$({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities

We establish $({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-dualities between quantum Gaudin models with irregular singularities. Specifically, for any $M, N \in {\mathbb Z}_{\geq 1}$ we consider two Gaudin models: the one associated with the Lie algebra ${\mathfrak{gl}}_M$ which has a double pole at infinity and $N$ poles, counting multiplicities, in the complex plane, and the same model but with the roles of $M$ and $N$ interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons; we establish that, in this realization, the algebras of integrals of motion of the two models coincide. At the classical level we establish two further generalizations of the duality. First, we show that there is also a duality for realizations in terms of free fermions. Second, in the bosonic realization we consider the classical cyclotomic Gaudin model associated with the Lie algebra ${\mathfrak{gl}}_M$ and its diagram automorphism, with a double pole at infinity and $2N$ poles, counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie algebra ${\mathfrak{sp}}_{2N}$, with a double pole at infinity and $M$ simple poles in the complex plane. In the special case $N=1$ we recover the well-known self-duality in the Neumann model.


Introduction
Fix a set of N distinct complex numbers {z i } N i=1 ⊂ C, and an element λ ∈ gl * M . The quadratic Hamiltonians of the quantum Gaudin model [Gau83,Gau14] associated to gl M are the following elements of U (gl M ) ⊗N : where {E ab } M a,b=1 denote the standard basis of gl M and E (i) ab means E ab in the ith tensor factor. The H i belong to a large commutative subalgebra Z ⊂ U (gl M ) ⊗N called the Gaudin [Fre05] or Bethe [MTV06a] subalgebra, for which an explicit set of generators is known [Tal11,MTV06a,CT06].
If the element λ ∈ gl * M is regular semisimple, i.e. if we can choose bases such that λ(E ab ) = λ a δ ab for some distinct numbers {λ a } M a=1 ⊂ C, then one can also consider the following elements of U (gl N ) ⊗M : So can Z. In fact their images in End(C N M ) coincide. This is the bispectral correspondence for quantum Gaudin models [MTV09,CF08]. (Under this realization the Hamiltonians H a ∈ Z of the dual model coincide with suitably defined dynamical Hamiltonians [FMTV00] of the original gl M Gaudin model. See [MTV09,MTV06b].) In this paper we generalize this bispectral relation in a number of ways, for both the quantum and classical Gaudin models. Let us describe first the main result. Two natural generalizations of the Gaudin model above are to (a) models in which the quadratic Hamiltonians (and the Lax matrix, see below) have higher order singularities at the marked points z i ∈ C. Such models are called Gaudin models with irregular singularities [FFTL10,VY17b]. 1 (b) models in which λ ∈ gl * M is not semisimple, i.e. has non-trivial Jordan blocks. 2 We show that these two generalizations are natural bispectral duals to one another. Namely, we show that there is a bispectral correspondence among models generalized in both directions, (a) and (b), and that under this correspondence the sizes of the Jordan blocks get exchanged with the degrees of the irregular singularities at the marked points in the complex plane. See Theorem 4.7 below.
The heart of the proof is the observation that the generating functions for the generators of both algebras Z and Z can be obtained by evaluating, in two different ways, the column-ordered determinant of a certain Manin matrix. (A similar trick was also used in [CF08,Proposition 8].) Given that observation, the duality between (a) and (b) above is a essentially a consequence of the simple fact that the inverse of a Jordan block matrix      where the Lie algebra gl ab M in the last summand is isomorphic to gl M as a vector space but endowed with the trivial Lie bracket. Henceforth we denote the copy of E ab in the 1 Strictly speaking the term λ(E ab )E ab in Hi is already an irregular singularity of order 2 at ∞. 2 Let us note in passing that the case of λ semisimple but not regular is very rich; see for example [Ryb06,FFR10,Ryb16]. ab . In terms of these data, the formal Lax matrix of the Gaudin model associated with gl M , with a double pole at infinity and simple poles at each z i , i = 1, . . . , N , is given by Regarding L(z) as an M × M matrix with entries in the symmetric algebra S gl M , the coefficients of its characteristic polynomial, or more precisely those of the polynomial in the variables λ and z given by M . Given a classical model described by a Poisson algebra P and Hamiltonian H ∈ P, the latter becomes of particular interest if we have a homomorphism of Poisson algebras π : S gl ⊂ P then consists of Poisson commuting integrals of motion of the model.
The Lax matrix (1.2) can also be used to describe quantum models by regarding it instead as an M × M matrix with entries in the universal enveloping algebra U gl In this case, a large commutative subalgebra Z (z i ) gl M , called the Gaudin algebra, is spanned by the coefficients of the polynomials S a (z), a = 1, . . . , M obtained as the coefficients in the differential operator where cdet is the column ordered determinant. Given a unital associative algebra U and a homomorphismπ : U gl provides a large commutative subalgebra of U.
Let U be the Weyl algebra generated by the commuting variables x a i for i = 1, . . . , N and a = 1, . . . , M together with their partial derivatives ∂ a i := ∂/∂x a i . We introduce another set {λ a } M a=1 ⊂ C of M distinct complex numbers. It is well known that is a commutative subalgebra of U. On the other hand, given the new set of complex numbers λ a , a = 1, . . . , M , we may now equally consider the Gaudin model associated with gl N , with a double pole at infinity and simple poles at each λ a for a = 1, . . . , M . LetL(λ) denote its formal Lax matrix defined as in (1.2). We can define another (Note here the order between ∂ a j and x a i as compared, for instance, to [MTV06b, §5.1] whereẼ (λa) ij is realised as x a i ∂ a j .) The bispectral duality between the above two Gaudin models associated with gl M and gl N can be formulated, in the present conventions, as the equality of differential polynomialŝ whose coefficients are U-valued polynomials in z. (See §4.2 for the precise definition of the differential operator appearing on the right hand side.) In the classical setting discussed above the same identity holds with ∂ z replaced everywhere by the spectral parameter λ, the Weyl algebra U is replaced by the Poisson algebra P defined as the polynomial algebra in the canonically conjugate variables (p a i , x a i ) and column ordered determinants replaced by ordinary determinants.
We generalise this statement in a number of directions. Firstly, in both the classical and quantum cases, we consider Gaudin models with irregular singularities. Specifically, fix a positive integer n ∈ Z ≥1 and let We consider a gl M -Gaudin model with a double pole at infinity and an irregular singularity of order τ i at each z i for i = 1, . . . , n. The direct sum of Lie algebras (1.1) is replaced in this case by a direct sum of Takiff Lie algebras where D is a divisor encoding the collection of points z i for i = 1, . . . , n weighted by the integers τ i for i = 1, . . . , n. The formal Lax matrix L(z) of this Gaudin model is an M × M matrix with entries in the Lie algebra gl D M , and the Gaudin algebra Z (z i ) (gl D M ) is spanned by the coefficients of the differential operator Let U be the same unital associative algebra as above. In order to define a suitable homomorphismπ : be a direct sum of m Jordan blocks of sizeτ a ∈ Z ≥1 with λ a ∈ C along the diagonal for a = 1, . . . , m, such that m a=1τ a = M , then the dual Gaudin model associated with gl N will have a double pole at infinity and an irregular singularity at each λ a of orderτ a for a = 1, . . . , m. LetD be the divisor corresponding to these data and glD N the associated direct sum of Takiff algebras, cf. (1.5). After defining a corresponding homomorphism π : U glD N → U for this Gaudin model, we prove a bispectrality relation similar to the one stated above for the regular singularity case, see Theorem 4.7. As before, a similar result also holds in the classical setting where π andπ in this case are homomorphisms from the symmetric algebras S gl D M and S glD N , respectively, to the Poisson algebra P, see Theorem 3.2.
In the classical setup of §3 we also consider fermionic generalisations of bispectral duality. Specifically, for the Poisson algebra P we take instead the even part of the Z 2graded Poisson algebra generated by canonically conjugate Grassmann variable pairs (π a i , ψ a i ). The corresponding homomorphisms of Poisson algebras π f : S gl D M → P and π f : S glD N → P are defined in Lemma 3.3. In this case we establish a different type of bispectral duality between the same Gaudin models with irregular singularities and associated with gl M and gl N as above. Denoting by L(z) andL(λ) their respective Lax matrices, it takes the form See Theorem 3.4, the proof of which is completely analogous to that of Theorem 3.2 in the bosonic setting, using basic properties of the Berezinian of an (M |N ) × (M |N ) supermatrix. We leave the possible generalisation of such a fermionic bispectrality to the quantum setting for future work.
Finally, in §5 we consider extensions of these results to cyclotomic Gaudin models also in the classical setting. Specifically, we consider a Z 2 -cyclotomic gl M -Gaudin model with a double pole at infinity as usual and with irregular singularities at the origin of order τ 0 and at points z i ∈ C × , with disjoint orbits under z → −z, of order τ i for each i = 1, . . . , n. Let N = τ 0 + n i=1 τ i . Using the bosonic Poisson algebra P generated by canonically conjugate variables (p a i , x a i ) we prove that this model is in bispectral duality with a Gaudin model associated with the Lie algebra sp 2N , with a double pole at infinity and regular singularities at M points λ a , a = 1, . . . , M , see Theorem 5.2. We show that the well know bispectral self-duality in the Neumann model is a particular example of the latter with N = 1. Generalisations of such bispectral dualities involving cyclotomic Gaudin models to the quantum case are less obvious since it is known [VY16] that in this case the cyclotomic Gaudin algebra is not generated by a cdet-type formula as in (1.6), see remark 2. Let z i ∈ C for i = 1, . . . , n and λ a ∈ C for a = 1, . . . , m be such that z i = z j for i = j and λ a = λ b for a = b. Pick and fix integers τ i ∈ Z ≥1 for each i = 1, . . . , n andτ a ∈ Z ≥1 for each a = 1, . . . , m. We call these the Takiff degrees at z i and λ a , respectively. Consider the effective divisors (Recall that an effective divisor is a finite formal linear combination of points in some Riemann surface, here the Riemann sphere C ∪ {∞}, with coefficients in Z ≥0 .) We require that deg D = N + 2 and degD = M + 2 or in other words, (1, . . . , N ) = (1, . . . , τ 1 ; ν 2 + 1, . . . , ν 2 + τ 2 ; . . . ; ν n + 1, . . . , ν n + τ n ), (1, . . . , M ) = (1, . . . ,τ 1 ;ν 2 + 1, . . . ,ν 2 +τ 2 ; . . . ;ν m + 1, . . . ,ν m +τ m ).
is called a Takiff Lie algebra over gl M . When k ∈ Z ≥2 , for every n ∈ Z ≥1 with n < k we have a non-trivial ideal in , which by abuse of terminology we shall also refer to as a Takiff Lie algebra. We define direct sums of Takiff Lie algebras over gl M and gl N , respectively, as We use the abbreviated notation X ε k for an element X ⊗ ε k ∈ gl M [ε] where X ∈ gl M and k ∈ Z ≥0 , and likewise for elements of gl The set of non-trivial Lie brackets of these basis elements read , for any i, j = 1, . . . , n and a, b, c, d = 1, . . . , M , and for any i, j, k, l = 1, . . . , N and a, b = 1, . . . , m. Note, in particular, that E are Casimirs of the Lie algebras gl D M and glD N , respectively. 2.2. Lax matrices. Let ρ : gl M → Mat M ×M (C) andρ : gl N → Mat N ×N (C) denote the defining representations of gl M and gl N , respectively. We write E ab := ρ(E ab ) and The sets {E ab } M a,b=1 and {E ba } M a,b=1 form dual bases of gl M with respect to the trace in the representation ρ since tr(E ab E cd ) = δ ad δ bc for all a, b, c, d = 1, . . . , M . Likewise, dual bases of gl N with respect to the trace in the representationρ are given by . The Lax matrix of the Gaudin model associated with gl D M is given by It is an M × M matrix whose coefficients are rational functions of z valued in gl D M . Likewise, the Lax matrix of the Gaudin model associated with glD N reads and is an N × N matrix with entries rational functions of λ valued in glD N .
3. Classical (gl D M , glD N ) bispectrality 3.1. Classical Gaudin model. The algebra of observables of the classical Gaudin model associated with gl D M is the symmetric tensor algebra S(gl D M ). It is a Poisson algebra: the Poisson bracket is defined to be equal to the Lie bracket (2.2) on the subspace gl D M ֒→ S(gl D M ) and then extended by the Leibniz rule to the whole of S(gl D M ). Consider the quantity

Bosonic realisation. Introduce the Poisson algebra
. . , M and i, j = 1, . . . , N . In the following we shall regard P b as a Lie algebra under the Poisson bracket.
For any x ∈ C and k ∈ Z ≥1 we denote by J k (x) the Jordan block of size k × k with x along the diagonal and −1's below the diagonal, namely We note for later that if x = 0 then this is invertible and its inverse is given by for every r = 0, . . . , τ i − 1, i = 1, . . . , n and a, b = 1, . . . , M , and for every s = 0, . . . ,τ a − 1, i, j = 1, . . . , N and a = 1, . . . , m, are homomorphisms of Lie algebras. In particular, they extend uniquely to homomorphisms of Poisson algebras Proof. We will prove the corresponding result in the quantum case in detail below. See Lemma 4.6. That proof applies line-by-line here, with ∂ replaced by p.
Let C[λ, z] denote the algebra of polynomials in the variables λ and z. Given any Poisson algebra P we introduce the Poisson algebra P[λ, z] := P ⊗ C[λ, z] with Poisson bracket defined using multiplication in the second tensor factor. We extend the homomorphisms π b andπ b from Lemma 3.1 to homomorphisms of Poisson algebras respectively, by letting them act trivially on the tensor factor C[λ, z]. In particular, we may apply these homomorphisms respectively to the polynomials (3.1) and (3.2) in the variables λ and z. The coefficients of the resulting polynomials in P b [λ, z] span the images of the classical Gaudin algebras in P b , namely respectively. The following theorem establishes that these Poisson-commutative subalgebras of P b coincide.
Theorem 3.2. We have the following bispectrality relation Consider the block matrix with entries in the commutative algebra P b [λ, z]. We may evaluate its determinant in two ways. On the one hand, we have On the other hand, Hence we obtain the relation It remains to note that the square matrices Z and Λ can be written as with π b andπ b as defined in Lemma 3.1, and that their inverses are given by Thus we have which is nothing but λ1 − π b t L D (z) using Lemma 4.6, the expression (2.4a) for the Lax matrix L D (z) and (3.5) for the inverse of a Jordan block. Likewise which coincides with z1 −π b L (λ) , as required. Since det t A = det A for any square matrix A and noting that det Z = n i=1 (z − z i ) τ i and det Λ = m a=1 (λ − λ a )τ a , the result follows.
3.3. Fermionic realisation. Let V := span C {ψ a i , π b j } N M i,j=1 a,b=1 and define the exterior algebra P f := V = 2M N k=0 k V , whose skew-symmetric product we denote simply by juxtaposition. We refer to an element u ∈ k V as being homogeneous of degree k and write |u| = k. In particular, |ψ a i | = |π a i | = 1 for any a = 1, . . . , M and i = 1, . . . , N . We endow P f with a Z 2 -graded Poisson structure defined by for every i = 1, . . . , n and a, b = 1, . . . , M , and for every i, j = 1, . . . , N and a = 1, . . . , m, are homomorphisms of Lie algebras.
Proof. For each i, j = 1, . . . , n and a, b = 1, . . . , M we have N and a, b = 1, . . . , m one shows that , and all Poisson brackets involving the generators at infinity are also easily seen to be preserved by the linear maps π f andπ f since z i ∈ C and λ a ∈ C are central in P0 f . Theorem 3.4. We have the following bispectrality relation  [BF84]. Equating these two expressions of Ber M we obtain the relation Recalling the expressions for the square matrices Z and Λ and their inverses given in the proof of Theorem 3.2, we can write which is nothing but λ1 − π f L D (z) . Likewise which is z1 −π f t LD(λ) . The result now follows as in the proof of Theorem 3.2.
4. Quantum (gl D M , glD N ) bispectrality There is a natural quantum version of Theorem 3.2. In order to state it, we first need a short digression on Manin matrices. In this section we do not consider the fermionic counterpart of Theorem 3.2, namely Theorem 3.4, but leave this for future work. That is, elements of the same column must commute amongst themselves, and commutators of cross terms of 2 × 2 submatrices must be equal (for example [M 11 , M 22 ] = [M 21 , M 12 ]). Actually the second of these conditions implies the first (set j = l) but it is convenient to think of them separately.
In the literature Manin matrices have been also called right quantum matrices [Kon07, Kon08, KP07, MR14] or row-pseudo-commutative matrices [CSS09]. For a review of their properties, and further references, see [CFR09].   This has the following corollary which will be important for us. as an equality in A.
Proof. We work initially over A ′ . Suppose A has a right inverse. By Proposition 4.4 we have as an equality in A ′ . But cdet M belongs to A, so in fact this is an equality in A. This establishes part (i).
Recall the definition of the column-ordered determinant, Definition 4.2, and consider the quantity This is a differential operator in z of order M . For each 0 ≤ k ≤ M , the coefficient The quantum Gaudin algebra Z (glD N ) of the glD N -Gaudin model is defined in exactly the same way in terms of the N th order differential operator in λ, There is an automorphism of gl D N defined by LD(λ) → − t LD(λ). The Gaudin algebra is stabilized by this automorphism. (This statement follows from applying a tensor product of evaluation homomorphisms of Takiff algebras to the statement of [MTV06a, Proposition 8.4]). Therefore we may equivalently consider the N th order differential operator and define the quantum Gaudin algebra Z (glD N ) to be the unital subalgebra of U (glD N ) generated by the coefficients of the polynomialsS k (λ). It is a commutative subalgebra of U (glD N ).
To state our result on quantum bispectrality, it will be convenient to write (4.3) in the equivalent form Let us explain the meaning of the expression cdet − z1 N ×N + LD(∂ z ) . The quantity cdet ∂ λ 1 N ×N +LD(λ) , which appears in (4.3), belongs to the algebra U (glD N )(λ)[∂ λ ] of differential operators in λ whose coefficients are rational functions of λ with coefficients in U (glD N ). Here λ and ∂ λ can be regarded as formal generators obeying the commutation relation [∂ λ , λ] = 1. We can relabel these generators as we wish, provided we preserve this relation. In particular, we may send for a, b = 1, . . . , M and i, j = 1, . . . , N .
Lemma 4.6. The linear mapsπ b : for every r = 0, . . . , τ i − 1, i = 1, . . . , n and a, b = 1, . . . , M , and for every s = 0, . . . ,τ a − 1, i, j = 1, . . . , N and a = 1, . . . , m, are homomorphisms of Lie algebras. In particular, they extend uniquely to homomorphisms of associative algebraŝ Proof. For each i, j = 1, . . . , n and a, b = 1, . . . , M we have In the second equality we have used the fact that if i = j then all commutators vanish due to the restriction in the range of values in the sums over u and v.
Likewise, for all i, j = 1, . . . , N and a, b = 1, . . . , m we find kl[s] , as required. Moreover, all the commutators involving the generators at infinity are also easily seen to be preserved by the linear mapsπ b andπ b since z i ∈ C and λ a ∈ C are central in U b .
Given any unital associative algebra U we denote by U[z, ∂ z ] the tensor product of unital associative algebras U ⊗ C[z, ∂ z ]. By contrast with the classical setting of §3.2, we shall also need to consider the unital associative algebras , both containing U[z, ∂ z ] as a subalgebra. We extend the homomorphismsπ b andπ b from Lemma 4.6 to homomorphisms of tensor product algebras, , respectively. Applying these homomorphisms respectively to the differential operators in z given by (4.1) and (4.5), the coefficients of the resulting differential operators in z span the respective images of the quantum Gaudin algebras in U b , namelŷ The following theorem establishes that these commutative subalgebras of U b coincide.
Theorem 4.7. We havê as an equality of polynomial differential operators in z.
Proof. Introduce the M × M and N × N block diagonal matrices Also introduce the M × N matrices

Consider the block matrix
with entries in the noncommutative algebra A : The key observation is that this is a Manin matrix. Indeed, the only non-trivial check is for the 2 × 2 submatrices of the form and for these we have [∂ z − λ a , z − z i ] = 1 = [∂ a i , x a i ] as required. This fact means that we can follow the proof of Theorem 3.2, with suitable modifications, as follows.
The square matrices Z and Λ with entries in C[z, , respectively, both of which contain A as a subalgebra. These inverses are given explicitly by We are therefore in the setup of Proposition 4.5. We may apply it to evaluate cdet M in two different ways. We obtain as an equality in A = U b [z, ∂ z ], namely this is an equality of polynomial differential operators in z with coefficients in U b . It remains to evaluate both sides of (4.7) more explicitly. We have where the order of the products on the right of these equalities does not matter. Now Z and Λ can be written explicitly as follows withπ b andπ b as defined in Lemma 4.6. In terms of these expressions we can write The latter expression is exactly ∂ z 1−π b t L D (z) by virtue of Lemma 4.6, the expression (2.4a) for the Lax matrix L D (z) and the expression (3.5) for the inverse of a Jordan block. Likewise which coincides with z1 −π b LD(∂ z ) . The result now follows.
In the special case of no Jordan blocks and no non-trivial Takiff algebras, Theorem 4.7 can be found in [MTV09]. See also [CF08,Proposition 8], where it is noted that the relation cdet M = det Z cdet(Λ − XZ −1 t D) leads to a relation between the classical spectral curve and the "quantum spectral curve".

Z 2 -cyclotomic Gaudin models with irregular singularities
Another possible class of generalisations of Gaudin models are those whose Lax matrix is equivariant under an action of the cyclic group, determined by a choice of automorphism of the Lie algebra (here gl M ). Such models were considered in [Skr06,Skr07,Skr13] and in [CY07] for automorphisms of order 2, and for automorphisms of arbitrary finite order in [VY16,VY17b].
It is natural to ask whether bispectralities also exist, in the sense of §3, between cyclotomic Gaudin models. Theorem 5.2, which can be deduced from the results of [AHH90], establishes a bispectrality between a cyclotomic gl M -Gaudin model associated with the diagram automorphism of gl M and a non-cyclotomic sp N -Gaudin model. 5.1. Z 2 -cyclotomic Lax matrix for the diagram automorphism. Let z i ∈ C for i = 1, . . . , n be such that 0 = z i = ±z j for i = j. Pick and fix integers τ i ∈ Z ≥1 for i = 0 and for each i = 1, . . . , n. Consider the effective divisor Note, in particular, that the Takiff degree at the origin is always even. Let N ∈ Z ≥1 . We require that deg C = 2N + 2 or in other words, Let M ∈ Z ≥1 . As before, cf. §2.1, denote by E ab for a, b = 1, . . . , M the standard basis of gl M . There is an automorphism σ of gl M defined by We call this the diagram automorphism of gl M . The Lie algebra gl M decomposes into the direct sum of the ±1 eigenspaces of σ, Here the subalgebra of invariants, i.e. the (+1)-eigenspace, is a copy of the Lie algebra so M . The (−1)-eigenspace p M is a copy of the symmetric second rank tensor representation of so M . We shall write E ± ab := E ab ± E ba , so that E + ab ∈ so M and E − ab ∈ p M , for all a, b = 1, . . . , M . We introduce the pair of maps There is an extension of the automorphism σ to an automorphism of the polynomial algebra gl M [ε] defined by X ε k → σ(X)(−ε) k . Let gl M [ε] σ denote the subalgebra of invariants. As vector spaces, we have Define gl C M to be the direct sum of Takiff Lie algebras Note that as a vector space the Takiff algebra attached to the point at infinity is simply It obeys the following Lax algebra (5.2) L C 1 (z),L C 2 (w) = r 12 (z, w),L C 1 (z) − r 21 (w, z),L C 2 (w) where r 12 (z, w) denotes the (non-skew-symmetric) classical r-matrix Consider the quantity This  Introduce the direct sum of Lie algebras The Lax matrix of the classical Gaudin model associated with the divisorD is the 2N × 2N matrix of spD 2N -valued rational functions of λ given by where by abuse of notation we drop the subscript on the Takiff generators, namely we defineĒ IJ ⊗Ē IJ µ − λ .
To simplify the notation, introduce y ab r := τ 0 −r u=1 x a u+r p b u − (−1) r x b u+r p a u . We can then write We define a pair of M × 2N matrices P and X, whose columns are also indexed by the set I, as For each i = 1, . . . , n we note using the expression (3.5) for the inverse of a Jordan block together with Lemma 5.1 that (z + z i ) r+1 .
Next, for the two terms in the middle line above, corresponding to the origin, we find Finally, for the remaining term we have Putting all the above together we deduce that Λ − XZ −1 t P = λ1 − π b tLC (z) .
On the other hand, we have To see the second equality we note that setting z = 0 in Z − t P Λ −1 X yields a 2N × 2N symplectic matrix, i.e. of the block form Lastly, we clearly have det Λ = M a=1 (λ − λ a ) and det Z = z 2τ 0 n i=1 (z − z i ) τ i (z + z i ) τ i from which the result now follows, using again the fact that det t A = det A for any square matrix A, as in the proof of Theorem 3.2.
used in the proof of Theorem 5.2. The resulting square matrix with non-commutative entries is not Manin since, for example, the entries of the first column are not mutually commuting. Consequently, we do not immediately obtain a quantum analogue of the classical bispectrality relation in Theorem 5.2. A related remark is that in the quantum case, higher Gaudin Hamiltonians for cyclotomic Gaudin models do exist but they are not in general given by a simple cdet-type formula. See [VY16,VY17a] (and especially Remark 2.5 in [VY16]). ⊳ Remark 3. Note that we did not allow irregular singularities on the sp 2N side (appart from the double pole at infinity). From the point of view of bispectrality, the absence of irregular singularities in the sp 2N -Gaudin model is controlled by the fact that the matrix