z(r)) be the vector of its values at all the preimages of z. If σ is a closed loop starting∈ and ending at z and avoiding the critical values of Z,

12 then the analytic continuation of ~ϕ along σ yields the same vector up to a quasi-permutation matrix Pσ, where the multipliers are determined by the multiplier system. Consider then the matrix Φ inX (3.1): under analytic continuation along σ the matrix Φ undergoes the same transformation: σ Φ(z )=Φ(z)Pσ. (3.5) 1 Moreover Φ is invertible on C˙ because of Thm. 3.1. Thus we can consider F~(z) := ~ϕ(z)Φ− (z). It follows at once that this vector extends to a single valued function of z on C minus the critical points of Z, where it can have at most poles so that F~ (z) is at most a rational function of z. We now show that it has no poles at the critical points, and thus it is a polynomial whose degree is easily determined to be as claimed. The j-th entry of F~ (z) can be written as a ratio of determinants: detΦ f (z)= j (3.6) j detΦ where Φj is the matrix obtained by replacing the j-th row of Φ with the vector ~ϕ (Cramer’s rule). Then the Theorem 3.1 states that the ratio is locally analytic also at the critical values. It follows from the Liouville theorem that fj are polynomials. The degree follows from simple counting of the order of the poles. [2] As per Assumption 2.9 we have div(ψkr+ℓ 1) kΞ Rℓ. To simplify the considerations we write Ξ = (1) (r) − ≥ − −(1) (ℓ) + + (the points may be repeated), and Rℓ = + + , ℓ r 1 with R0 being the empty ∞ ··· ∞ 0 ∞ ··· ∞ ≤ − divisor. Then the sections ψ0,...,ψr 1 form a basis of H ( √ (Ξ)) and we can apply the result at point [1] − (k) Xk ⊗ K(k) of this proof. Let us denote by Fk the coeﬃcient matrix of z of F (z) of the polynomial matrix:

(k) k (k) (k) F (z)= z Fk + G (z). (3.7) k where G (z) is a polynomial matrix of degree k 1. Now, the sections ψkr,...,ψkr+r 1 span Pkr+r 1 mod Pk(r 1) − − (k)≤ − t − and the r sections given by the entries of G (z)[ψ0,...,ψr 1] fall within Pk(r 1) by simple degree counting. Thus the r entries of − − ψ0 k (k) . z Fk  .  (3.8) ψr 1  −    P P (k) must span the r–dimensional space kr+r 1 mod k(r 1) and hence Fk must be invertible. − −

Remark 3.3 If we consider an arbitrary increasing sequence of divisors Dn of degree n supported at Ξ but we waive the requirement that Dkr+ℓ = kΞ+ Rℓ, then the Proposition 3.2 still holds save for the statement about the degrees. In this more general case the sequence of degrees of the polynomials fj depends not just on the degree of 1 1 Dn but on the choice of points. As a simple example of this phenomenon consider the map Z : CP CP given by Z(t) = 1 (t 1 ). As basis of √ (Ξ) (there is no nontrivial flat bundle on CP1) we can choose ϕ→= √dt and 2 − t K 1 √dt D ϕ2 = t . If n = n t, for example then the degrees of the polynomials in (3.3) are clearly n and not n/2 because the function Z ∞has a simple pole at infinity. ⌊ ⌋ P Similar statements to those of Theorem 3.1 and Proposition 3.2 hold for sections of n∨ with the obvious modifi- cations. The following corollary is then immediate

Corollary 3.4 Let ψj , ψj∨ be the biorthogonal sections for the pairing (2.14). Denote by Ψk(z), Ψk∨(z) the following matrices (1) (r) (1) (1) ψrk(z ) ... ψrk(z ) ψrk∨ (z ) ... ψrk∨ +r 1(z ) − . . . . Ψk(z)=  . .  , Ψk∨(z)=  . .  (3.9) (1) (r) (r) (r) ψrk+r 1(z ) ... ψrk+r 1(z ) ψrk∨ (z ) ... ψrk∨ r (z )  − −   + 1     −  Then there exist matrix–valued polynomials Pk(z), Pk∨(z) of degree k such that

Ψk(z)= Pk(z)Φ(z) , Ψk∨(z)=Φ∨(z)Pk∨(z), (3.10)

r r where Φ, Φ∨ are the matrices constructed out of a pair of arbitrary holomorphic bases B = ϕℓ , B∨ = ϕ∨ { }ℓ=1 { ℓ }ℓ=1 of √ (Ξ), ∨ √ (Ξ) (respectively) as follows X ⊗ K X ⊗ K (b) (a) Φab(z)= ϕa(z ); Φab∨ = ϕb∨(z ). (3.11)

13 The matrix polynomials Pk, Pk∨ will be our matrix-orthogonal polynomials under a few additional assumptions. 1 Let γ0 C be an oriented contour in the plane and γ = Z− (γ0): let Y (z) as before, a smooth function on γ and deﬁne⊂ Λ(z) = diag Y (z(1)),...,Y (z(r)) . (3.12) Note that if γ consists of several connected components, we are free to choose the function Y on each of them, namely, we are not requiring that Y extends outside of γ but simply that it is smooth on each of its connected components. We also deﬁne the matrix weight on γ0 by the formula

W (z)dz := Φ(z)Λ(z)Φ∨(z). (3.13)

In general Λ is not the diagonalization of the weight matrix because Φ∨ is not the inverse of Φ. This reﬂects the arbitrariness of choices of bases B, B∨ which aﬀect W by the left/right action of GLn(C). We then have the simple theorem

P P Theorem 3.5 Assume that Dn = 0 in (2.16) so that the biorthogonal sections ψn, ψn∨ span n mod n 1 and P P 6 − n∨ mod n∨ 1 respectively. The matrix–valued polynomial sequences Pk(z) k 0, Pk∨(z) k 0 deﬁned in (3.10) are a biorthogonal− sequence for the matrix weight W (z)dz (3.13): { } ≥ { } ≥

Pk(z)W (z)Pℓ∨(z)dz = δℓkHℓ, (3.14) Zγ0 where Hℓ is the diagonal matrix

Hℓ = diag hℓr,...,hℓr+r 1 . (3.15) − We can normalize the two sequences to be monic

(k) 1 (k) 1 Qk(z) = (Fk )− Pk(z),Qk∨(z) = (Fk∨ )− Pk∨(z), (3.16)

(k) (k) where the matrices Fk and Fk are the leading coeﬃcients of Pn, Pn∨ (respectively), invertible by Prop. 3.2. In this case the “norming” matrices are:

(k) 1 (k) 1 Qk(z)W (z)Qℓ∨(z)dz = δℓkHℓ. Hℓ = (Fk )− Hℓ(Fk∨ )− . (3.17) Zγ e e

Proof. By construction we have Pk(z)W (z)Pℓ∨(z)=Ψk(z)Λ(z)Ψℓ∨(z). (3.18) Consider the entry (a,b) of the above matrix;

r (j) (j) (j) (Ψk(z)Λ(z)Ψℓ∨(z))a,b = ψkr+a(z )Y (z )ψℓr∨ +b(z ). (3.19) j=1 X

Integrating this expression on γ is the same as the integral over γ of ψkr+aY (p)ψℓr∨ +b(p), which gives δℓkδa,bhℓr+a. b Z2 characters and orthogonality: symmetric weight matrices. Let us consider again the case : π1 Z2. t t X →t In this case we can choose the same bases ϕℓ = ϕℓ∨ so that Φ∨ = Φ . Then we have Ψk∨ =Ψk and Pk(z)= Pk(z). The weight matrix W is then a symmetric matrix. Under a few additional assumptions one can easily consider a setup where W is actually positive definite. It suffices to consider curves admitting an antiholomorphic involution that fixes γ and positive–valued weight function Y . We can only considerC meromorphic functions Z : CP1 that intertwine the anti-involution on with some antiholomorphic involution on CP1 (i.e. the conjugateC → by an C arbitrary Möbiusb transformation of the ordinary complex conjugation).

14 3.1 Matrix Christoffel-Darboux from scalar Christoffel–Darboux–Szegö As observed in the special case where genus( ) = 0 in [10] (see Thm. 1.16 ibidem) the matrix-Christoffel-Darboux (MCD) reproducing kernel can be expressed inC terms of the scalar one. Specifically we recall the standard definition

Deﬁnition 3.6 The matrix Christoﬀel–Darboux kernel is

ℓ 1 − K 1 ℓ(z, w) := Pj∨(z)Hj− Pj (w), ℓ =1,.... (3.20) j=0 X

The kernel satisﬁes

Kℓ(z,t)W (t)Kℓ(t, w)dt = Kℓ(z, w), Kℓ(z,z)W (z)dz = ℓ 1r. (3.21) Zγ Zγ

It follows directly from the deﬁnitions of Pj (z), Pj∨(z) that we have

Corollary 3.7 The matrix Christoffel-Darboux kernel Kℓ(z, w) in (3.21) and the (scalar) Christoffel–Darboux– Szegökernel Kn in (2.47) are related by K (a) (b) [Φ∨(z) ℓ(z, w)Φ(w)]ab = Kℓr(w ,z ). (3.22)

3.2 Change of multiplier system X We briefly recall, following ideas explored in [6], of the effect of change of multiplier system . X → X Under our assumptions that h0( √ )=0= h0( √ ) (which, we remind, is equivalent to Θ( ) =0 = X ⊗ K X ⊗ K X 6 6 Θ( ))), there are sections, ϕ and ϕ , of the two bundles with a simple pole and g zeros. We choosee the pole to X 0 0 ϕ0 e be at the same point p0. Then the ratio, F , = ϕ , is a meromorphic section of the flat bundle ∨ with g X X e0 X ⊗ X zerose and g poles. Namely a multivaluede meromorphice function on such that, under analytic continuation around C a loop σ π , it satisfies e ∈ 1 σ (σ) F , (p )= X F , (p). (3.23) X X (σ) X X e Xe e This function can be simply written in terms of the Riemann Theta function as

Θ p p + g A 0 j=1(αj αj ) j (p) − X F , (p)=eP − (3.24) X X e Θ (p p0 + ) e − Xe where we recall that we identify the multiplier system ( ) with the point in the Jacobian J( ) (of the same symbol) X X C 2iπαj 2iπβj = β τ α,, (aj )=e e , (bj )=e , (3.25) X − X X and similar formulas for . Let X ϕ e div 0 = D D, deg D = deg D = g. (3.26) ϕ − 0 e 0 e e Now, any holomorphic ϕ section in H ( √ (Ξ)) can be written as ϕ(p) = f(p)ϕ0 with f(p) a meromorphic function with divf Ξ+ D. Let us setX ⊗ K ≥− (1) (r) D(z) := diag F , (z ),...,F , (z ) . (3.27) X X X X e e

15 Proposition 3.8 Given two flat bundles , and corresponding bases B, B∨, B, B∨ of the corresponding spaces, X X let Φ, Φ∨ be defined as in (3.11) and similar expressions defining the tilde counterparts. For a given weight function Y : γ C the weight matrices W (z), W (z) aree related by e e → f W (z)= L(z)W (z)R(z), (3.28) with f 1 1 1 L(z) := Φ(z)D− (z)Φ− (z), R(z) := (Φ∨(z))− D(z)Φ∨(z). (3.29) Both L(z), R(z) are rational functions with poles at Z(D) and Z(D), respectively. e e e Proof. Since both D and Λ are diagonal matrices we can write the weight matrix as

1 W (z) := Φ(z)D(z)Λ(z)D− (z)Φ∨(z). (3.30)

Then (3.28) immediately follows. The important information that the theorem is conveying is that both L(z) := 1 1 1 Φ(z)D− (z)Φ− (z) and R(z) := (Φ∨(z))− D(z)Φ∨(z) are rational (i.e. single–valued) functions of z with poles only at the Z–projection of the two divisors D, D of degree g. We inspect L(z) with the considerations extending toe R(z) by swapping the roles of , . The matrixe Φ(z)D(z) is of the form X X e (b) e (b) ϕ0∨(z ) [Φ(z)D(z)]ab = ϕa(z ) (b) , a,b =1, . . . , r. (3.31) ϕ0(z )

ϕ0(p) 1 For each a = 1,...,r the a-th row is obtained by evaluating ̟a(p) := ϕa(p) at the points Z− (z). It is ϕe0(p) apparent that ̟a is a meromorphic section of √ (Ξ) with poles at D coming from the zeros of ϕ0. Then 1 X ⊗ K Φ(z)D(z)Φ(z)− is single–valued by Theorem 3.1 (applied to the basis B). e 3.3 Abelianization:e from matrix to scalar orthogonalitye The above construction starts with a Riemann surface and a projection Z. It is interesting to give a general framework where we start from the weight matrix W (z) itself. This can be done at least under the assumption that W (z) is a rational matrix; in this case there exists a well established general spectral theory which we brieﬂy summarize below. In the context of MOPs, this precise approach has been investigated recently in [10] and hence we will not delve into all details. The reader that would like to have a geometrical approach to the so–called “(inverse) spectral problem” (such is the issue at hand) should consult for example Chapter 5 of [3] for a recent review, or the papers [1, 2, 6] We preface this construction by warning that it is not quite the inverse construction to the previous one; the reason is that we will use as weight on the Riemann surface the eigenvalues of W while in the previous section the matrix Λ is not the matrix of eigenvalues of W in (3.13) in general.

The spectral curve is simply the locus of the characteristic polynomial of W : C 2 := (y,z) C : det(y1r W (z))=0 . (3.32) C ∈ − n o The function Y used in the previous section is now the eigenvalues of W on the spectral curve, and Z is the natural projection to the z–plane. The curve can be compactified under general assumptions and we denote by g the genus of this compactification. C If (y,z) then the matrix M(y,z) := y1r W (z) is singular and generically its adjugate matrix M(y,z) ∈ C − (the transposed matrix of cofactors) has rank one. Any nonzero column of M is then a right eigenvector and any nonzero row is a left eigenvector. Note that we are not assuming that W be a symmetric matrix. f Choose one such row R~ and column C~ : choose one of the entries of eachf (say, the first) and then it can be seen that they have g + r 1 zeros. The divisors of these zeros determine line bundles of the same degree L , L ∨ dual to each other. Consider− the case of L ; the divisor is linearly equivalent to a divisor of the form D (1) + Ξ, where D is positive and of degree g and (1) is one of the preimages of z = . Let us fix a normalized− ∞ basis of ∞ ∞

16 holomorphic diﬀerentials on and the corresponding Abel map A with basepoint (1). We denote by α, β the (real) characteristics of D asC follows, ∞

g A(D)= β τ α, τ ij = ωj = ωi, β, α R . (3.33) − ∈ Ibi Ibj Let us denote by = A(D) = β τ α; this point defines also a multiplier system which we denote by the same symbol : π ( ) X U(1) with slight− abuse of notation according to the formula: X 1 C → 2iπαℓ 2iπβℓ (aℓ)=e , (bℓ)=e , ℓ =1, . . . , g. (3.34) X X 1 r (j) For simplicity let us assume at first that Ξ = Z− ( ) = j=1 is a simple divisor; if S(p, q) := Sα,β(p, q) denotes the Szegökernel with the indicated characteristics∞ (see Appendix∞ A) then we can follow [18] and define the matrix P S(z(a), (j)) S( (k),z(b)) Φj,a(z)= ∞ , Φb,k∨ := ∞ (3.35) dζj √dζk (j) where dζj denotes any local coordinate near p in which to trivialize the semicanonical bundle √ . If we choose 1 ∞ K ζ = z (for simple divisor Ξ) then they satisfy

Φ(z)Φ∨(z)= 1rdz. (3.36) To see this (see [18]) one computes directly the j, k entry: r S( (k),z(a)) S(z(a), (j)) (Φ(z)Φ∨(z))jk = ∞ ∞ (3.37) √dζ a=1 k dζj X Since the expression is clearly invariant under permutation of the sheets,p and single valued, it must be a differential on the target curve of the z–plane; as such it can only have poles at due to the simple pole along the diagonal of the Szegökernel; however if j = k the pole is of order at most 1 and∞ since there are no nontrivial differentials with only one simple pole on the Riemann6 sphere, we conclude that it vanishes. For j = k the expression has a double pole at z = due to the fact that the Szegökernel has a simple pole on the diagonal and hence it is proportional to dz. To show∞ that the proportionality is 1 we recall that if p, q are in the same coordinate patch ζ then (2.22) 1 applies. Then, near j the coordinate is ζj = and hence one finds that the coefficient is 1 on all sheets. ∞ z More general divisor Ξ. If Ξ is not simple, then (see also App. A) we proceed as follows to construct sections (j) νj of √ (Ξ). Suppose that has multiplicity νj : then take local coordinate ζj = z− and consider the expressionsX ⊗ K ∞ 1 1 S(p, ζj− (ζj )) S(ζj− (ζj ), q) ϕ(p, ζj ) := ; ϕ∨(q, ζj ) := . (3.38) dζj dζj

Namely, we evaluate the Szegökernel in thep coordinate ζj with respect thep second variable (first, respectively). 1 Consider the expression ϕ(p, w); it is a section of √ in the variable p with a simple pole at ζj− (w). Thus, it depends parametrically on the complex variable wXin ⊗ a neighbourhoodK of w = . Similar considerations apply to ∞ ϕ(q, w) as a section of ∨ √ . X ⊗ K 1 ν The dependence on z is that of a Taylor series in ζj := w− j (i.e. a Puiseux Laurent series). Then one defines

ℓ 1 ℓ 1 1 d − ( 1)ℓ d − ϕℓ;j (p) := ϕ(p, ζj ) , ϕℓ;j (p)∨ := − ϕ∨(p, ζj ) ℓ =1,...,νj . (3.39) (ℓ 1)! dζj (ℓ 1)! dζj ζj =0 ζj =0 − −

It follows from the properties of the Szeg¨okernel that these are holomorphic sections of √ (ℓ j ) (and hence X ⊗ K ∞ also holomorphic sections of √ (Ξ)) with a single pole of order ℓ at j and normalized as X ⊗ K ∞ ℓ ϕℓ j (p)= ζj (p)− (1 + (ζj (p))) dζj (p). (3.40) ; O q In particular they are clearly linearly independent. Similar considerations apply to ϕ∨. Thus, for a general divisor k (j) Ξ= νj a pair of bases of holomorphic sections of √ (Ξ) and ∨ √ (Ξ) are j=1 ∞ X ⊗ K X ⊗ K P B := ϕℓ;j (p), j =1,...,k; ℓ =1,...,νj B∨ := ϕℓ∨;j (p), j =1,...,k; ℓ =1,...,νj . (3.41)

17 We arrange the bases in the natural order implied by the indexing and construct the matrices

(1) (r) ϕ1;1(z ) ... ϕ1;1(z ) . .  . .  (1) (r) ϕ1;ν1 (z ) ... ϕ1;ν1 (z ) (1) (1) (1) (1)   ϕ∨ (z ) ...ϕ∨ν (z ) ... ϕ∨ (z ) ...ϕ∨ (z )   1;1 1; 1 k;1 k;νk Φ(z) :=   Φ (z) := . .  .  ∨  . .   .  (r) (r) (r) (r)   ϕ∨ (z ) ...ϕ∨ (z ) ... ϕ∨ (z ) ...ϕ∨ (z )  ϕ (z(1)) ... ϕ (z(r))   1;1 1;ν1 k;1 k;νk   k;1 k;1     . .   . .   . .   ϕ (z(1)) ... ϕ (z(r))   k;νk k;νk    (3.42) Then we have

Proposition 3.9 The product Φ(z)Φ∨(z) is a constant matrix times dz of the form:

Φ(z)Φ∨(z) = diag Jν1 ,...,Jνk dz = Jdz (3.43) where each block Jℓ is a constant (in z) antitriangular matrix with unit on the antidiagonal entries: (Jℓ)a,r+1 b = 1 and nonzero entries only for a + b r +1. − − ≥

Proof. Without great loss of generality and to avoid a plethora of indices, we consider the case Ξ = r (i.e. ∞ with k = 1 and ν1 = r). The basis ϕj deﬁned in (3.39) behaves as (we write it directly in the coordinate form)

j ϕj (z)= ζ− 1+ (ζ) dζ, j =1,...r (3.44) O 1 p and ϕj∨ has the same behaviour. Here ζ = z− r is the local coordinate near . On the diﬀerent sheets the diﬀerent behaviours are obtained by multiplying ζ by the r roots of unity. Thus∞∈C

r r (a) (a) j k 2iπa (j+k 1) 2iπa (j+k 2) 2 ϕj (z )ϕk(z )= ζ− − e r − + c e r − ζ + (ζ ) dζ. (3.45) 1 O a=1 a=1 ! X X We know, a priori, that the result is a series in integer powers of z (this can also be seen by the fact that the sum of the roots of unity is zero). Moreover we know that the result must be a polynomial times dz because there are no singularities in the ﬁnite z-plane. Thus we conclude that we can have a nonzero result only for 2r 1 j +k 1 r. dζ − ≥ − ≥ For j + k 1= r we get r r = dz. − ζ +1 − Remark 3.10 It is clear that by a triangular change of basis we can arrange for basis sections in such a way that ΦΦ∨ = 1dz.

1 We now assume that the bases have been chosen so that the Remark 3.10 applies and hence ΦΦ∨ = 1dz, Φ− = 1 dz Φ∨. In this case the matrix Λ is indeed the diagonal matrix of eigenvalues of W = Φ(z)Λ(z)Φ∨(z) (while in the previous construction it needed not be). The construction of the matrix biorthogonal polynomials then proceeds by construction the (scalar) orthogonal sections as in Section 2 and then the resulting matrix biorthogonal polynomials as in Section 3.

4 Examples: genus zero and one

The reader without familiarity with the above algebro-geometric language may ﬁnd respite in contemplating the following two classes of examples in genus zero and one.

18 4.1 Genus zero cover We assume that = CP1, with aﬃne coordinate t and Z(t) is simply a rational function of degree r. C 1 We choose a path γ0 in the z-plane and its pullback γ = Z− (γ0). Given any smooth function Y (t) deﬁned on P P γ one can orthogonalize the spaces n, n∨ and produce instances of matrix orthogonal polynomials. To keep the example concrete and simple, we take Z(t) to be a polynomial of degree r:

r r 1 Z(t)= t + a t − + + ar. (4.1) 1 ···

Then Ξ = r t (we use the subscript to indicate that this is the point at inﬁnity in the t–plane). Since the map Z ∞ has a single pole, we necessarily must set Dn = n t so that Pn is identiﬁed with polynomials of degree n 1 in the t–variable. ∞ − Then the matrix weight on the z plane is given by

W (z)dz = Φ(z)Λ(z)Φ∨(z), z γ , (4.2) ∈ 0 where Φ, Φ∨ are constructed as in (3.42) and

Λ(z) := diag Y (z(1)),...,Y (z(r)) . (4.3)

Note that in general Λ is not the matrix of eigenvalues of W . Now the Proposition 3.2 translates to a simple construction in commutative algebra; a polynomial P (t) of degree n r can be written as ≥ P (t)= Q (t)Z(t)+ R (t), deg R (t) r 1, deg Q = n r. (4.4) 0 0 0 ≤ − 0 − by long division. If n r r then we can perform the long division on Q(t) and obtain − ≥ P (t)= Q (t)Z2(t)+ R (t)Z(t)+ R (t), deg Q = n 2r, deg R r 1. (4.5) 1 1 0 1 − 1 ≤ − n We clearly can repeat this procedure until we reach a step k = r where deg Qk r 1; thus we can uniquely write ⌊ ⌋ ≤ − k j P (t)= Rj (t)Z (t), deg Rj r 1. (4.6) ≤ − j=0 X We can now express each of the polynomials Rj (t) as a linear combination with constant coeﬃcients of the polynomials p0,...,pr 1 and rearrange the sum so as to get an expression of the form − r

P (t)√dt = fℓ(Z)ψℓ 1(t), ψℓ(t)= pℓ(t)√dt. (4.7) − ℓ X=1 This is the elementary version of the Proposition 3.2. Thus we are implicitly choosing the bases ϕℓ = ψℓ 1, ℓ =1,...,r. − Now consider the orthogonal polynomial pnr+a with 0 a r 1; by the above considerations we can write it as ≤ ≤ − r

ψnr+a 1(t)= pnr+a 1(t)√dt = Pn;a,ℓ(z)ϕℓ(t), z = Z(t), (4.8) − − ℓ X=1 and thus r (ℓ) t Ψn(z)= ψnr+a 1(z ) = Pn(z)Ψ0(z), Ψn∨(z)=Ψn(z) . (4.9) − a,ℓ=1 h i We thus have

Proposition 4.1 The matrix polynomials Pn(z) are orthogonal with respect to the matrix weight W (z)dz given in (4.2): t Pn(z)W (z)Pm(z)dz = δnmHn , Hn = diag hnr,...,hnr+r 1 . (4.10) − Zγ0

19 4.1.1 Laguerre matrix polynomials We give a simple example of the above scheme, but clearly one can devise inﬁnitely many such simple examples. Consider the map Z(t) = (t c)2, c 0. We can invert the map Z as − ≤ dz t , (z)= c √z, dt = ± (4.11) 1 2 ± 2√z

We consider the Laguerre polynomials Lα on R CP1: j + ⊂ t α t n α t− e d t n+α L (t)= e− t (4.12) n n! dtn α α t α Γ(n + α + 1) Ln(t)Lm(t)e− t dt = δnm . (4.13) R n! Z + We choose as basis for √ (Ξ) the sections ψ (t)= √dt and ψ (t)=Lα(t)√dt = (1+ α t)√dt, which we need to K 0 1 1 − evaluate at t1,2. Then we ﬁnd

1 i √dz t Φ(z)= 1 , Φ∨(z)=Φ (z). (4.14) α +1 c √z i(α +1 c + √z) √ 4 − − − 2z We note that 0 1 Φ(z)Φ (z)= dz (4.15) ∨ 1 2(c −1 α) − − − The half-line is mapped to [c2, ) CP1. We then set ∞ ⊂ z t (z)αe t1(z) 0 Λ(z) := 1 − (4.16) 0 0 2 2 Note that since c< 0 we have t1(c )= c + √c = 0. A short computation yields the weight matrix:

c √z 1 α +1 c √z α e− − WL(z)dz = Φ(z)Λ(z)Φ∨(z)= − − (c + √z) dz. (4.17) α +1 c √z (α +1 c √z)2 2√z − − − − α α Finally we express an arbitrary polynomial p(t) in terms of a linear combination of L0 (t) = 1 and L1 (t)= α +1 t with coeﬃcients being polynomials in z = Z(t). A direct veriﬁcation shows − p(s)(s +1+ α 2c)ds p(s)ds p(t)= res − +(1+ α t) res (4.18) s= z Z(s) − s= (Z(s) z) ∞ − ∞ − which allows to explicitly compute the MOPs:

Lα (s)(s+1+α 2c)ds Lα (s)ds 2j − 2j res z Z(s) res Z(s) z s= − s= − Pj (z)= ∞Lα (s)(s+1+α 2c)ds ∞Lα (s)ds . (4.19)  2j+1 − 2j+1  res z Z(s) res Z(s) z s= − s= −  ∞ ∞  By construction these polynomials satisfy

Γ(2j+α+1) ∞ t (2j)! 0 Pj (z)WL(z)Pk(z)dz = δjk Γ(2j+α+2) . (4.20) 2 0 Zc " (2j+1)! #

Note that Λ in (4.16) is not the matrix of eigenvalues for WL. This weight matrix (as well as the one in the next example) does not fall within the purview of [10] because it is not a rational matrix.

4.1.2 Hermite matrix polynomials To give another completely explicit example let us consider the Hermite polynomials in the t–plane and the same degree 2 map Z(t) = (t c)2, c R: − ∈ n n t2 d t2 hn(t) = ( 1) e e− , h (t)=1, h (t)=2t, . . . (4.21) − dtn 0 1

20 We choose as basis for √ (Ξ) the sections ψ (t)= √dt and ψ (t) = h (t)√dt =2t√dt, which we need to evaluate K 0 1 1 at t1,2. We then ﬁnd 1 i √dz Φ(z)= 1 (4.22) 2c +2√z 2i(c √z) √ 4 − 2z We note that 0 2 Φ(z)Φ(z)t = dz (4.23) 2 8c Note that the matrix in the right side can be factored as MM t with

i 1 − 0 2 M = √2c √2c ,MM t = (4.24) 0 √8c 2 8c " #

1 T This would allow us to re-deﬁne Φ = M − Φ and Φ∨ Φ∨M − in such a way that actually ΦΦ∨ = 1dz. However the ﬁnal weight matrix would not7→ be real–valued. 7→ 1 2 1 The image of the oriented path γ = R CPt under Z is the half-line γ0 = [c , ) CPz covered twice in opposite directions. We then set ⊂ ∞ ⊂

(c+√z)2 (c2+z+2c√z) e− 0 e− 0 Λ(z) := (c √z)2 = (c2+z 2c√z) (4.25) 0 e− − 0 e− − " − # " − # The minus sign is due to the fact that the orientation is ﬂipped on the second sheet. A short computation using Φ as in (4.22) yields the weight matrix:

t cosh(2c√z) 1 2c 0 1 z c2 W (z)dz = Φ(z)Λ(z)Φ (z)= 2sinh(2c√z) e− − dz. (4.26) √z 2c 4c2 +4z − 1 4c Note that for c = 0 the matrix W is diagonal and also the matrix orthogonal polynomials become diagonal. This is simply a manifestation that in this case z = t2 and the Hermite polynomials respect the parity. Finally we express an arbitrary polynomial p(t) in terms of a linear combination of h0(t) =1 and h1(t)=2t with coeﬃcients being polynomials in z = Z(t). A direct veriﬁcation shows

p(s)(s 2c)ds p(s)ds p(t) = h0(t) res − + h1(t) res (4.27) s= z Z(s) s= 2(z Z(s)) ∞ − ∞ − which allows to explicitly compute the MOPs;

h j (s)(s 2c)ds h j (s)ds 2 − 2 res z Z(s) res 2(z Z(s)) s= − s= − Pj (z)= ∞h j (s)(s 2c)ds ∞ h j (s)ds . (4.28)  res 2 +1 − res 2 +1  s= z Z(s) s= 2(z Z(s)) ∞ − ∞ −   The ﬁrst few are P0 = 1 and

4 c2 +4 z 2 4 c P1 = − − (4.29) " 16 c3 + 16 cz 12 c2 +4 z 6 # − − 16 z2 + 32 c2 48 z 48 c4 + 48 c2 +12 32 c3 + 32 cz 48 c P2 = − − − (4.30) " 128c5 + 320 c3 + 128 cz2 320 cz 16 z2 + 160 c2 80 z + 80 c4 240 c2 + 60 # − − − − 3 2 2 4 2 6 4 2 2 3 5 3  64z +576c −480 z +−320c −960c +720 z−320c +1440c −720c −120 192cz +640c −960c z+192c −960c +720c  P = . 3    3 3 2 5 7 5 3 3 2 2 4 2 6 4 2   768cz +1792c −5376c z +−1792c +6720c z−768c +5376c −6720c 64z +1344c −672 z +2240c −6720c +1680 z+448c −3360c +5040c −840  (4.31)

By construction these polynomials satisfy

2j ∞ t 2 (2j)! 0 Pj (z)W (z)Pk(z)dz = √πδjk 2j+1 . (4.32) 2 0 2 (2j + 1)! Zc

21 4.2 Elliptic covers 4.2.1 Genus 1

ω2 Let be an elliptic curve, which we represent as the quotient v C/Z + τZ with τ = H+. The curve can also ω1 be representedC in Weierstraß form: ∈ ∈ y2 =4z3 g z g = 4(z e )(z e )(z e ). (4.33) − 1 − 3 − 1 − 2 − 3 3 2 The uniformization is given by the Weierstraß’s ℘ function as y = (2ω1)− ℘′(v), z = (2ω1)− ℘(v) and

e1 e2 dz dz ω2 ω1 = , ω2 = , τ := (4.34) y e y ω1 Z∞ Z 1 1 1 1 ℘(v) := + (4.35) v2 (v + m + nτ)2 − (m + nτ)2 m2+n2=06 m,nX∈Z and then the curve (4.33) is identified with the 2–torus (i.e. the Jacobian J) v C mod Z + τZ, with the point ∈ 1 τ 1+τ z = corresponding to v = 0 and the three points e1,e2,e3 corresponding with the half-periods v = 2 , 2 , 2 , respectively.∞ The point at infinity in the z–plane is a double point corresponding to v = 0. Since the function z = Z(p)= ℘(v) has a single (double) pole at v = 0, the only choice of sequence of divisors is Dn = n . Let θ be the Jacobi (odd) theta function ∞ 1 iπ τ+iπ(v+ 1 ) iπn2τ+2iπn(v+ 1 + τ ) θ1(v; τ)=e 4 2 e 2 2 . (4.36) n Z X∈ The function θ1 is odd, vanishes at v = n + mτ of simple order and satisfies

2iπv iπτ θ (v +1)= θ (v), θ (v + τ)= e− − θ (v). (4.37) 1 − 1 1 − 1 Szegökernel. The Szegökernel for a multiplier system is expressed using the Jacobi theta function (4.36) as follows: X 2iπα(v w) θ1′ (0)θ1(v w ) S(v, w)=e − − − X √dv√dw, := β τα, (4.38) θ ( )θ (w v) X − 1 X 1 − where the multiplier system4 is (a)=e2iπα and (b)=e2iπβ, with a,b denoting the two cycles that correspond to v v + 1 and v v + τ, respectively.X The SzegökernelX consequently satisfies 7→ 7→ 2iπα 2iπα S(v +1, w)=e S(v, w), S(v, w +1)=e− S(v, w), 2iπβ 2iπβ S(v + τ, w)=e S(v, w), S(v, w + τ)=e− S(v, w). (4.39)

P 0 √ P 0 Following the general strategy, we can write a basis of = n 0 H (n+1) , ∨ = n 0 H ∨ ≥ X⊗ K ∞ ≥ X ⊗ √ (n + 1) as follows: L L K ∞ dℓ S(v, w) ( 1)ℓ+1dℓ S(w, v) ϕ (w)= ϕ∨ = − , ℓ =0, 1,... (4.40) ℓ dwℓ √ ℓ dwℓ √ dw w=0 dw w=0

For example, the sections of P are given by linear combinations of the following form (n 1) ≥ n θ (v aj ) ϕ(v)=e2iπvα 1 − √dv (4.41) θ (v) j=1 1 Y where a ,...,an are any numbers with aj = = β ατ mod Z + τZ. The Weyl sections are 1 X − 2 P θ′ (0) θ (v w ) θ (w ) (v)= √dv e2iπαv 1 1 − − X 1 − X Y (w)dw (4.42) W θ ( )2θ (w v) θ (w) Zγ 1 X 1 − 1 2 2iπαv θ1′ (0) θ1(w v ) θ1 (w + ) ∨(v)= √dv e− − − X X Y (w)dw (4.43) W θ ( )2θ (v w) θ (w) Zγ 1 X 1 − 1 4We make here an abuse of notation where X denotes at the same time the point in the Jacobian and the corresponding character × X : π1(C) → C .

22 Consider the matrices

ϕ0(v) ϕ0( v) ϕ0∨(v) ϕ1∨(v) Φ(z) := − , Φ∨(z) := , z = ℘(v). (4.44) ϕ (v) ϕ ( v) ϕ∨( v) ϕ∨( v) 1 1 − 0 − 1 − The following Lemma clariﬁes their mutual relationship: Lemma 4.2 The inverse of Φ(z) is given by

1 1 0 1 1 ϕ1∨(v) ϕ0∨(v) Φ(z)− = −2 Φ∨(z) = −2 . (4.45) 4ω dz 1 0 4ω dz ϕ∨( v) ϕ∨( v) 1 1 1 − 0 −

Proof. The matrices Φ, Φ∨ satisfy 2 0 1 Φ(z)Φ∨(z)= (2ω ) dz. (4.46) − 1 1 0 This can be seen as follows; by a direct computation one has the expansions near v = 0;

1 1 2 ϕ{∨}(v)= (1 + (v))√dv, ϕ{∨}(v)= (1 + (v ))√dv. (4.47) 0 v O 1 v2 O 3 Then one needs to ﬁnd the coeﬃcient of v− in the expansion of ΦΦ∨ at the origin. A direct computation yields

0 1 dv 2 ΦΦ∨ =2 (1 + (v )). (4.48) 1 0 v3 O We then use that dz 3 2 dv dv = , y = (2ω1)− ℘′(v) , 2ω1dz = 3 (1 + (v)). (4.49) 2ω1y ⇒− v O So we conclude that (4.46) holds which implies that the inverse of Φ is indeed given by (4.45)

The Proposition 3.2 states that for any section ϕ Pn we can write ∈

ϕ(v)= A(z)ϕ0(v)+ B(z)ϕ1(v), z = ℘(v)= Z. (4.50) with A, B some polynomials in z. It is seen directly that det Φ is an elliptic function of v with a triple pole at v =0 1 τ τ+1 and zeros at 2 , 2 , 2 and hence it is proportional to ℘′(v). Then 1 ϕ(v) ϕ( v) 1 ϕ (v) ϕ ( v) A(z)= det − , B(z)= det 0 0 − . (4.51) detΦ ϕ1(v) ϕ1( v) detΦ ϕ(v) ϕ( v) − − Here A, B are polynomials in z = ℘(v) because the determinant in the numerator vanishes at the same half periods (to order at least equal to) and the ratio is an even elliptic function with pole only at v = 0. Unfortunately there are no known explicit orthogonal systems on elliptic curves. We consider some examples corresponding to ﬁnite matrix orthogonality that are relevant for applications originating from the study of the Aztec diamond [13, 17] in the next section.

4.3 2R–torsion points on elliptic curves and ﬁnite (matrix) orthogonality The following class of examples is prompted by the geometric interpretation of the nice example contained in [13] studied in the context of the periodic tilings of the aztec diamond. We propose here a generalization of that setup. We comment in Remark 4.10 on how this is a generalization and how it compares to loc. cit. Consider the elliptic curve (4.33). On its Jacobian J, viewed as an abelian group, the 2R–torsion points are the points q J such that 2Rq 0. Namely they are the points of the form ∈ ≡ b a q = + τ , a,b Z. (4.52) 2R 2R ∈ Up to the equivalence q q +n+mτ (for m,n Z), we can assume a,b 0,..., 2R 1 . Another characterization of the R–torsion points∼ which is relevant for us∈ is the following: they are∈{ in correspondence− } with functions on the elliptic curve that have at most a unique pole of order R, a unique zero of the same multiplicity and such that both these points map to the same z–value (i.e. on the two sheets of the cover).

23 τ τ +1

Figure 1: The 2R-torsion points (here depicted for R = 4) are the points at the intersection of the dashed grid, modulo Z + τZ. The points that correspond to our functions are the torsion points that are not half-periods and modulo the hyperelliptic involution v v. They are indicated by dots. Their total number is 2R2 2. →− −

Amongst those, the ones that are non-trivial (i.e. non-constant) correspond to the torsion points that are not periods or half-periods; moreover any non-trivial such function Y has the property that the elliptic involution maps 1 Y Y − and hence we want to consider these torsion points modulo the elliptic involution as well. →The 2R–torsion points are actually algebraic points on the curve (meaning that they can be written as the solution of an algebraic equation in the coefficients of the curve). To find them one writes the most general elliptic function with pole of order R at z⋆: pR(z) y(z)qR 2(z) − F := − R (4.53) (z z⋆) − (ℓ) where p, q are polynomials of the indicated degrees. The condition is that the numerator, N(z), satisfies N (z⋆)=0 for ℓ =0, 1,..., 2R 1 with the determination of y(z) at z⋆ being the one corresponding to the chosen sheet. For − (R+ℓ) F to be non-constant we need to have qR 2 0. The equations N (z⋆)=0 for ℓ =1,...,R 1 give a linear − 6≡ − homogeneous system for the coefficients of qR 1 and hence the condition is that the determinant of this system vanishes. A direct computation shows that −

(R+1) (R) (R 1) (R+1)! (3) y (R + 1)y R(R + 1)y − ... (R 3)! y (R+2) (R+1) (R) (R−+1)! (4)  y (R + 2)y (R + 1)(R + 2)y ... (R 2)! y  det − = 0 (4.54) . .  . .   (2R 1) (R) (2R 1) (2R 1)! (R+1)   y − (2R 1)y (2R 2)(2R 1)y − ... − y   − − − (2R 3)!   −  2 where all derivatives are evaluated at z⋆. This gives a polynomial equation of degree 2R 2 in z⋆ and this is the number of (inequivalent) 2R torsion points. See Fig. 1. − Here is how these functions can be expressed using the Jacobi theta function (up to multiplicative scalar):

R b+aτ 2iπ a v θ1 v + 2R Y (v)= e R (4.55) θ v b+aτ 1 − 2R ! They also have the evident property that Y ( v)= 1 , namely, the elliptic involution that exchanges the sheets − Y (v) of the z–projection acts as reciprocal on these functions. This latter condition ﬁxes the multiplicative scalar to be 1. ± Remark 4.3 (“Prime” torsion points) Note that the expression in the bracket in (4.55) is not an elliptic func- 2iπa 2iπb tion but rather a meromorphic section of a ﬂat bundle a,b. This is the bundle with multiplier system e R , e− R on the A, B cycles, respectively. Namely, although it hasX pole and zero of multiplicity R, the function Y (v) is not the R power of a single–valued function with simple pole and zero, which cannot possibly exist on a curve of genus g 1. ≥

24 However, for example, if R is not a prime number and a,b have the same common divisor with R, then some of these functions are actually powers of that common divisor. We call “prime” those torsion points (4.52) where at least one of a,b is prime relative to 2R.

Remark 4.4 The case R = 1 of formula (4.55) gives constant functions. The case R = 2 is the ﬁrst interesting case, corresponding to quarter periods. This is the case implicitly used in [13] (see Sec. 4.10).

b+aτ (j) Let us set z⋆ := ℘( 2R ). According to our notation z⋆ , j = 1, 2 are the two points on the elliptic curve b+aτ corresponding to ℘(v)= z⋆ and hence they are (in the v–plane) . We now deﬁne ± 2R R R (z z⋆) (z z⋆) ψ (v) := − ϕ (v), ψ (v) := − ϕ (v), z = ℘(v), (4.56) 2R Y (v) 0 2R+1 Y (v) 1

(1) with similar expressions for ψ2∨R, ψ2∨R+1. Note that all these sections have a zero of order at least 2R at z⋆ on one (2) the main sheet, and are analytic at z⋆ . We deﬁne now the pairing (2.14) in this case to be

2 ϕ, ϕ∨ = ϕ(v)ϕ∨(v)Y (v), ϕ P, ϕ∨ P∨. (4.57) h i v z(1) =ǫ ∈ ∈ I| − ⋆ | (2) Note that we may integrate also on a circle around z⋆ but it would yield zero because there is no pole of the integrand there. It is evident, since Y has a pole of order R, that this pairing is of rank 2R and ψ2R,2R+1 (and their counterparts) P P ∨ span the 2–dimensional kernel of the pairing in 2R+1, 2∨R+1, respectively.

Matrix biorthogonality. We deﬁne, according to the general picture, the weight matrix W (z) by the formula

Y (z(1)) 0 W (z)=Φ(z) Φ(z) 1, (4.58) 0 Y (z(2)) − p 1 with Φ as in (4.44) and Φ− given as in Lemma 4.2. Note that √W is a rational function of z with a pole of order R at z = z⋆ and with unit determinant. Moreover it is bounded as z . The matrix → ∞

ψ2R(v) ψ2R( v) ΨR(z)= − , z = ℘(v), (4.59) ψ R (v) ψ R ( v) 2 +1 2 +1 − can then be written as per Prop. 3.2

ΨR(z)= PR(z)Φ(z), (4.60) 1 R 1 R Y − (v) 0 1 PR(z) = (z z⋆) Ψ2R(z)Φ− (z) = (z zK ) Φ(z) 1 Φ− (z)= − − 0 Y − ( v) − R 1 = (z z⋆) √W (z) (4.61) − − By construction, or by inspection, one concludes with the following

Lemma 4.5 The polynomial matrix PR of degree R satisﬁes

PR(z)W (z)Q(z)dz = 0 (4.62) z z⋆ =ǫ I| − | for all polynomial matrices Q(z).

Following the ideas in [13, 17] can ﬁnd explicitly the matrix polynomial PR 1 which is biorthogonal to all powers of zj1, j =0,...,R 2. − Interestingly, following− the idea in [13], one can write the MOP of degree R 1 in a rather explicit form as in the following Lemma. −

25 Lemma 4.6 The matrix–valued function dw R E 1 √ 1 PR 1(z) = (z z⋆) Ψ0(w) 11Ψ0− (w) W − (z) (4.63) − 2R − w z⋆ =ǫ (w z⋆) (w z)2iπ I| − | − − is a polynomial of degree R 1. − Proof. Since √W is a rational matrix and bounded at infinity, it is clear from the expression that PR 1 is of R 1 1 − growth bounded by z − at infinity. The residue at z = z⋆ is a Laurent polynomial in (z z⋆)− of degree at most − 2R 1. We now show that the total expression (4.63) does not have a pole at z⋆. To see this we use Cauchy’s − residue theorem on the integral, taking a circle centered at z⋆ that contains z. Then we find dw R E 1 √ 1 PR 1(z) = (z z⋆) Ψ0(w) 11Ψ0− (w) W − (z) − 2R − w z⋆ > z z⋆ (w z⋆) (w z)2iπ I| − | | − | − − 1 E 1 √ 1 Ψ0(z) 11Ψ0− (z) R W − (z). (4.64) − (z z⋆) − 1 R The term on the first line of (4.64) is analytic at z = z⋆ because √W − (z)(z z⋆) is analytic and so is the integral. The term on the second line of (4.64) may still, in principle, have a pole of− order R. However, by the definition (4.58) we have that it matches

1 1 R (1) 0 E 1 √ 1 (z z⋆) Y (z ) 1 Ψ0(z) 11Ψ0− (z) R W − (z)=Ψ0(z) − Ψ0(z)− . (4.65) (z z⋆) 0 0 − (1) 1 Since Y has a pole at z of order R, we see that R (1) is analytic at z = z⋆. (z z⋆) Y (z ) −

We now show that PR 1 is also a biorthogonal matrix polynomial. −

Proposition 4.7 The polynomial PR 1 is orthogonal to all matrix polynomials of degree R 2: − ≤ −

ℓ dz PR 1(z)W (z)z =0, ℓ =0,...,R 2. (4.66) − z z⋆ =ǫ − I| − |

Proof. We compute directly, j R√ j E 1 z (z z⋆) W (z) dz PR 1(z)W (z)z = dz dw Ψ0(w) 11Ψ0− (w) − (4.67) − 2R z z⋆ =2ǫ z z⋆ =2ǫ w z⋆ =ǫ (w z⋆) (w z)2iπ I| − | I| − | I| − | − − By Fubini’s theorem we can exchange the order of integration. Then, using the Cauchy theorem we can replace the ǫ integration on z z⋆ =2ǫ by an integration on z z⋆ = while also picking up the residue at z = w: | − | | − 2 j R√ E 1 z (z z⋆) W (z) (4.67)= dw dzΨ0(w) 11Ψ0− (w) − ǫ 2R w z⋆ =ǫ z z⋆ = (w z⋆) (w z)2iπ I| − | I| − | 2 − − wj √W (w)dw E 1 res Ψ0(w) 11Ψ0− (w) R . (4.68) − w=z⋆ (w z⋆) − R The term on the ﬁrst line of (4.68) vanishes because (z z⋆) √W is locally analytic. We now show that also the residue on the second line vanishes. Indeed we are computing− the residue of

Y (w(1)) j R 0 1 (w z⋆) w Ψ0(w) − Ψ0(w)− . (4.69) " 0 0 # The residue is the same as the residue of

(1) Y (w ) j R 0 j (w z⋆) 1 w w Ψ (w) − Ψ (w)− = √W (w), (4.70) 0 Y (w(2)) 0 R  0 R  (w z⋆) (w z⋆) − −   26 Y (w(2)) (2) because R is locally analytic at z⋆ thanks to the fact that Y has a zero of order R at z⋆ . But now, √W (w) (w z⋆) is a rational− matrix, bounded at inﬁnity. So, the residue of the expression (4.70) can be computed as the residue at inﬁnity. The latter is zero if j R 2. ≤ − 2 1 1+√1 a2 √ 2 Example 4.8 For the family of curves y = z(z a)(z a− ) the points z = ± a − ,a a 1 are quarter periods. − − ± − 2 For y = z(z 1)(z 8+4√3) one of the 16 sixth periods (R =3) is z⋆ =2. − − 4.4 A more general example in genus 1. Riding the same train of thoughts, we can consider for weight the elliptic functions having R distinct pairs of simple zeros/poles at points zj instead of (4.55):

R θ (v + v ) Y (v) := e2iπav 1 j (4.71) θ (v vj ) j=1 1 Y − where R b + τa vj = , a,b Z. (4.72) 2 ∈ j=1 X The function has simple poles at zj := ℘(vj ), j =1,...,R on one sheet and simple zeros at the same points on the other sheet; the values of these zj are not independent because of the constraint (4.72). Remark 4.9 To clarify the constraint alluded at above, we note that the function Y can be written as A(z) B(z)y(z) Y (p)= R− , (4.73) (z zj ) j=1 − where A, B are polynomials of degrees R, R 2, respectively,Q with B not identically zero and A monic (so as to dispose of the freedom of multiplication by a scalar).− Then the constraints are that the numerator and its derivative must vanish at zj: A(zj ) B(zj )y(zj )=0, A′(zj ) B′(zj )y(zj ) B(zj )y′(zj)=0. (4.74) − − − This gives 2R equations for the coeﬃcients of A (monic), B and the R positions of the points zj. So we see that the solutions form a family of dimension (R + R 1+ R) 2R = R 1. − − − △ Deﬁne

R qr(z) qr(z) ψ R(v) := ϕ (v), ψ R (v) := ϕ (v), z = ℘(v), qR := (z zj), (4.75) 2 Y (v) 0 2 +1 Y (v) 1 − j=1 Y with similar expressions for ψ2∨R, ψ2∨R+1. Once again, ψ2R,2R+1 (and the versions) have zeros at zj’s on one sheet and are holomorphic at the same points on the second sheet. We then∨ define the weight matrix W by the same formula (4.58). Observe that √W is still a rational function of z with unit determinant and with poles at the zj ’s and bounded at z = (v = 0). The pairing is defined by the same (4.57) with the contour of integration taken as ∞ the union of small circles around z = zj ’s (or a loop containing all of them at once). In the v–plane, we have R-pairs of circles surrounding v = vj (modulo Z + τZ) but the circles around vj do not contribute to the pairing. The ± − matrix ΨR is defined by the same (4.59) but now the matrix-polynomial PR is of the form:

1 PR(z)= qR(z)√W − (z) (4.76) In lieu of Lemma 4.6 and formula (4.63) we have the formula dw E 1 √ 1 PR 1(z)= qR Ψ0(w) 11Ψ0− (w) W − (z). (4.77) − 2 w z⋆ =ǫ qR(w) (w z)2iπ I| − | −

The proof that PR 1 is indeed a polynomial proceeds in the same way as for Lemma 4.6. The orthogonality to all j − R powers of z , j =0,...,R 2 is also shown as in Prop. 4.7, simply replacing expressions of the form (z z⋆) by − − qR(z).

27 Remark 4.10 (Comparison with [13]) In loc. cit. the authors consider the weight matrix (in their paper N = 2k)

2 2 1 2k 1 (z + 1) +4α z 2α(α + α− )(z + 1) W (z)= W1(z) , W1(z) := 1 1 2 2 , α> 0. (4.78) (z 1)2 2α− (α + α− )z(z +1) (z + 1) +4α− z − k Note that det W1 =1 and √W = W1 is a rational matrix as in our description. The spectral analysis of W1 is as follows. The spectral curve is of genus 1 and given by (see (2.11) in loc. cit.) C 2 2 2 y = z(z + α )(z + α− ). (4.79)

2 4 2 4 The quarter-period points of the curve (4.79) are z⋆ = 1, α √α 1, α− √α 1. In particular, note ± ± − − ± − − that the pole is at the quarter period z⋆ =1. The eigenvalue of W1 on the spectral curve is the function

2(a4 + 1)z + a2(z + 1)2 2a(a2 + 1)y Y = − 1 a2(z 1)2 − 2 2 which can be seen to have a double zero at (z⋆,y⋆)=(1, (1 + α )(1 + α )) and double pole at (z⋆, y⋆). This is − − a reﬂection of the fact that z⋆ = 1 is a quarter period as explained in the previous section. The weight W (z) has 2 2k p eigenvalue Y = Y1 which has pole of order 4k and hence R =2k in the notation of Section 4.3. We see here that the function Y is a perfect k power , i.e. does not correspond to “prime” torsion points (in the terminology of Rem. 4.3). The divisor for the eigenvector line bundle D of Sec. 3.3 is obtained by looking at the poles of the eigenvector normalized to have 1 in the ﬁrst entry and then found to be D = (0) + ( 1)(1). This divisor is equivalent to ( ) + ( 1)(2) which is seen by considering the function − ∞ − zc y α2 1 f(z,y) := − , c = (α2 1)(1 α 2)= − z(z + 1) − − − α p which has divisor of poles at (0) + ( 1)(1) and divisor of zeros at ( ) + ( 1)(2). So the multiplier system corresponds to the quarter period ( 1)−(2). ∞ − X − Acknowledgements. The work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06660.

A Theta functions and the Szeg¨okernel

We choose a Torelli marking a1,...,ag,b1,...,bg for in terms of which we construct the classical Riemann Theta functions. We refer to [{14], Ch. 1-2 for a review} ofC these classical notions. The corresponding normalized Abelian diﬀerentials, period matrix and Abel map will be denoted by ωj , τ , A, respectively:

p ωk = δjk, τ ij = ωi = ωj , Ap (p)= ~ω : J( ). (A.1) 0 C×C→ C Iaj Ibj Ibi Zp0 1 In our setting the basepoint, p0, of the Abel map will be chosen amongst the support of Ξ = Z− ( ). The basepoint will not be indicated except when necessary and the Abel map is naturally extended to a map on∞ divisors. For given characteristics [α, β] Rg Rg corresponding to the point e = β τ α J( ) in the Jacobian we ∈ × − ∈ C denote with S(p, q)= S[α,β](p, q) the Szeg¨o kernel:

Θ[α,β](p q) S(p, q)= S[α,β](p, q)= − (A.2) Θ[α,β](0)E(p, q) where E(p, q) is the Klein prime form [[14], pag. 16] and Θ[α,β] denotes the Riemann Theta function with characteristics (see loc. cit.). Following tradition, the Abel map of points or divisors used in arguments of Theta functions is understood for brevity.

28 The celebrated Fay identity, generalizing the tri-secant one, is the following equality

K K Θ[α,β] j=1(pj qj ) E(pj ,pk)E(qj , qk) − j

2iπαj 2iπβj (aj )=e , (bj )=e , j =1, . . . , g, (A.4) X X where αj ,βj are the (real) characteristics of the point e. Then the condition Θ(e) = 0 has the geometric interpretation that h0( √ ) 1. Furthermore the order of vanishing of Θ at e is precisely the value of h0( √ ). Q XK ⊗ K ≥ X ⊗ K Let := j=1 qj . Then the identity (A.3) can be interpreted in the same way as the Vandermonde determinant; 0 namely, it states whether the K spanning sections ϕℓ(p) := S(p, qℓ) of H √ (Q) can interpolate any local P X ⊗ K section deﬁned at the points of the divisor P = pj. j A small technical issue is when the divisor Q has points of higher multiplicity. As an extreme example if Q = Kq then one proceeds as follows to obtainP holomorphic sections of √ (Q); ﬁxing a local coordinate ζ 1 X ⊗ K near q1 (ζ(q1) = 0) one can write, for q is the coordinate patch

S(p, q)= ϕ(p, ζ)√dζ (A.5) where ϕ(p, ζ) is a section of the bundle √K with a simple pole at p = q (or, equivalently, a holomorphic section of √K(q)). Then one obtains sectionsX⊗ with higher order poles at q simply by X ⊗ 1 dj 1 ϕ (p) := − ϕ(p, ζ) , j =1,...,K (A.6) j dζj 1 − ζ=0

which can be interpreted as holomorphic spanning sections of √ (Kq1) In this case the Fay identity A.3 survives by taking appropriate limits and using l’Hopital’s rule andX ⊗ givesK

K Θ α β pj Kq K [ , ] j=1 − 1 j

r Θ ; (z(j) (j)) =Θ (0) . (A.8) [α,β]  − ∞  [α,β] j=1 X   Thus the Fay identity (A.3) (or the likes of it as (A.7) if Ξ has multiplicities) shows that the determinant, as a function of z, vanishes only at the critical points of Z.

29 Let c be a critical value of Z and pℓ the points above it. If pℓ has multiplicity νℓ then a local coordinate is 1 ν ζℓ = (z c) ℓ . − 1 We trivialize the bundle √ near the points Z− (c) using the above coordinates; this simply means that we X⊗ K write every section ϕj (p)= fj (ζℓ)√dζℓ and similarly ψj (p)= gj (ζℓ)√dζℓ where fj ,gj are analytic functions near 0 in the respective variables. The function F is thus the ratio of the determinants of two alternant matrices of the sets of functions fj ,gj. (j) (k) Near z = c the term responsible for the vanishing of det Φ is the Vandermonde–like term j

0 1 1 0 1 2 gj (p)= g + g (Z(p) c) ν + = g + g ζ + (ζ ). (A.10) j j − ··· j j O The diﬀerent sheets are distinguished by multiplying ζ by a ν–root of unity;

(a) 0 1 2iπa 2 gj(z )= g + g e ν ζ + (ζ ), a =1, . . . , ν. (A.11) j j O Thus the alternant matrix has the following expansion

0 0 2 G F + ζ Ω 0r (r ν) + (ζ ) (A.12) × − O 1 2iπa 0 0 0 where Ω Matr ν is the matrix Ωja = gj e ν and G Matr ν is the matrix Gja = gj of rank 1 (all columns ∈ × ∈ × are equal). The matrix F Matr (r ν) is in general of maximal rank r ν and it is irrelevant for the present ∈ × − − (a) consideration (it consists of the evaluations of the functions gj at the distinct points z , a = ν +1,...,r). ν 1 ν−1 It is then clear that in the evaluation of the determinant the lowest order term are of order ζ − = (z c) ν . This proves that the ratio F (z) is bounded at z = c; since it is single–valued, the singularity is removable and− F (z) extends to an analytic function at z = c. The case where there are several branchpoints in the ﬁber of the critical value z = c is only slightly more complicated and left to the reader.

Remark A.1 The arguments above apply also if Z maps to a higher genus curve; in that case the ratio F will be simply a meromorphic function on the target curve.

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