An elliptic hypergeometric function approach to branching rules

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty $2$-cores.

The natural q, t-analogues of the Schur functions s λ are the Macdonald polynomials P λ (q, t) [18]. Similarly, the natural analogues of the symplectic Schur function sp 2n,λ are the C n Macdonald polynomials [19] P (Cn,Bn) λ (q, t, s) and P (Cn,Cn) λ (q, t, s).
In fact, the lifted Koornwinder polynomials (which are not polynomials) are the unique symmetric functions such that (1.6) holds, so that the above may serve as the definition of theK λ . 1 The Macdonald polynomials P (Bn,Cn) λ (q, t, t 2 ) may also be defined for half-partitions λ = (λ 1 , . . . , λn) where λ 1 . . . λn 0 and λ i ∈ 1 2 + Z, in which case the right-hand side of (1.5) needs to be slightly modified [35].
It is an elementary exercise to do the same for d 2λ and e λ . For t = q, (1.7a) and (1.7b) simplify to the p = 0 instances of (1.3) and [16, Equation where so λ is a universal special orthogonal character. By (1.4)-(1.6), Theorem 1.2 has the following corollary.
and for m, r, n nonnegative integers such that r n, .
The remainder of the paper is organised as follows. The next section covers some introductory material on partitions, various kinds of shifted factorial, and elliptic hypergeometric series. Then, in Section 3, we introduce some of the standard bases of the ring of symmetric functions and discuss the various types of classical Schur functions and classical branching rules. The final introductory section is Section 4, in which we survey material from Macdonald-Koornwinder theory, including the elliptic generalisation of this theory. In Sections 5-7 we prove a number of new results needed for our proof of Theorem 1.2 and Corollary 1.3, which is presented in Section 6.2. These new results include the evaluations of two quadratic elliptic beta integrals over elliptic interpolation functions (Theorems 5.1 and 5.4), their corresponding discrete analogues (Corollaries 5.3 and 5.6), a formula for the transition coefficients between Okounkov's BC n -symmetric Macdonald interpolation polynomials and ordinary Macdonald polynomials (Theorem 6.3), and a number of quadratic summations for a new type of elliptic hypergeometric series (Theorems 7.3 and 7.5). Finally, in Section 9 we propose a number of conjectures in the spirit of Conjecture 1.4. As a simple example, we conjecture the new Littlewood-type identity

Partitions.
A partition λ = (λ 1 , λ 2 , . . . ) is a sequence of weakly decreasing nonnegative integers such that |λ| := λ 1 + λ 2 + · · · is finite. If |λ| = n we say that λ is a partition of n, written as λ ⊢ n. The number of strictly positive λ i (the parts of λ) is called the length of the partition λ and denoted by l(λ). We also use l o (λ) to denote the number of odd parts of λ. If l o (λ) = 0 we say that λ is an even partition. The set of all partitions of length at most n is denoted by P + (n), and typically we write λ = (λ 1 , . . . , λ n ) for partitions in P + (n). The multiplicity of parts of size i in the partition λ is denoted by m i = m i (λ). We sometimes use the multiplicities to write a partition λ as (1 m1 2 m2 . . . ). When only a single multiplicity arises, i.e., a partition is of the form (m n ), we refer to it as a rectangle. When m i (λ) 1 for all i we say that λ is a distinct partition. Given a partition λ ∈ P + (n) such that λ 1 m, we write (m n ) − λ for the complement of λ with respect to the rectangle (m n ). That is, (m n ) − λ := (m − λ n , . . . , m − λ 1 ) ∈ P + (n). The partition λ ′ = (λ ′ 1 , λ ′ 2 , . . . ) such that λ ′ i = j i m j (λ) is called the conjugate of λ. Perhaps more simply, if we identify a partition λ with its Young diagram (in which the parts are represented by l(λ) left-aligned rows of boxes or squares, with ith row containing λ i squares) then the parts λ ′ correspond to the columns of λ. Other special notation for partitions that we will employ is 2λ := (2λ 1 , 2λ 2 , . . . ) and λ 2 := (λ 1 , λ 1 , λ 2 , λ 2 , . . . ), so that 2λ (resp. λ 2 ) corresponds to the partition in which the length of each row (resp. column) of λ has been doubled. The partition µ is contained in λ, denoted as µ ⊂ λ, if µ i λ i for all i, i.e., if the diagram of µ fits in the diagram of λ. We write µ ≺ λ if µ ⊂ λ such that the interlacing condition λ 1 µ 1 λ 2 µ 2 · · · holds. (Alternatively, µ ≺ λ if the skew shape λ/µ is a horizontal strip, see [18].) A partition λ has empty 2-core, written as 2-core(λ) = 0, if its diagram can be tiled by dominoes. For example, the partition (5, 4, 4, 1) has empty 2-core since it admits the tiling as well as four other such tilings.
Lemma 2.1. Let λ ∈ P + (n) and m an even integer such that m n. Then 2-core(λ) = 0 if and only if the ordered set contains m even and m odd integers.
Proof. The claim is (almost) trivially true by induction on the size of λ. Either we cannot remove a single domino from the border of λ, in which case λ does not have a trivial 2-core (it is in fact a 2-core itself) and is of the form (n, n − 1, . . . , 1) for some positive n, or it is possible to remove a domino from the border of λ to form a partition of size |λ| − 2. The removal of a domino of shape simply decreases one of the elements of A λ by 2, whereas the removal of a domino of shape decreases two consecutive elements of A λ by 1. The claim thus follows.
Given a square s = (i, j) ∈ λ the arm-length, arm-colength, leg-length and leg-colength of s are defined as Extending this to type-C, we also set The rationale for denoting the set of partitions of length at most n as P + (n) is that we identify such partitions with the dominant (integral) weights of GL(n, C). Frequently we also require the superset P (n) := {(λ 1 , . . . , λ n ) ∈ Z n : λ 1 λ 2 · · · λ n } of all (integral) weights. By mild abuse of notation, we sometimes write for µ ∈ P (n + 1) with µ n+1 = 0 that µ ∈ P + (n), i.e., we consider P + (n) not just as a subset of P (n) but also of P (n + 1).
For λ a partition, define the statistic This extends to skew shapes λ/µ in the obvious manner: n(λ/µ) := i (i − 1)(λ i − µ i ) = n(λ) − n(µ). By further abuse of notation (since conjugation no longer makes sense) we will also use n(λ) and n(λ ′ ) for λ ∈ P (n), defined as Two describe some of our conjectures we also require three types of even and odd analogues of n(λ) for λ a partition: it is generally not true that n e/o (λ),n e/o (λ) andn e/o (λ) (for fixed parity) coincide. In fact,n e/o (λ) can take half-integer values.
We will in fact only use the above six functions for partitions that have empty 2-core. In that case, we have the following simple relation.
For λ a partition such that 2-core(λ) = 0, Proof. We again prove the claim by induction on the size of the partition λ. For λ = 0 the claim is trivially true. Now let λ be a partition of size at least two. Because λ can be tiled by dominoes, it is always possible to remove a domino from its border to form a partition µ of size |λ| − 2.
First assume it is possible to remove a domino of shape such that the row-coordinate of the two boxes of the domino is i. Then The above gives irrespective of the choice of the parities p 1 , p 2 and p 3 . (This is consistent with the trivial fact that for λ an even partition, n p1 (λ) =n p2 (λ) =n p3 (λ).) Next assume that λ ′ is a distinct partition so that it is impossible to remove a domino of shape . We now only need to consider the first column of λ from which a domino of shape can be removed. If this is the jth column, then λ ′ is a distinct partition such that 2. (Of course, not each such a partition necessarily has an empty 2-core.) From a case-by-case analysis it follows that is even. Hence (2.2) again holds, but now with p 1 = p 2 = p 3 .

2.2.
Generalised shifted factorials. In this paper we require several types of shifted factorials. For complex q such that |q| < 1 the ordinary q-shifted factorial (z; q) ∞ is defined as This may be used to defined (z; q) N for arbitrary integer N as In particular, if N is nonnegative, (z; q) N = N k=1 (1 − zq k−1 ) and if N is a negative integer 1/(q; q) N = 0. To generalise both definitions to the elliptic case, we need the elliptic gamma function [39] where z ∈ C * and p, q ∈ C such that |p|, |q| < 1. This function is symmetric in p and q, satisfies the reflection formula Γ p,q (z)Γ p,q (pq/z) = 1 and functional equation where θ(z; p) is the modified theta function θ(z; p) := (z; p) ∞ (p/z; p) ∞ .
Since lim p→0 1/Γ p,q (z) = (z; q) ∞ the reciprocal of the elliptic gamma function can be viewed as an elliptic analogue of (2.3), and the elliptic analogue of (2.4) is then which for nonnegative N can also be expressed as Clearly, (z; q, 0) N = (z; q) N . Three important generalisations of (z; q, p) N to the case of partitions are given by [29,44] C 0 λ (z; q, t; p) := where it is noted that C 0 (N ) (z; q, t; p) = C − (N ) (z; q, t; p) = (z; q, p) N and C + (N ) (z; q, t; p) = (z; q, p) 2N (z; q, p) N .
From the simple functional equations for the theta function it follows that the elliptic C-symbols satisfy the quasi-periodicities as well as a long list of other simple identities, such as and C 0 λ+(N n ) (z; q, t; p) = C 0 (N n ) (z; q, t; p)C 0 λ (zq N ; q, t; p), (2.13a) where in the final set of identities it is assumed that λ ∈ P + (n) and N is a nonnegative integer. Expressing C 0,± (N n ) (z; q, t; p) in terms of the elliptic shifted-factorial (2.6), we may use (2.13) to extend the elliptic C-symbols to arbitrary weights λ ∈ P (n): (zq λn t 2−n−i ; q, p) λn , (2.14c) where µ := (λ 1 − λ n , . . . , λ n−1 − λ n , 0) ∈ P + (n). All of the above identities, with the exception of (2.10) remain valid for non-dominant weights.
They satisfy the obvious analogues of (2.10)-(2.13), where it is noted that in the case of (2.10a), (2.10c) and (2.11) one first needs to eliminate an explicit p in the argument on the right using (2.9) before setting p to 0. To avoid having to produce another five identities, we will always refer to the above relations -even in the non-elliptic case -when manipulating C-symbols. The reader should have no trouble writing down the explicit p = 0 versions. For instance, in the case of (2.10a) on finds , and so on.
For all the shifted factorials as well as the elliptic gamma and modified theta functions adopt the usual multiplicative and plus-minus notations, such as C 0 λ (z 1 , . . . , z k ; q, t; p) := C 0 λ (z 1 ; q, t; p) · · · C 0 λ (z k ; q, t; p) and . To further shorten some of our expressions we also introduce the multiplicative well-poised ratio . . , apq/b k ; q, t; p) and the non-multiplicative Finally, there are six more non-elliptic C-symbols needed to describe some of our conjectures. They are defined as where λ e/o := |{(i, j) ∈ λ : i+j even/odd}| and |λ| e/o := |{s ∈ λ : a(s)+l(s) even/odd}|.
We will again only be using the above for λ a partition with empty 2-core, in which case we simply have λ e/o = |λ| e/o = |λ|/2.
Proof. We will only show the last of the three identities. Applying definition (2.16c) to its left-hand side and interchanging i and j in the product leads to By (2.1) the result now follows.
In a similar manner it may be shown that (the p = 0 case of) (2.11) dissects into even and odd cases as follows.
where in the final line it is assumed that λ is a partition with empty 2-core.

Elliptic hypergeometric series.
Our proof of Theorem 1.2 relies on (the p → 0 limit of) two higher-dimensional quadratic summation formulas for elliptic hypergeometric series. In the one-dimensional case the simplest form an elliptic hypergeometric series can take is [11,42,44] (2.17) where, for reasons of convergence, it is assumed that one of the b i is of the form q −N with N a nonnegative integer so that the series terminates. If the upper and lower parameters satisfy ab 1 · · · b r (pq) 3 = c 1 · · · c r the series (2.17) is said to be balanced, and if b i c i = apq for all i it is said to be very-well poised. If both these conditions are satisfied then (2.17) is an elliptic function (in multiplicative form) in each of the variables a, b 1 , . . . , b r , see e.g., [42]. The most important identity for one-dimensional elliptic hypergeometric series corresponds to (2.17) for r = 5, and is given by Frenkel and Turaev's elliptic analogue of Jackson's sum [10]: where bcdeq −N = a 2 pq. Four other balanced, very-well poised instances of (2.17) for r = 7 that admit closed-form evaluations are given by Because of the occurrence of base q and q 2 (and nomes p and p 2 ), the above identities are commonly referred to as quadratic summation formulas. In Section 7 we obtain higherdimensional analogues of (2.19a), (2.19b) and (2.19d). Two of these play a key role in our proof of Theorem 1.2. We remark that higher-dimensional analogues of a different type of quadratic elliptic hypergeometric series, in which the term (apq; q, p) 2k /(a; q, p) 2k in (2.17) is replaced by (apq; q, p) 3k /(a; q, p) 3k , were recently considered in [37].

Schur functions and classical branching rules
In this section we briefly review the definitions of the Schur functions of classical type as well as their occurrence in some of the branching rules stated in Section 1. For a more in-depth treatment we refer the reader to [14, 16-18, 23, 35].
Let Λ n := Z[x 1 , . . . , x n ] Sn denote the ring of symmetric functions in n variables, and Λ the ring of symmetric functions in countably many variables, see [18,43]. The monomial symmetric functions {m λ } λ∈P+(n) and {m λ } λ , where form Z-bases of Λ n and Λ respectively. The elementary, complete and power-sum symmetric functions e r , h r and p r are defined in terms of the monomial symmetric functions as These functions form algebraic bases of either Λ (in the case of the e r and h r ) or of Λ Q := Λ ⊗ Z Q (in the case of the power sums). A number of classical branching rules for universal characters discussed below are related by the involution ω on Λ defined by ω(h r ) = e r or ω(p r ) = (−1) r−1 p r for all r 1.
The ordinary (or GL(n)) Schur function indexed by the partition λ is defined as n and 0 otherwise. To simultaneously extend this to Λ as well as skew shapes, we use the Jacobi-Trudi identity or its dual [18, pp. 70&71]: , where n and m are arbitrary integers such that n l(λ) and m λ 1 . Obviously, ω(s λ/µ ) = s λ ′ /µ ′ . The Littlewood-Richardson coefficients c λ µν may now be defined by for λ ⊂ (m n ) so that the Littlewood-Richardson coefficients satisfy the complementation symmetry For λ a partition, the universal orthogonal and symplectic characters indexed by λ are given by [ with n and m as above. Hence ω(o λ ) = sp λ ′ . In particular, where g = sp (resp. g = o) corresponds to an orthogonal or symplectic character indexed by λ. We add to the above the universal special orthogonal character indexed by λ as so that ω(so λ ) = so λ ′ . The character so λ is the unique symmetric function such that where so 2n+1,λ is the odd-orthogonal Schur function indexed by λ. The character so λ may readily be related to the universal symplectic and orthogonal characters as For the actual symplectic, orthogonal and odd-orthogonal Schur functions we have [17] sp 2n,λ (x 1 , . . . , , where f n,n = 1 and f i,n = 1/2 if i < n, and Littlewood [17] and Koike and Terada [14] proved some very general branching formulas for the classical groups. For example, in the universal case [ By the e-Pieri rule [18, p. 73], We also remark that the three universal branching rules (1.3), (1.9) and (1.10) are not independent. Obviously, (1.9) follows from (1.3) by application of ω and vice versa. Also, the rectangular (i.e., p = 0) cases of each of the branching rules are related via (3.4). For example, from (1.10) and (3.4a), and the fact that where in the last step we have used Conversely, from (1.9) and (3.4b), since there is a unique even partition λ such that λ ′ ≺ µ ′ .
In the q, t-case, we neither have analogues of (3.4) nor of (3.6), making the proof of Theorem 1.2 much harder than in the classical case. As we shall see, however, (1.7a) and (1.7b) are related by the q, t-analogue of the involution ω.

Macdonald-Koornwinder theory
In this section we survey some necessary background material from the theory of Macdonald and Koornwinder polynomials, covering Macdonald polynomials, BC n -symmetric (Macdonald) interpolation polynomials, (lifted) Koornwinder polynomials and elliptic interpolation functions.
The Macdonald polynomials satisfy the symmetry Since they can be extended from λ ∈ P + (n) to arbitrary weights λ ∈ P (n) via where µ := (λ 1 − λ n , . . . , λ n−1 − λ n , 0) ∈ P + (n). Then {P λ (q, t)} λ∈P (n) forms a Q(q, t)basis of the ring of S n -symmetric Laurent polynomials, which in the following we will denote by Λ GL(n) . Since, where m = min{λ n , µ n }, the orthogonality (4.3) and evaluation (4.4) extend to all λ, µ ∈ P (n). In the case of (4.4) for λ a non-dominant weight, the first expression on the right should be used as both C − λ (zq; q, t) and 1/C 0 λ (zqt n−1 ; q, t) have a pole (of order one) at z = 1.
For later use we note that by (4.7) and the complementation symmetry for Macdonald polynomials (see e.g., [3]) we have the further symmetry If the involution ω on Λ is extended to the following automorphism of Λ Q(q,t) : Any non-constant BC n -symmetric polynomial is necessarily inhomogeneous. It will thus be convenient to extend the dominance order from partitions of the same size to all partitions in the obvious way: λ µ, if λ 1 + · · · + λ i µ 1 + · · · + µ i for all i 1.
Let R = Q(q, t)[s ± ], λ, µ ∈ P + (n) and λ n;q,t := q λ1 t n−1 , . . . , q λn−1 t, q λn a spectral vector. Then the BC n -symmetric (Macdonald) interpolation polynomial satisfying the vanishing conditions (4.12) P µ s λ n;q,t ; q, t, s = 0 if µ ⊂ λ, see [25,28]. Since any triangular BC n -symmetric polynomial with leading term m W µ is uniquely determined by its values at z λ n;q,t for λ < µ and some arbitrary nonzero z, the above vanishing conditions in fact lead to an overdetermined linear system for the coefficients c µλ . One of the main results of [25] is the actual existence of the interpolation polynomials. 2 The interpolation polynomialP * µ (q, t, s), whose top-degree term coincides with the Macdonald polynomial P µ (q, t), satisfies the symmetries for N an arbitrary integer such that N −µ n . Like the Macdonald polynomials, this can be used to extend the BC n interpolation polynomials to arbitrary weights µ ∈ P (n): where ν := (µ 1 − µ n , . . . , µ n−1 − µ n , 0) ∈ P + (n). Of course, for µ not dominant, i.e., for µ ∈ P + (n),P * µ (x; q, t, s) is not a Laurent polynomial but a rational function in x. The interpolation polynomials also admit a closed-form evaluation at x = s µ n;q,t (4.16) P µ s µ n;q,t ; q, t, s , as well as a principal specialisation formula (4.17)P * µ z 0 n;q,t ; q, t, s for µ ∈ P (n).
By (4.13) and (4.20) it follows that where 1/t is shorthand for (1/t 0 , 1/t 1 , 1/t 2 , 1/t 3 ). One drawback of the above definition of the Koornwinder polynomials is that it hides the S 4 -symmetry in the parameters t 0 , t 1 , t 2 , t 3 . It however follows from the connection coefficient formula for the interpolation polynomials [ where y := (y 1 , . . . , y m ).
As mentioned in the introduction, the lifted Koornwinder polynomial [28] (4.27) is the unique symmetric function such that Some care is required when dealing with this function since the above equation requires t to be generic. Issues may arise for t such that C + λ (t 2n−2 t 0 t 1 t 2 t 3 /q; q, t) = 0. This for example happens for the parameter choice t = (1, −1, t 1/2 , −t 1/2 ) in which case it is important to specialise T = t n before specialising t.

4.4.
Elliptic interpolation functions. The (BC n -symmetric) elliptic interpolation functions R * µ (a, b; q, t; p) [6,29,32] are an elliptic analogue of the BC n -symmetric interpolation polynomialsP * µ (q, t, s). Although they satisfy analogous vanishing conditions, their definition is more complicated. Below we follow the characterisation of these function given in [29].
The elliptic binomial coefficients satisfy a large number of symmetries and identities, and for a complete list of these the reader is referred to the original papers [29,31,32] or the survey [38]. Here we state a selection of result needed later. By (4.38) and (4.39) it follows that the special case λ = (N n ) and ν = 0 of (4.40) corresponds to the (B)C n -analogue of the elliptic Jackson sum (2.18), see e.g., [6,22,29,36,44]. Two important symmetries we will rely on in Section 7 are the reciprocity and conjugation symmetries Finally, as follows from (4.18) and (4.32), in the limit the elliptic binomial coefficients reduce to the binomial coefficients (4.18):

5.2.
The interpolation kernel. Let x, y ∈ (C * ) n and c, p, q, t ∈ C * such that |p|, |q| < 1. Then the interpolation kernel K c (x; y; t; p, q) is a meromorphic BC n -symmetric function in both x and y, satisfying (5.3) K c (x; y; t; p, q) = K c (x; y; t; q, p) = K c (y; x; t; p, q) = K −c (−x; y; t; p, q) such that [33] (5.4) K c (x; a µ n;t;p,q /c; t; p, q) = R * µ (x; a, b; q, t; p) where c 2 = abt n−1 and µ n;t;p,q := p µ (1) 1 q µ (2) 1 t n−1 , . . . , p µ (1) n−1 q µ (2) n−1 t, p µ (1) n q µ (2) n is an elliptic spectral vector. The interpolation kernel may recursively be defined using the initial conditions and branching rule ×Ĉ K c/t 1/2 (z;ŷ; t; p, q)∆ D t 1/2 x ± , pqy ± n /ct 1/2 ; z; p, q dz 1 z 1 · · · dz n−1 z n−1 , for x, y ∈ (C * ) n ,ŷ = (y 1 , . . . , y n−1 ), z ∈ (C * ) n−1 and a suitable subset of the parameter space. By (5.3), making the substitution (and also negating the integration variables on the right) leaves (5.6) unchanged. Hence there is no need to fix a branch of t 1/2 . We further note that the symmetry of the kernel in y is not manifest from the recursive definition, but follows from a similar such symmetry for the "formal interpolation kernel" K c (x; y; q, t; p) defined in [33] as a generalisation of the connection coefficients identity for R * µ (x; a, b; q, t). (By (5.5), for n = 2 the symmetry is an immediate consequence of a special case of the elliptic integral transformation [31, Theorem 4.1].) 5.3. The dual Littlewood kernel. Following [33] we consider two further kernels, known as the dual Littlewood kernel and Kawanaka kernel. For the dual Littlewood kernel, L ′ c (x; t; p, q), let x ∈ (C * ) n and c, p, q, t, u, v ∈ C * such that c 4 uv = p and |p|, |q| < 1.

The Kawanaka kernel.
For the Kawanaka kernel, L − c (x; t; p, q), we have a very similar definition and set of results as for the dual Littlewood kernel.
Let x ∈ (C * ) n and c, p, q, t, u, v ∈ C * such that c 2 uv = pq and |p|, |q| < 1. Then the Kawanaka kernel is defined as [33] which once again does not depend on the individual choice of u and v. The Kawanaka kernel satisfies the symmetry and factorisation formula Evaluating (5.14) at x = a µ n;t 2 ;p 2 ,q 2 /c and proceeded exactly as in the dual Littlewood case, also using we find L − c (a µ n;t 2 ;p 2 ,q 2 /c; t; p, q) L − c (a 0 n;t 2 ;p 2 ,q 2 /c; t; p, q) where, c 2 uv = pq and c 2 = abt 2n−2 . Replacing t → −t and then specialising c = (pq/t) 1/2 , the left-hand side factors by (5.15), leading to our next theorem.
Here we note that a in (5.17) has been replaced by apqt 1−2n .
6. Transition coefficients via elliptic hypergeometric integrals 6.1. Transition coefficients. Recall that the Macdonald polynomials P λ indexed by weights λ ∈ P (n) form a basis Λ GL(n) . In particular, any BC n -symmetric polynomial can be expanded in terms of Macdonald polynomials. If {f λ } (for λ ∈ P + (n) or λ ∈ P (n)) is a basis of Λ BC(n) and g an arbitrary element of Λ BC(n) which expands in this basis as we will write [f λ ]g to denote the coefficient c λ . By (4.9) it then follows that (6.1) P (λ1,...,λn) (q, t) g = P (−λn,...,−λ1) (q, t) g, so that it suffices to consider P λ (q, t) g for λ ∈ P (n) such that λ 1 0. We are concerned with computing the transition coefficients for λ ∈ P (n) and µ ∈ P + (n). Apart from It will be convenient to scale c (n) λµ (q, t, s) to a function that depends polynomially on s 2 . To this end we define , where we recall that n(λ) and n(λ ′ ) are defined for arbitrary weights λ on page 6. Some of the above symmetries for c Proof. According to [28,Theorem 6.16], for λ, µ ∈ P + (n), , andC µµ = 1. We also have, for λ ∈ P + (n) and ν ∈ P (n), thatĈ νλ := P ν (x; q, t) P λ (x ± ; q, t) = 0 if |λ| − |ν| is odd, or if there exists an 1 i n such that |ν i | > λ i (whereĈ νλ ∈ Q(q, t)), andĈ λλ = 1. Combining these two results the claim immediately follows.
Corollary 6.2. Let N ∈ Z and µ ∈ P + (n). Then It seems to be a hard problem to get a handle on the general form of C (n) λµ (q, t, s). When n = 1 it is a straightforward exercise in basic hypergeometric series to show that for N an integer and k a nonnegative integer Here k i q is the standard q-binomial coefficient For s = 1 the sum in (6.5) can be performed by the q-Chu-Vandermonde summation [11, Equation (II.7)] to give which generalises nicely for λ = (N n ).
Theorem 6.3. For N a nonnegative integer and µ ∈ P + (n) such that (N n ) ⊂ µ, By (6.4) with s 2 = q, the restriction that N is nonnegative is non-essential, and it is of course clear that the right-hand side of (6.6) is invariant under negation of N . Replacing µ → µ + (N n ) and using (2.13) and (4.14), it follows that (6.6) may also be stated as .
By the connection coefficient formula forP * µ (q, t, s), see [28, Theorem 3.12], (6.6) leads to an expression for the more general transition coefficient C (n) (N n ),µ (q, t, sq 1/2 ). Since this result is not needed later, we omit the details. We do remark, however, that this expression does not (in an obvious manner) generalise (6.5), but instead extends the alternative form which obscures the fact that this is polynomial in s 2 . The equality of (6.5) and (6.8) follows from the transformation formula which we have not yet succeeded in generalising to the multivariable setting. The more general expression for C (n) (N n ),µ+(N n ) (q, t, sq 1/2 ) does however show it to be a rational function in Q(q, t, s, t n , q N t n ). 6.2. Proof of Theorem 6.3. In this section we give a proof of Theorem 6.3 based on the elliptic hypergeometric integral (5.12). Proposition 6.4. For µ ∈ P (n) and q, t, s ∈ C such that 0 < |q|, |t| < 1 and |sq µn | < 1, The integrand on the left has simple poles at z i = (sq µn+k ) σ for σ ∈ {−1, 1}, 1 i n and k a nonnegative integer. The condition |sq µn | < 1 ensures that the poles for σ = 1 (σ = −1) lie in the interior (exterior) of T n .
Before showing how (6.9) follows from (5.12), we first use the former to prove Theorem 6.3.
Taking the same limit in the right-hand side of (6.11) yields Equating the right-hand sides of (6.12) and (6.13), and replacing v → q/v results in (6.9) with (q, t) → (q 2 , t 2 ), completing the proof.

The elliptic hypergeometric function Φ λ
In this section we define a new elliptic hypergeometric function, Φ λ = Φ λ (q, t; p) = Φ λ (a; b, c, d; q, t; p), study its symmetries and prove two summation formulas for one-parameter specialisations of {a, b, c, d}. The p → 0 limit of Φ λ (q, t) will play an important role in proving the q, tbranching rules (1.7) and (1.11). For λ a partition and a, b, c, d, p, q, t ∈ C * such that |p| < 1, the elliptic hypergeometric function Φ λ (q, t; p) is defined as Φ λ (a; b, c, d; q, t; p) where e := bcd/aq 2 and f := bcdq/a 3 pt. The equality of the two expressions on the right of (7.1) is a direct consequence of the quasi-periodicity (2.9) of the elliptic C-symbols. We also note that Φ λ (a; −a, c, d; q, t; p) is a function of a 2 only, so that Φ λ (a; −a, c, d; q, t; p) = Φ λ (−a; a, c, d; q, t; p).
Proof. In the following we consider the right-hand side of (7.8) with (p 2 , q 2 , t 2 ) → (p, q, t).
We note that for λ = (N ) and (a, t) → (−a, b/a) this is (7.13) and that in going from (7.12) to (7.14) the parameter t has been replaced by −t.
The proof of Corollary 7.6 proceeds along the same lines as the proof of Corollary 7.4, and we omit the details.

Proof of Theorem 1.2
In Section 1 we stated (1.11) as a corollary of (1.7), but in fact both results are equivalent, and proving (1.11) for fixed m, r and all n r is the same as proving (1.7) for fixed m, r. To avoid the use of virtual Koornwinder polynomials in our proof, we will in the following establish Corollary 1.3 instead of Theorem 1.2.
The first step in our proof is to dualise the three claims of Corollary 1.3, an approach that was also utilised in [35] to prove bounded Littlewood identities for Macdonald polynomials.
Let x := (x 1 , . . . , x n ) and y := (y 1 , . . . , y m ). By the complementation symmetry (4.8) and homogeneity of the Macdonald polynomials, the (dual) Cauchy identity (4.6) can be written in the form (see also [25]) Replacing n → 2n and then specialising x i+n = x −1 i for all 1 i n yields µ⊂(m 2n ) Up to the simple factor (−1) mn (y 1 · · · y m ) n the right-hand side coincides with the righthand side of the Cauchy identity (4.26) for Koornwinder polynomials. Correcting for this factor, we can thus equate the respective left-hand sides, resulting in Let r be an integer such that 0 r n, λ a partition contained in (m r ), and s := 2n − r n. Extracting the coefficient of K (m r )−λ (x; q, t; t)P (s m ) (y; t, q) where N := n − r. Hence where the reader is warned that in the above right-hand side we have not followed our earlier convention, and P λ (y; t, q) f (y) denotes the coefficient of P λ (t, q) of f ∈ Λ GL(m) .
Similarly, the dual case of (1.11b) translates to with N and λ as above, and .
Finally, in the case of (1.11c) we get .
In the large-m, n limit, Conjecture 9.4 then simplifies to the following pair of unbounded Littlewood identities.