Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

The super-Macdonald polynomials, introduced by Sergeev and Veselov (Commun Math Phys 288: 653–675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald–Ruijsenaars operators introduced by the same authors in Sergeev and Veselov (Commun Math Phys 245: 249–278, 2004). We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald–Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald–Ruijsenaars operators. Motivated by recent results in the nonrelativistic (q→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\rightarrow 1$$\end{document}) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.


Introduction
As is well-known, the Macdonald polynomials [Mac95] can be viewed as eigenfunctions of a commuting family of difference operators associated with a relativistic generalisation of the integrable quantum Calogero-Moser-Sutherland systems of trigonometric A-type [Rui87].Such relativistic quantum systems were originally conceived by Ruijsenaars as an integrable quantum mechanical description of a relativistic quantum field theory in 1+1 spacetime dimensions known as the quantum sine-Gordon theory, restricted to sectors where the particle number is fixed [RS86,Rui01].
While the standard Ruijsenaars systems account for one particle type, a relativistic quantum field theory typically has two kinds of particle: particles and anti-particles.This strongly suggests to us that Ruijsenaars' systems should have generalisations allowing for two particle types, and we propose that, in the trigonometric regime, such a generalisation is given by the so-called deformed Macdonald-Ruijsenaars operators M n,m;q,t and M n,m;q −1 ,t −1 (specified in (1) below), and their joint eigenfunctions, the super-Macdonald polynomials, introduced and studied by Sergeev and Veselov [SV04,SV09a].The super-Macdonald polynomials SP λ ((x 1 , . . ., x n ), (y 1 , . . ., y m ); q, t) depend on arbitrary numbers, n and m, of two types of variables, x i and y j , and we expect that these two variable types correspond to particles and anti-particles in an underlying quantum field theory.
In any quantum mechanical model, there is a scalar product providing the space of wave functions with a Hilbert space structure, and this structure is essential for the physical interpretation of the model.For the trigonometric Ruijsenaars systems and the Macdonald polynomials such a Hilbert space structure is provided by the scalar product denoted as •, • ′ n in Macdonald's book [Mac95].In particular, with respect to this scalar product, the commuting family of difference operators alluded to above, which include operators that define the Hamiltonian and momentum operator in the model, are self-adjoint and the Macdonald polynomials form an orthogonal system with explicitly known Hilbert space norms [Mac95].By contrast, for the deformed Macdonald-Ruijsenaars operators and the super-Macdonald polynomials, such a Hilbert space structure has been missing.Our main purpose with this paper is to provide this missing Hilbert space structure and thereby substantiate our proposal, as formulated above.Moreover, recent quantum field theory results in the nonrelativistic case [AL17,BLL20], discussed in Section 5, provide further support in favour of our proposal.
To describe our results in more detail, we recall from [SV09a] that the super-Macdonald polynomials are joint eigenfunctions of a large commutative algebra of difference-operators, containing the so-called deformed Macdonald-Ruijsenaars operator introduced in [SV04]: 1 M n,m;q,t = t 1−n 1 − q n i=1 A i (x, y; q, t) T q,xi − 1 with coefficients A i (x, y; q, t) t 1/2 x i − q 1/2 y j t 1/2 x i − q −1/2 y j , B j (x, y; q, t) = m j ′ =j q −1 y j − y j ′ y j − y j ′ • n i=1 q −1/2 y j − t −1/2 x i q −1/2 y j − t 1/2 x i , and where T q,xi and T t −1 ,yj act on functions f (x, y) of x = (x 1 , . . ., x n ) ∈ C n and y = (y 1 , . . ., y m ) ∈ C m by shifting x i → qx i and y j → t −1 y j , respectively, while leaving the remaining variables unaffected.
For our Hilbert space results, it will be important to restrict attention to parameter values 0 < q < 1, 0 < t < 1.
However, as discussed briefly in the final paragraph of Section 5, some of our results extend analytically to complex q and t (with modulus in (0, 1)).Deformed Macdonald-Ruijsenaars operators first appeared in the m = 1 case in work by Chalykh [Cha97,Cha00].Further examples, including deformed Koornwinder operators, were later obtained and studied by Feigin [Fei05], Sergeev and Veselov [SV09b] and Feigin and Silantyev [FS14].
Taking m = 0, the operator given by (1)-(2) reduces to (3) and the super-Macdonald polynomials reduce to the ordinary (monic symmetric) Macdonald polynomials P λ ((x 1 , . . ., x n ); q, t).Note that, up to the overall factor t/(1 − q), M n;q,t (3) coincides with the operator E n in Madonald's book [Mac95,Section VI.4].Moreover, M n;q,t is closely related to the trigonometric limit of the elliptic operator S 1 introduced by Ruijsenaars in [Rui87].The precise relationship, which was first observed by Koornwinder (in unpublished notes), is, e.g., detailed in [vDie95, Section 5.2] and [Has97, Section 5.1].We recall that S 1 ± S −1 , where S −1 is similarly related to M n;q −1 ,t As is well known, the Macdonald polynomials form an orthogonal system on the n-dimensional torus T n ≡ T n 1 , where (4) with respect to the weight function where (a; q) ∞ = ∞ k=0 (1 − aq k ) is the usual q-Pochhammer symbol.Moreover, the corresponding (quadratic) norms are given by remarkably simple and explicit formulas [Mac95, Section VI.9]; see (25)-( 27).
These orthogonality results, together with the corresponding Hilbert space structure, entail a natural quantum mechanical interpretation of the Macdonald polynomials P λ ((x 1 , . . ., x n ); q, t) and the commuting Macdonald-Ruijsenaars operators M n;q,t and M n;q −1 ,t −1 ; this is the trigonometric Ruijsenaars model.
In this paper, we obtain analogous results for the the super-Macdonald polynomials SP λ (x, y; q, t).More specifically, we establish orthogonality relations with respect to a sesquilinear form given by with weight function , and where P, Q are polynomials in the space spanned by the super-Macdonald polynomials, the bar denotes complex conjugation and Furthermore, in order to ensure that we avoid the poles of the weight function, we integrate x and y over tori T n ξ and T m ξ ′ with radii ξ, ξ ′ > 0 that are sufficiently separated.Our main results are: (I) The expression (6) defines a Hermitian product that is independent of ξ, ξ ′ > 0 provided | log(ξ/ξ ′ )| > 1 2 | log(q/t)|, (II) the orthogonality relations SP λ , SP µ ′ n,m;q,t = 0 hold true for all λ = µ, (III) the (squared) norms SP λ , SP λ ′ n,m;q,t are given by the simple and explicit formulas (47)-(48).
Remark 1.1.The attentive reader might wonder why we do not simply integrate over T n × T m in (6) since, clearly, poles in the denominator of (7) would also be avoided by choosing ξ = ξ ′ = 1.This can be readily understood in the simplest non-trivial case n = m = 1 since, in this case, the integral in (6) can be easily computed; see Appendix D for details.One finds that the integral is the same for ξ ≫ ξ ′ and ξ ′ ≫ ξ, but the integral for ξ = ξ ′ differs by a non-trivial residue term which spoils our orthogonality results, as described above.
Remarkably, even though we are working with a complex-valued weight function (since the denominator in (7) is only real if ξ = ξ ′ ), we find that all norms are given by non-negative real numbers.In addition, the super-Macdonald polynomials with non-zero norms are characterised by the simple condition λ n ≥ m ≥ λ n+1 .As discussed in Section 3.2, the product •, • ′ n,m;q,t therefore provides the space spanned by the super-Macdonald polynomials with non-zero norm with a Hilbert space structure allowing for a quantum mechanical interpretation of the model defined by the commuting deformed Macdonald-Ruijsenaars operators M n,m,q,t and M n,m,q −1 ,t −1 .
The results in this paper can be considered as natural q-deformations of the orthogonality relations and norm formula we obtained in [AHL19] for the super-Jack polynomials.As compared to loc.cit., significant simplifications occur: Since the eigenvalues of M n,m;q,t separate the super-Macdonald polynomials SP λ , there is no need to involve higher order eigenoperators; and the fact that ∆ n,m is a meromorphic function simplifies arguments involving contour deformations.
Our plan is as follows.In Section 2, we briefly review known facts about the Macdonald functions (Section 2.1) and super-Macdonald polynomials (Section 2.2) that we need.Our results can be found in Section 3: a precise formulation of our orthogonality result is given in Theorem 3.1 (Section 3.1), followed by a discussion of the Hilbert space interpretation of the super-Macdonald polynomials suggested by this (Section 3.2).The proof of Theorem 3.1 is given in Section 4. We conclude with a short discussion of research questions motivated by our results in Section 5. Three appendices explain how the conventions on super-Macdonald polynomials we use are related to the ones of Sergeev and Veselov [SV09a] (Appendix A), prove the relativistic invariance of the generalized Ruijsenaars model (Appendix B), give proof details to make this paper self-contained (Appendix C), and shortly discuss the special case n = m = 1 (Appendix D).
Notation.We denote as P the space of all partitions, i.e., λ ∈ P means that λ = (λ 1 , λ 2 , . ..) with integers λ i ≥ 0 satisfying λ i ≥ λ i+1 , i = 1, 2, . .., and only finitely many λ i 's non-zero; the non-zero λ i 's are called parts of λ, and partitions differing only by a string of zeros at the end are not distinguished.For any partition λ, ℓ(λ) is the number of parts of λ, and |λ| is the sum of its parts; ℓ(λ) and |λ| are called length and weight of λ, respectively.Moreover, for λ ∈ P, λ ′ denotes the conjugate of λ (so that the Young diagrams of λ and λ ′ are transformed into each other by reflection in the main diagonal).We also recall the definition of the dominance partial ordering on the set of partitions of a fixed weight: for λ, µ ∈ P such that |λ| = |µ|, In addition, for λ, µ ∈ P, µ ⊆ λ is short for µ i ≤ λ i for all i, and λ ∪ µ denotes the partition obtained by merging and re-ordering the parts of λ and µ.

Prerequisites
We collect definitions and results we need, following Macdonald [Mac95] in Section 2.1 and Sergeev and Veselov [SV09a] in Section 2.2.
2.1.1.Ring of symmetric functions.We consider the complex vector space Λ = Λ C of symmetric functions in infinitely many variables x = (x 1 , x 2 , . ..) (we work over C since we are motivated by quantum mechanics).It can be defined as the space of all finite linear combinations, with complex coefficients, of the symmetric monomial functions m λ , labeled by partitions λ, and defined as follows: where the sum is over all distinct permutations a = (a 1 , a 2 , . ..) of λ = (λ 1 , λ 2 , . ..).Thus, the symmetric monomial functions constitute a (vector space) basis in Λ labeled by partitions.Another such basis is given by the products (10) of the Newton sums The space Λ has a natural ring structure and, as such, is freely generated by the Newton sums p r , r ∈ Z ≥1 .
2.1.2.Macdonald functions.The space Λ becomes a (pre-)Hilbert space when equipped with the scalar product •, • q,t characterised by linearity in its first (and antilinearity in its second) argument and 2 (12) p λ , p µ q,t = δ λµ z λ ℓ(λ) where z λ := λ1 i=1 i mi m i ! with m i = m i (λ) the number of parts of λ equal to i (setting i 0 0! = 1), and δ λµ the Kronecker delta.
It is known that the Macdonald functions P λ are eigenfunctions of the inverse limit M q,t of the operators M n;q,t (3): (14) M q,t P λ (x; q, t) = d λ (q, t)P λ (x; q, t), and the corresponding eigenvalues are given by (15) 2 To avoid possible confusion, we stress that this product is different from the one allowing for a quantum mechanical interpretation of the Macdonald polynomials.
In the following Lemma, we state a well-known technical result that we need.
Lemma 2.1.The coefficients and in the extremal cases they are given by (For the convenience of the reader, we give a proof in Appendix C.1.)2.1.4.Macdonald polynomials.The Macdonald polynomials P λ ((x 1 , . . ., x n ); q, t) are obtained from the Macdonald functions P λ ((x 1 , x 2 , . ..); q, t) by setting x i = 0 for all i > n, and similarly for Q λ and P λ/µ .It is know that P λ ((x 1 , . . ., x n ); q, t) is non-zero only for partitions λ = (λ 1 , . . ., λ n ) of length less or equal to n.Moreover, as already discussed in the introduction, the Macdonald polynomials are orthogonal with respect to the following scalar product, (25) P, Q ′ n;q,t := as in (4), and ∆ n (x; q, t) in ( 5): for all P λ = P λ (x; q, t) We also need 2.2.Super-Macdonald polynomials.Following Sergeev and Veselov [SV09a], we define Λ n,m;q,t as the algebra of complex polynomials P (x, y) in n + m variables (x, y) = (x 1 , . . ., x n , y 1 , . . ., y m ) ∈ C n × C m that are symmetric in each set of variables separately, i.e., (29) where S n is the group of permutations of n objects, and, furthermore, that satisfy the symmetry conditions3 (30) T q,xi − T t −1 ,yj P (x, y) = 0 at q 1/2 x i = t −1/2 y j (∀i, j).
This algebra, Λ n,m;q,t , is generated by the following deformed Newton sums, Remark 2.1.Many results in [SV09a] require a restriction to so-called non-special parameters q, t, i.e., q i t j = 1 for all i, j ∈ Z ≥0 such that i + j ≥ 1; see e.g.[SV09a, Theorem 5.8]. 4 However, since we assume 0 < q, t < 1, we can ignore this restriction.
Thus, the super-Macdonald polynomials SP λ ((x 1 , . . ., x n ), (y 1 , . . ., y m ); q, t) are also eigenfunctions of the deformed Macdonald-Ruijsenaars operator M n,m;q −1 ,t −1 with eigenvalue d λ (q −1 , t −1 ).We also recall that SP λ (x, y; q, t) for (x, y) ∈ C n × C m is non-zero if and only if λ belongs to the following set of partitions, Below we give an explicit representation of the super-Macdonald polynomials needed in the proof of our main result (this is a slight refinement of a result in [SV09a]).
Lemma 2.2.For (x, y) ∈ C n × C m and λ ∈ H n,m , we have where the sum runs over all partitions µ such that where k := max(k, 0).

Results
We now turn to our results.In Subsection 3.1, we introduce the relevant scalar product on the space Λ n,m;q,t , spanned by the super-Macdonald polynomials, and state our main results in Theorem 3.1.The proof of this theorem is deferred to Section 4. The Hilbert space interpretation of deformed Macdonald-Ruijsenaars operators and super-Macdonald polynomials, as provided by this scalar product, is discussed in Section 3.2.
In what follows, we use the short-hand notation for variables x = (x 1 , . . ., x n ) ∈ C n and n ∈ Z ≥1 ; we also recall the definition of the n-torus T n ξ of radius ξ > 0 in (4).
As already described in the introduction, the Hermitian product of P, Q ∈ Λ n,m;q,t is obtained by integrating the product of P (x, y)Q * (x, y) with the weight function ∆ n,m (x, y; q, t) in ( 5)-( 7) over the n + m-dimensional torus T n ξ × T m ξ ′ with suitable radii ξ, ξ ′ > 0; see (6).To see that we need to restrict the radii, we note that, while P (x, y)Q * (x, y) ∈ L n,m , and thus is holomorphic for (x, y) ∈ (C * ) n × (C * ) m , the weight function ∆ n,m (x, y; q, t) is meromorphic with simple poles located along the hyperplanes (42a) Clearly, T n ξ × T m ξ ′ is contained in the complement of these hyperplanes provided the radii ξ, ξ ′ > 0 are constrained as follows: if we restrict ourselves to such radii, we avoid all singularities of the integrand and thus obtain well-defined integrals; see Remark 1.1.Note that the condition in (43) can be written in a more compact way as follows, | log(ξ/ξ ′ )| > 1 2 | log(q/t)|.Definition 3.1.For ξ, ξ ′ > 0 satisfying either of the two conditions in (43), we define a sesquilinear form •, • ′ n,m;q,t on Λ n,m;q,t by for arbitrary P, Q ∈ Λ n,m;q,t .
Using that the integrand in (44) is analytic everywhere except along the hyperplanes (42), it is not difficult to prove that this sequilinar form does not depend on ξ, ξ ′ as long as they vary over only one of the two regions in (43); see Lemma 4.2.This argument applies to any Laurent polynomials P, Q ∈ L n,m , but it does not rule out the possibility that the value of P, Q ′ n,m;q,t in the former region ξ/ξ ′ < min δ=±1 q However, as we will show, if P and Q belong to Λ n,m;q,t , then the value of P, Q ′ n,m;q,t is the same in both regions.In order to appreciate the significance of the conditions (29)-(30), it is instructive to consider the simplest non-trivial case n = m = 1, in which the above claim can be verified by direct computations; the interested reader can find the details in Appendix D.
To state our main result in Theorem 3.1 below, we need two mappings e and s on partitions.For that, we observe that a partition λ ∈ H n,m such that (m n ) ⊆ λ satisfies the conditions and, for this reason, it can be written as with two partitions e(n, m; λ) and s(n, m; λ) of lengths less or equal to n and m, respectively, and determined by λ = (λ 1 , λ 2 , . ..) as follows, see [AHL19, Section 2.2] for more details on these mappings e (short for east) and s (short for south), including the motivation for these names.To simplify notation, we write e(λ) short for e(n, m; λ) and s(λ) short for s(n, m; λ) if no confusion can arise.
3.2.Hilbert space interpretation.From Theorem 3.1, we see that the kernel of the Hermitian product (44) is spanned by the super-Macdonald polynomials with zero norm: Since the remaining norms N n,m (λ; q, t), where (m n ) ⊆ λ, are positive, we have the following result.
Moreover, since the deformed Macdonald-Ruijsenaars operators M n,m;q,t and M n,m;q −1 ,t −1 leave K n,m;q,t invariant and their eigenvalues are all real (cf.(36)-(37)), they define (essentially) self-adjoint operators in V n,m;q,t .Hence, we have assembled everything needed for a quantum mechanical interpretation of the model defined by M n,m;q,t and M n,m;q −1 ,t −1 .
From a physics point of view, it would be natural to express wave functions and operators in terms of the "additive" variables (u, v) = (u 1 , . . ., u n , v 1 , . . ., v m ) defined as follows, (49) x i = e 2πiui/L , y j = e 2πivj/L (L > 0) and parameters We note that w(z) = w(r, a, b; z) is a (globally) meromorphic function with simple poles located at and zeros at z = jπ/r − ia(2k − 1) (j ∈ Z, k ∈ Z ≥1 ).Due to the manifest complex conjugation property G(r, a; z) = G(r, a; −z), it follows, in particular, that w(z) is a regular and (strictly) positive function in R. Assuming that u ∈ (R + iǫ) n and v ∈ (R + iǫ ′ ) m for some ǫ, ǫ ′ ∈ R, we can thus introduce the (formal) groundstate wave function , (where we take the positive square roots), and obtain a natural factorisation of the weight function ∆ n,m (7), as detailed in the following Lemma.
Proof.For x and z complex variables related as x = e 2πiz/L , we use (50) to deduce and From ( 7) and ( 51)-( 52), the statement can now be inferred by a straightforward computation.
If we consider wave functions of the form with P ∈ Λ n,m;q,t , then we can use Lemma 3.1 to rewrite our Hermitian form as a suitably regularised version of a conventional Hilbert space product for a quantum mechanical model describing particles moving on the circle for all P, Q ∈ Λ n,m;q,t , where Changing variables according to (49)-( 50) in (44) and invoking Lemma 3.1, the equality (53) results.
Remark 3.1.Note that, from a physics point of view, the positions u i and v j are real, but one has to continue the arguments of the super-Macdonald polynomials to the complex plane in order to compute their scalar product.This bears some resemblance to the fact that an eigenfunction of Ruijsenaars' (analytic) difference operators needs to have sufficient analyticity in order for the corresponding eigenvalue equations to make sense, see e.g.[Rui01].
As we demonstrate in Appendix C.3, the terms in the deformed Macdonald-Ruijsenaars operator M n,m;q,t in (1)-( 2) not involving a shift operator add upp to a constant.Dropping this overall constant, we get the operator (54) M n,m;q,t := Changing variables and parameters according to (49) and performing a similarity transformation with Ψ 0 , a direct computation, using the difference equation satisfied by the trigonometric Gamma function (cf.[Rui97, Section III.C]), yields , and where ∂ ui = ∂/∂u i and ∂ vj = ∂/∂v j .The structure of these operators occupies a sort of middle-ground between the trigonometric degeneration of Ruijsenaars' original (undeformed elliptic) operator Ŝ−1 and a similarity transform A −1 = U −1/2 Ŝ−1 U 1/2 with a trigonometric 'scattering function' U .(Explicit expressions for the latter operator and the pertinent scattering function (in the hyperbolic case) can, e.g., be found in [HR14].)In particular, when m = 0 we recover the trigonometric instance of Ŝ−1 .

Proofs
This section is devoted to the proof of Theorem 3.1.In place of (44), we write dω m (y)∆ n,m (x, y; q, t)P (x, y)Q * (x, y) for P, Q ∈ Λ n,m;q,t , so that we easily can keep track of the choice of integration radii ξ, ξ ′ > 0. Introducing the maximum function we note that the conditions (43) can be expressed as (56) ξ/ξ ′ > M (q, t) or ξ/ξ ′ < 1/M (q, t).
Observing that ∆ n,m (x, y) = ∆ n,m (x −1 , y −1 ), we thus rewrite the left-hand side of (57) as Finally, using the observation we see that this integral is equal to the right-hand side of (57).
We proceed to show that (55) is invariant under continuous deformations of the integration radii as long as they satisfy (56).
Proof.We note that ξ/ξ ′ < 1/M (q, t) if and only if (1/ξ)/(1/ξ ′ ) > M (q, t).Hence, thanks to Lemma 4.1, we may and shall restrict attention to the region ξ/ξ ′ > M (q, t).By Cauchy's theorem, we can deform the integration contours in (55) one at a time, without changing the value of the integral, as long as we do not encounter any of the poles (42a)-(42c).In particular, taking To this end, we observe that and using this, we can write (63) as follows, We now change variables x i → qx i in the latter integral to obtain (64) where the equality holds true due to Cauchy's theorem, since the integrand, which is the same in both integrals, is an analytic function of x i in the region ξ ≤ |x i | ≤ ξ/q when (65) and ( 61) is satisfied.A proof of this analyticity property of the integrand can be found in Appendix C.4.We have thus established (63) and, as previously noted, self-adjointness of A i (x, y)T q,xi immediately follows.

Conclusions and outlook
We introduced a Hermitian product •, • ′ n,m;q,t , given by (6), on the algebra Λ n,m;q,t , in which the super-Macdonald polynomials constitute an orthogonal basis (cf.[SV09a, Theorem 5.6] and Theorem 3.1), and we proved, in particular, that this product endows the factor space V n,m;q,t = Λ n,m;q,t /K n,m;q,t , where K n,m;q,t denotes the kernel of •, • ′ n,m;q,t , with a Hilbert space structure.Furthermore, we argued that these results provides the means for a quantum mechanical interpretation of the model defined by the deformed Macdonald operators M n,m;q,t and M n,m;q −1 ,t −1 , cf. (1)-(2).This model describes two kinds of particles, and we proposed that they represent particles and anti-particles in an underlying relativistic quantum field theory, which is the same theory that inspired the Ruijsenaars models [RS86,Rui01].
As mentioned in the introduction, from the quantum field theory point of view, it would be interesting to generalise our results to the elliptic case.The elliptic generalisation of the deformed Macdonald-Ruijsenaars operator is known from [AHL14].Specifically, rewriting the additive difference operator in Eq. (70) in multiplicative form, we obtain M n,m;p,q,t = t 1−n q m θ(q; p) with the elliptic deformation parameter, p, in the range 0 ≤ p < 1 and coefficients θ(t 1/2 x i /q 1/2 y j ; p) θ(t 1/2 q 1/2 x i /y j ; p) , B j (x, y; p, q, t) = m j ′ =j θ(q −1 y j /y j ′ ; p) θ(y j /y j ′ ; p) • n i=1 θ(q −1/2 t 1/2 y j /x i ; p) θ(q −1/2 y j /t 1/2 x i ; p) , where θ(z; p) := (z; p) ∞ (p/z; p) ∞ .We note that M n,m;p,q,t reduces, up to an additive constant, to the deformed Macdonald-Ruijsenaars operator M n,m;q,t from (54) in the trigonometric limit p → 0.Moreover, the results in [AHL14] suggest that a natural elliptic generalisation of our Hermitian product is as in (6) but with the weight function ∆ n,m (x, y; p, q, t) = ∆ n (x; p, q, t)∆ m (y; p, t, q) n i=1 m j=1 θ(q −1/2 t 1/2 x i /y j ; p)θ(q −1/2 t 1/2 y j /x i ; p) , ∆ n (x; p, q, t) = 1≤i =j≤n Γ(tx i /x j ; p, q) Γ(x i /x j ; p, q) , where Γ(z; p, q) := ∞ k=0 (p k+1 q/z; q) ∞ /(p k z; q) ∞ is the elliptic Gamma function; note that, in the limiting case (n, m) = (n, 0), this reduces to the operator and weight function of the elliptic Ruijsenaars model (see e.g.[Has97, Section 5]).However, at this point, very little is known about the eigenfunctions of the deformed elliptic Macdonald-Ruijsenaars operator M n,m;p,q,t .In fact, even in the ordinary m = 0 case, the understanding of these eigenfunctions is still not complete; see, however, [Shi19,LNS20] for recent progress in this direction.Our results provide further motivation for any attempt at developing a theory of eigenfunctions at the deformed elliptic level, and generalising the results in [Shi19,LNS20] to the deformed case could be an interesting starting point.
In the non-relativistic limit q → 1, the trigonometric Ruijsenaars model reduces to the trigonometric Calogero-Sutherland model and, in this case, a quantum field theory formulation is known, which naturally includes the deformed models [AL17].Moreover, parts of this construction were extended recently to the elliptic case [BLL20, Section III.A].Our results in this paper suggest that these quantum field theory results can be generalised to the Ruijsenaars case.A natural starting point would be a well-established quantum field theory description of the trigonometric Ruijsenaars model [SKA92], which allows for an elliptic generalisation [FHH09].
It is interesting to note that, while the deformed elliptic Calogero-Sutherland (eCS) system first appeared more than 10 years ago in a systematic search for kernel functions for eCS-type systems [Lan10], this very model recently appeared in the context of super-symmetric gauge theories [Nek17, CKL20].In our abovementioned paper [AHL14], we obtained the deformed elliptic Ruijsenaars model by generalising the former results to the relativistic case; it would be interesting to also establish relativistic generalisations of the latter results.
Finally, we note that the result in Eqs. ( 46)-(48) remains true even for complex q and t such that 0 < |q| < 1 and 0 < |t| < 1, provided that the definition of (i.e.no complex conjugation).However, then •, • ′ n,m;q,t is not sesquilinear and (in general) not positive (semi)definite, and thus no longer provides V n,m;q,t with a Hilbert space structure.

Appendix A. Conventions used by Sergeev and Veselov
Here we explain the relation between the conventions for the super-Macdonald polynomials used in this paper, and the ones used by Sergeev and Veselov (SV) [SV09a].As will be made clear, it is easy to translate from one convention to the other.Moreover, both conventions have their advantages and disadvantages.More specificially, the advantages of our conventions are that the super-Macdonald polynomials are manifestly invariant under (q, t) → (q −1 , t −1 ), and that Hilbert space adjungation agrees with what one would naively expect; cf.(37) vs. ( 76) and (41) vs. (78).The advantage of the SV-conventions is that factors t ±1/2 and q ±1/2 are avoided, and that some formulas look somewhat more symmetric; cf.(2) vs. (80), (30) vs. (81), and (39) vs. (82).
It is interesting to note that if we were to deform the integration radii ξ, ξ ′ > 0 from the region ξ/ξ ′ > M (q, t) or ξ/ξ ′ < 1/M (q, t) into the excluded region 1/M (q, t) < ξ/ξ ′ < M (q, t), then our sesquilinear form P, Q ′ 1,1;q,t would be changed by the addition of a residue term such as the integral in (88).By considering specific examples, is readily seen that this spoils orthogonality of the super-Macdonald polynomials as well as non-negativity.