Symmetry algebra for the generic superintegrable system on the sphere

The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the $d$-sphere. Appropriately normalized, the symmetry operators preserve the space of polynomials. Under mild conditions on the free parameters, maximal abelian subalgebras of the symmetry algebra, generated by Jucys-Murphy elements, have unique common eigenfunctions consisting of families of Jacobi polynomials in $d$ variables. We describe the action of the symmetries on the basis of Jacobi polynomials in terms of multivariable Racah operators, and combine this with different embeddings of symmetry algebras of lower dimensions to prove that the representations restricted on the space of polynomials of a fixed total degree are irreducible.


Introduction
Recall that a quantum superintegrable system is an integrable Hamiltonian system on a d-dimensional Riemannian or pseudo-Riemannian manifold with potential: H = ∆ + V that admits the maximum possible 2d − 1 algebraically independent partial differential operators H k commuting with H, i.e.
These systems appear in a wide variety of modern physical and mathematical theories, from semiconductors to black holes. For a thorough account of the general theory and its numerous applications we refer the reader to the review article by Miller, Post and Winternitz [15].
Many important examples of superintegrable systems can be obtained through limits from the so called generic quantum superintegrable system on the sphere, with potential where y belongs to the d-sphere, and {b k } k=1,...,d+1 are free parameters. The representation theory of the algebra generated by the symmetries of this system has attracted a lot of attention recently and it turned out to be closely related to multivariate extensions of the Askey scheme of hypergeometric orthogonal polynomials and their bispectral properties, the Racah problem for su (1,1), representations of the Kohno-Drinfeld algebra, the Laplace-Dunkl operator associated with Z d+1 2 root system; see for instance [2,5,6,8,11,12,16,18] and the references therein.
The goal of the present paper is two fold. First, we describe the representations of the symmetry algebra for the generic quantum superintegrable system on the sphere in terms of the multivariable Racah operators introduced in [7] on the space of polynomials in several variables. We provide a detailed account of the theory, building and expanding on the recent work [8]. In dimensions 2 and 3, the approach is based on first principles which naturally leads to precise constraints on the free parameters {b k } for which these constructions can be applied. In particular, it explains how these constraints fix uniquely the underlying basis of Jacobi polynomials as eigenfunctions of maximal abelian subalgebras generated by Jucys-Murphy elements. The second goal of the paper is to combine these results and constructions with different embeddings of the symmetry algebras of lower dimensions to prove that the representations are irreducible.
While the arguments apply in arbitrary dimension, the difficulty increases significantly as the dimension grows, and dimension d ≥ 4 requires conceptually new ingredients. For that reason, we treat separately the 2-dimensional and the 3dimensional sphere, where we can write simple closed formulas for all symmetry operators. The explicit formulas here serve as important building blocks for the constructions in higher dimensions, and provide an alternative approach to several earlier works. In dimension d ≥ 4, the picture changes drastically -linear combinations of the Racah operators are no longer sufficient to describe the representations of the symmetry algebra. To see this, we discuss the linear independence of the second-order symmetries, which combined with the total number of available Racah operators explains the need to search for algebraic generators. Thus, we arrive at a fourth-order relation which allows to construct an explicit set consisting of 2d − 1 algebraic generators for the symmetry algebra.
The paper is organized as follows. In the next section, we introduce the generic quantum superintegrable system on the sphere, its symmetry algebra and an appropriate gauge transformation, which allows to induce representations on the space of polynomials of several variables. In Section 3 and 4, we describe the representations of the symmetry algebra for the 2-sphere and the 3-sphere, respectively, and we prove that these representations are irreducible. In Section 5, we outline the constructions in arbitrary dimension together with a detailed proof of the irreducibility.

The model and its normalization
We denote by S d = {y ∈ R d+1 : y 2 1 + · · · + y 2 d+1 = 1} the d-dimensional sphere in R d+1 and we set ∂ yj = ∂ ∂yj for the partial derivative with respect to y j , for j = 1, . . . , d + 1. The generic superintegrable system on the sphere is the quantum system with Hamiltonian is the Laplace-Beltrami operator on the sphere and {b k } k=1,...,d+1 are free parameters. If we set Next, we pick appropriate coordinates and a gauge factor, so that the symmetry operators preserve the space of polynomials. Let In the remaining part of the paper, we will work with the variables x i and the parameters γ i , related to the original variables and parameters by (2.3). In particular, we will impose later mild restrictions on the parameters γ i , which can easily be translated onto the parameters b i using the connection in (2.3). If G(x) denotes the gauge factor then it is straightforward to check that For a vector z ∈ R s we denote by |z| = z 1 + · · · + z s the sum of its coordinates. If we fix x = (x 1 , . . . , x d ) as coordinates, then the fact that y ∈ S d implies that x d+1 = 1 − |x|. If we denote by L i,j the operator on the right-hand side of equation (2.4), we obtain the following expression in the coordinates x = (x 1 , . . . , x d ): and With the above notations we also have and |γ| = γ 1 + · · · + γ d+1 . The above computations show that we can replace the Hamiltonian H and its symmetry operators H i,j by the operator L and its symmetry operators L i,j . Note that the operators L i,j have polynomial coefficients and therefore they preserve the space of polynomials of x 1 , . . . , x d . In the next sections, we use this fact to construct irreducible representations of the algebra A d (γ) generated by the operators L i,j , which in view of the above computations, corresponds to the symmetry algebra for the generic quantum superintegrable system on the sphere S d . Therefore, we refer to A d (γ) as the symmetry algebra for the generic system on the sphere. The operator L defined in (2.8) has a long history in the mathematical literature. It first appeared in the monograph [1] in the case d = 2 in connection with differential equations satisfied by the Lauricella functions. Its link to the superintegrable system on the sphere in arbitrary dimension was revealed in [13].
Finally, we note yet another link which plays an important role in the constructions. From the explicit formulas in equation (2.5)-(2.6) it is easy to see that the operators L i,j satisfy the following commutativity relations: These relations show that the symmetry operators provide a representation of the Kohno-Drinfeld algebra which appears in the structure of the holonomy of the Knizhnik-Zamolodchikov connection and the representation of the braid group, see for instance [14].
3. Symmetry algebra for the 2-sphere 3.1. Construction of the module. When d = 2, the symmetry algebra is generated by the 3 operators: L 1,2 , L 1,3 , L 2,3 which commute with L = L 1,2 +L 1,3 +L 2,3 . The operators L and L 2,3 can be simultaneously diagonalized on the space of polynomials and the eigenfunctions can be written explicitly in terms of appropriate two-variable Jacobi polynomials. This was discovered by Proriol [19], building on the work [17] where an important particular case appeared in the study of the Schrödinger equation for helium. To describe this construction, we define the classical Jacobi polynomial of degree n with parameters α and β via the formula where (a) k denotes the Pochhammer symbol: (a) 0 = 1 and (a) k = a(a + 1) · · · (a + k − 1) for k ∈ N.
We use Z ≤k to denote the set of all integers less than or equal to k. Note that the Jacobi polynomial p (α,β) n (t) is a well-defined polynomial of degree n when the parameters α and β satisfy the following conditions These conditions are automatically satisfied when α > −1 and β > −1. In this case, the polynomials p (α,β) n (t) are mutually orthogonal with respect to the weight (1 − t) α (1 + t) β on (−1, 1).
Next, we define polynomials in two variables x = (x 1 , x 2 ), with degree indices ν 1 and ν 2 , depending on the three parameters γ = (γ 1 , γ 2 , γ 3 ) in terms of the one-variable Jacobi polynomials as These polynomials are well-defined if the parameters γ 1 , γ 2 , γ 3 satisfy the following conditions The polynomials P ν1,ν2 (x; γ) are eigenfunctions of the commuting operators L and L 2,3 and satisfy the spectral equations For n ∈ N 0 we denote by V 2 n (γ) the space spanned by the two-variable polynomials in (3.2) of total degree n, i.e. we set We already know that the operators L and L 2,3 act diagonally on the basis P ν1,ν2 . It turns out that the operators L 1,2 and L 1,3 preserve this space and therefore V 2 n (γ) is a module of the symmetry algebra A 2 (γ) generated by the operators L 1,2 , L 1,3 and L 2,3 . To describe their action, we denote by Id the identity operator and we define shift operators E νj acting on functions of ν = (ν 1 , ν 2 ) by and E ν2 f ν1,ν2 = f ν1,ν2+1 .
With these notations, we consider the recurrence operator acting on functions of ν = (ν 1 , ν 2 ) by with coefficients Note that for n ∈ N 0 we can restrict its action on functions defined for ν 1 + ν 2 = n since both operators E −1 ν1 E ν2 and E ν1 E −1 ν2 preserve this condition. Using the explicit formulas above, one can show that this operator represents the action of L 1,2 on the basis P ν1,ν2 (x; γ) of V 2 n (γ), i.e. we have L 1,2 P ν1,ν2 (x; γ) = B 1,2 P ν1,ν2 (x; γ). (3.8) This shows that the space V 2 n (γ) is a module over the symmetry algebra A 2 (γ). The explicit action of L 1,3 on the basis P ν1,ν2 (x; γ) can be easily deduced by writing L 1,3 as and using equations (3.4), (3.5) and (3.8).
We note that, appropriately normalized, the operator B 1,2 corresponds to the recurrence operator for the Racah polynomials. This fact was discovered in the work of Kalnins, Miller and Post [11] in different notations, where the representations of the symmetry algebra for the 2-sphere were investigated. A different interpretation, related to representations of su(1, 1) was obtained by Genest, Vinet and Zhedanov [6]. We outline yet another explanation based on orthogonal polynomials, which is close to the presentation above, and which was used in [8] to obtain explicit formulas in arbitrary dimension in terms of the Racah operators introduced in [7]. If γ j > −1 for j = 1, 2, 3, then the polynomials P ν1,ν2 (x; γ) in (3.2) are mutually orthogonal with respect to the weight Moreover, one can show that the operators L 1,2 , L 1,3 , L 2,3 are self-adjoint with respect to the inner product induced by w 2 . This means that the symmetry algebra will preserve the space V 2 n (γ) of orthogonal polynomials of fixed total degree n. Note also that w 2 is invariant if we permute simultaneously (γ 1 , γ 2 , γ 3 ) and (x 1 , This shows that we can construct other bases of V 2 n (γ) by applying permutations. The transition matrices between these different orthogonal bases can be expressed in terms of the Racah weight and polynomials as shown by Dunkl in [3]. In particular, the result of Dunkl which connects P ν1,ν2 (x; γ) to the basis on which L 1,2 acts diagonally can be used to express B 1,2 in terms of the Racah operator.

4.2.
Irreducibility. The goal of this subsection is to show that V 3 n (γ) is an irreducible module over A 3 (γ).
First, we consider the operators L 2,3 , L 2,4 , L 3,4 which generate the algebra A 2 (γ 2 , γ 3 , γ 4 ). Since these operators do not contain differentiation with respect to x 1 , it follows that they commute with the first factor on the right-hand side of equation (4.2). Therefore, if we fix k, n ∈ N 0 such that 0 ≤ k ≤ n, then the operators L 2,3 , L 2,4 , L 3,4 preserve the subspace of V 3 n (γ). It is not hard to see now that W 1 n,k (γ) can be identified with the module V 2 n−k (γ 2 , γ 3 , γ 4 ) constructed in the previous section. One way to see this is to note that, up to a factor independent of x 2 and x 3 , the product consisting of the last two terms in (4.2): coincides with the two-variable polynomial P ν2,ν3 (y 1 , y 2 ; γ 2 , γ 3 , γ 4 ) in (3.2) in the variables y 1 and y 2 defined by Moreover, if make the same change of variables in the operators L 2,3 , L 2,4 , L 3,4 for the 3-sphere, we obtain in the y variables the operators L 1,2 , L 1,3 , L 2,3 for the 2-sphere. It is also useful to compare equations (4.9), (4.6), (4.5) in the threedimensional setting with equations (3.7), (3.5), (3.4) in the two-dimensional setting, respectively. Next, we consider the subspace W 0 n (γ) = span{P ν1,ν2,0 (x; γ) : ν 1 + ν 2 = n}, (4.17) of V 3 n (γ). We want to show that the operatorsL 1,2 = L 1,2 ,L 1,3 = L 1,3 + L 1,4 , L 2,3 = L 2,3 + L 2,4 preserve this space, and we can identify the subspace W 0 n (γ) under their action with the module V 2 n (γ 1 , γ 2 , γ 3 +γ 4 +1) constructed in the previous section. Clearly, W 0 n (γ) is a space of polynomials in the variables x 1 and x 2 and therefore ∂ x3 will act as the zero operator on this space. Using equation (2.5) with i = 1, j = 3 and ignoring ∂ x3 , we see that Similarly, using equation (2.6) with j = 1 and d = 3 we deduce Adding the last two equations, we see that which coincides with the operator in L 1,3 defined by (2.6) in dimension d = 2 with parameter γ 3 replaced by γ 3 + γ 4 + 1. A similar computation shows that coincides with the operator in L 2,3 defined by (2.6) in dimension d = 2 with parameter γ 3 replaced by γ 3 + γ 4 + 1. Finally, it is easy to see that if we put ν 3 = 0 in equation (4.2), the expression on the second line is 1 and P ν1,ν2,0 (x; γ) coincides with the two-variable polynomials defined by (3.2) with parameter γ 3 replaced by γ 3 + γ 4 + 1. Applying the results in Section 3, we conclude that W 0 n (γ) is an irreducible module over the algebra generated byL 1,2 ,L 1,3 ,L 2,3 .

Symmetry algebra for the d-sphere
Recall that A d (γ) is the algebra generated by the operators L i,j , where 1 ≤ i < j ≤ d + 1. For every n ∈ N 0 , we construct in this section a module over A d (γ) consisting of polynomials of total degree n in d variables, and we show that this module is irreducible. 5.1. Construction of the module.
For n ∈ N 0 , we define the space V d n (γ) spanned by the polynomials in (5.1) of total degree n, i.e. we set We will show that V d n (γ) is an irreducible module of the algebra A d (γ) generated by the symmetry operators L i,j , 1 ≤ i < j ≤ d + 1. We postpone the proof of the irreducibility for later, and we describe first the action of all operators L i,j on the basis {P ν (x; γ)}.
We denote by R(z) the field of rational functions of finitely many of the z j 's and for k ∈ N we define an involution I k on R(z), by For k ∈ N we denote by E z k the forward shift operator acting on the variable z k , i.e. if f (z) ∈ R(z) then The inverse E −1 z k of E z k corresponds to the backward shift in the variable z k defined by . . ) : ν j = 0 for finitely many j} be the additive group consisting of semi-infinite vectors having finitely many nonzero integer entries. For ν ∈ Z ∞ we have a well-defined shift operator z3 · · · , since the right-hand side has only finitely many terms different from the identity operator. We denote by D z the associative algebra of difference operators of the form where S is a finite subset of Z ∞ and l ν (z) ∈ R(z). The involution I k can be extended to an involution on D z by defining I k (E z k ) = E −1 z k and I k (E zj ) = E zj for j = k. We say that an operator L ∈ D z is I-invariant, if it is invariant under the action of all involutions I k , k ∈ N.

Action of generators for
and β − j = |γ d+1−j | + j, for j = 0, 1, . . . , d. Then for every ν ∈ N d 0 with |ν| = n and j = 2, . . . , d we have and The operator B j−1 (ν + ; β + ) in (5.10) is a difference operator in the variables ν 1 , . . . , ν d obtained from the operator in (5.7) by changing the variables. Explicitly, we replace z l by |ν l | = ν 1 + · · · + ν l in the coefficients, and we replace E z l by E ν l E −1 ν l+1 . The operator B d+1−j (ν − ; β − ) in equation (5.11) is defined in a similar manner. Equations (5.10) and (5.11) follow from Proposition 5.1 in [8]. Note that these formulas extend the operators constructed in the previous sections. For instance, when d = 3, equation (5.10) with j = 2 corresponds to (4.11), while equation (5.11) with j = 2 and j = 3 corresponds to equations (4.13) and (4.9), respectively. It turns out, however, that there is an explicit fourth-order algebraic relation which allows to express all L i,j 's in terms of the operators M k , M ± k , k = 1, . . . , d. First, note that we can obtain the operators L 1,j and L i,d+1 via the formulas

Action of all elements in
for j = 2, . . . , d + 1, (5.14) with the convention that M k = M ± k = 0 for k = d + 1, d + 2. Next, one can check that if i, j, k, l are distinct, then (5.17) In the above formula, {A, B} = AB + BA denotes the anticommutator of the operators A and B. Equation (5.16) provides and interesting link to the 5 ⇒ 6 Theorem in the general structure theory for 3D second-order superintegrable systems with nondegenerate potentials [10]. According to this theorem, 5 algebraically independent second-order symmetries guarantee the existence of an additional second-order symmetry, such that the 6 symmetries are linearly independent and generate a quadratic algebra. If we think of the 6 operators L i,j , L i,k , L i,l , L j,k , L j,l , L k,l as symmetries for the generic superintegrable system on the 3-sphere, then (5.16) gives an explicit formula for each one of these symmetries in terms of the other five.
Note now that equations (5.5), (5.10), (5.11), (5.14), (5.15) and (5.16) allow us to compute the action of all operators L i,j on the basis P ν (x; γ) of V d n (γ). First, we compute the action of the operators L 1,j and L i,d+1 using (5.5), (5.10), (5.11), (5.14) and (5.15). Then, for all 1 < i < j < d + 1, we can compute the action of L i,j using (5.16) with k = 1 and l = d + 1, via the formula Since we assume that γ k = −1 for all k, the only subtle point when we use the above formula arises when γ 1 = 1 or γ d+1 = 1. But there is a simple argument which shows that, after we substitute the Racah operators in F (L 1,i , L i,d+1 , L 1,j , L j,d+1 , L 1,d+1 ), each coefficient of the resulting difference operator must be divisible by (1 − γ 1 )(1 − γ d+1 ). Thus, we can cancel these terms and we obtain a well-defined explicit formula even when γ 1 = 1 or γ d+1 = 1. Indeed, if we assume that at least one of the recurrence coefficients in the difference operator F (L 1,i , L i,d+1 , L 1,j , L j,d+1 , L 1,d+1 ) is not zero when, say, γ 1 = 1, then the equation will imply that the polynomials P ν (x; γ) satisfy a nontrivial recurrence relation in the indices ν, with coefficients depending only on ν. This means that the polynomials P ν (x; γ) are linearly dependent, which is impossible since they form a basis for the space of polynomials in the variables x 1 , . . . , x d , as long as the parameters γ = (γ 1 , . . . , γ d+1 ) satisfy the conditions in equation (5.3).

Irreducibility.
Let denote the eigenvalue of the operator M j in equation (5.5). Using that fact that the parameters satisfy the conditions in equation (5.3), we see that for ν, µ ∈ N d Therefore, the spectral equations (5.3) fix the polynomials P ν (x; γ), up to unessential factors depending only on ν.
We prove below that the A d (γ)-module V d n (γ) is irreducible, by induction on d, extending the constructions in Section 4.2. Since we have already proved the statement in dimensions 2 and 3, we assume below that d ≥ 4 and that the statement is true in dimension d − 1.
First, we identify two subspaces of V d n (γ), which are irreducible modules over subalgebras of A d (γ) isomorphic to A 2 and A d−1 , respectively.