Jack-Laurent symmetric functions

We develop the general theory of Jack-Laurent symmetric functions, which are certain generalisations of the Jack symmetric functions, depending on an additional parameter p_0.


Introduction
In the late 1960s Henry Jack [6,7] introduced certain symmetric polynomials Z(λ, α) depending on a partition λ and an additional parameter α, which are now known as Jack polynomials. When α = 1, they reduce to the classical Schur polynomials, so the Jack polynomials can be considered as a one-parameter generalization of Schur polynomials, whose theory goes back to Jacobi and Frobenius. When α = 2, they are naturally related to zonal spherical functions on the symmetric spaces U (n)/O(n), which was the main initial motivation for Jack. The theory of Jack polynomials was further developed by Stanley [25] and by Macdonald, who also extended them to the symmetric polynomials depending on two parameters, nowadays named after him [8]. Approximately at the same time, Calogero [2] and Sutherland [26] initiated the theory of quantum integrable models, describing the interaction particles on the line, which in the classical case were studied by Moser [12]. Although it was not recognized at the time, Jack polynomials can be defined as symmetric polynomial eigenfunctions of (properly gauged) version L k,N of the Calogero-Moser-Sutherland (CMS) operator which in the exponential coordinates x i = e 2zi has the form where the parameter k is related to Jack's α by k = −1/α. A remarkable property of Jack polynomials is stability, which corresponds to the fact that the dependence of L k,N on the dimension N can be eliminated by adding a multiple of the momentum (which is an integral of the system): the operators are stable in the sense that they commute with the natural homomorphisms φ M,N : Λ M → Λ N , sending x i with i > N to zero, where Λ N = C[x 1 , . . . , x N ] SN is the algebra of symmetric polynomials. This allows one to define the Jack symmetric functions P (k) λ as elements of Λ defined as the inverse limit of Λ N in the category of graded algebras (see [8]). The corresponding infinitedimensional version of the CMS operator has the following explicit form in power sums p a = x a 1 + x a 2 + · · · , a ∈ N (see [1,25]): where ∂ a = a(∂/∂p a ). Some new explicit formulas for the higher-order CMS integrals at infinity were recently found by Nazarov and Sklyanin [13,14].
In the present paper, we define and study a Laurent version of Jack symmetric functions, Jack-Laurent symmetric functions and the corresponding infinite-dimensional Laurent analogue of the CMS operator acting on the algebra Λ ± freely generated by p a with a ∈ Z \ {0} being both positive and negative. The variable p 0 plays a special role and will be considered as an additional parameter.
The idea to consider the Laurent polynomial eigenfunctions of CMS operator (1) is quite natural and was proposed already by Sutherland [27]. The corresponding Laurent polynomials were later discussed in more detail by Sogo [22][23][24]. However, as pointed out by Forrester in his MathSciNet review of the paper [22], in finite dimension it does not have much sense since the corresponding Laurent polynomials always can be reduced to the usual Jack polynomials simply by multiplication by a suitable power of the determinant Δ = x 1 · · · x N .
In the infinite-dimensional case one cannot do this since the infinite product x 1 x 2 · · · does not belong to Λ. Moreover, in the Laurent case there is no stability (at least in the same sense as above, since one cannot set x i to zero), so the corresponding Jack-Laurent symmetric functions essentially depend on both k and additional parameter p 0 , which can be viewed as 'dimension'. Such a parameter appeared already in Jack's paper [7] as S 0 (see page 9 there) and Sogo's papers, but its importance probably first became clear after the work of Rains [17], who considered BC-case (see also [18,19]).
Our main motivation for studying the Jack-Laurent symmetric functions came from the representation theory of Lie superalgebra gl(m, n) and related spherical functions, where these functions play an important role. We will discuss this in a separate publication.
The structure of the paper is as follows. In Section 2, we introduce the infinite-dimensional Laurent version of the CMS operator depending on an additional parameter p 0 , as well as its quantum integrals, acting on Λ ± . Our approach is based on an infinite-dimensional version of the Dunkl operator [20] and is different from that of [13,14] (see although the discussion of possible relations in [20]). In Section 3, we consider the Jack-Laurent polynomials P (k) SN parametrized by non-increasing sequences of integers χ = (χ 1 , . . . , χ N ). We study their properties, which essentially follow from the usual case.
In Section 4, we define our main object, Jack-Laurent symmetric functions P (k,p0) α ∈ Λ ± , rationally depending on the parameters k and p 0 and labelled by bipartitions α = (λ, μ), which are pairs of the usual partitions λ and μ. The defining property is that their images under natural homomorphisms ϕ N : Λ ± → Λ ± N give the corresponding Jack-Laurent polynomials. An alternative construction of Jack-Laurent symmetric functions, using the monomial symmetric functions, was proposed in [19]. We prove the existence of P (k,p0) α for all k / ∈ Q and kp 0 = n + km, m, n ∈ Z >0 . The usual Jack symmetric functions are particular cases corresponding to empty second partition μ: P The simplest Laurent example corresponding to two one-box Young diagrams is given by In Sections 5-8, we study the Laurent analogues of Harish-Chandra homomorphism, Pieri and evaluation formulas and compute the square norms of P (k,p0) α for the corresponding symmetric bilinear form on Λ ± . Section 9 is devoted to an important special case k = −1, corresponding to Schur-Laurent symmetric functions. We show that the limit S λ,μ of Jack-Laurent symmetric functions P (k,p0) λ,μ when k → −1 for generic p 0 does exist, does not depend on p 0 and can be given by an analogue of the Jacobi-Trudi formula. The related symmetric Laurent polynomials (called sometimes symmetric Schur polynomials indexed by a composite partition sμ ,λ (x)) and their supersymmetric versions play an important role in representation theory of Lie superalgebra gl(m, n) (see [3,4,9]).
In the last section, we discuss some conjectures and open problems.

Laurent version of CMS operators in infinite dimension
The finite-dimensional CMS operators (1) preserve the algebra of symmetric Laurent polynomials Let us define its infinite-dimensional version, the algebra of Laurent symmetric functions Λ ± as the commutative algebra with the free generators p i , i ∈ Z \ {0}. The dimension p 0 = 1 + 1 + · · · + 1 = N does not make sense in the infinite-dimensional case, so we will add it as an additional formal parameter, which will play a very essential role in what will follow.
The algebra Λ ± has a natural Z-grading, where the degree of p i is i. There is a natural involution * : Λ ± → Λ ± defined by This algebra can be also represented as Λ ± = Λ + ⊗ Λ − , where Λ + is generated by p i with positive i and Λ − by p i with negative i. Note that the involution * swaps Λ + and Λ − . For every natural N there is a homomorphism ϕ N : Λ ± → Λ ± N : The involution * under this homomorphism goes to the natural involution on Λ ± N mapping x i to x −1 i . Define also the following algebra homomorphism θ : then this map becomes an involution. Now we are going to construct explicitly the infinite-dimensional version of the CMS operator and higher integrals. Our main tool is an infinite-dimensional version of the Dunkl-Heckman operator [5].
Let us remind that the Dunkl-Heckman operator for the root system of the type A n has the form where s ij is a transposition, acting on the functions by permuting the coordinates x i and x j . Heckman proved [5] that the differential operators where Res means the operation of restriction on the space of symmetric polynomials, commute and give the integrals for the quantum CMS system with the Hamiltonian H N = L (2) k,N : We have the following simple, but important lemma.
acts trivially on the algebra Λ ± N and has the property Therefore, it is enough to prove that Δ i, i ] for l > 0, which follows from the identity Let Λ ± [x, x −1 ] be the algebra of Laurent polynomials in x with coefficients from Λ ± . Define the differentiation ∂ in Λ ± [x, x −1 ] by the formulae Define the infinite-dimensional analogue of the Dunkl-Heckman operator D k,p0 : 1 + · · · + x l N , l ∈ Z and set p 0 = N . We claim that the following diagram: where D i,N are Dunkl-Heckman operators (7), is commutative. This follows from the relations Introduce now a linear operator E p0 : and the operators L where the action of the right-hand side is restricted to Λ ± . We claim that these operators give a Laurent version of quantum CMS integrals at infinity. More precisely, we have the following result. k,p0 is a differential operator of order r with polynomial dependence on p 0 and the following properties: where θ is defined by (6). The operator L (2) k,p0 is the Laurent version of the CMS operator at infinity given by formula (3).
The operators L Proof. Consider f ∈ Λ ± . Since E p0 and Δ p0 commute with multiplication by f , we have k,p0 is a differential operator of order r. The formulae (17), (18) follow from the symmetries The explicit form (3) easily follows from a direct calculation.
To prove the commutativity of the integrals note that from (14) it follows that the diagram is commutative, where L (r) k,N are the CMS integrals given by Heckman's construction (8) and the homomorphism ϕ N : Indeed, for any f ∈ Λ ± we have D r k,p0 (f ) = l x l g l , g l ∈ Λ ± , where the sum is finite. We have which proves the commutativity of the diagram. This implies that since the integrals (8) commute [5]. Now the commutativity of the operators L (r) k,p0 follows from the following useful lemma. Proof of lemma. By definition g is a polynomial in a finite number of generators p r , 0 < |r| M for some M with coefficients polynomially depending on p 0 . Take N larger than 2M .
Since the corresponding ϕ N (p r ) with 0 < |r| M are algebraically independent and ϕ N (g) = 0, all the coefficients of g are zero at p 0 = N . Since this is true for all N > 2M, the coefficients must be identically zero, and therefore g = 0.

Jack-Laurent symmetric polynomials
As we have already mentioned above, the Laurent polynomial eigenfunctions for CMS operators were considered already by Sutherland [27] and later in more detail by Sogo [22][23][24], who parametrized these eigenfunctions by the so-called extended Young diagrams, when the negative entries are also allowed. Alternatively, one can use two Young diagrams, corresponding to positive and negative parts. However, in finite dimension 1 can always reduce them to the usual Jack polynomials simply by multiplication by a suitable power of the determinant Δ = x 1 · · · x N (see, for example, Forrester's comment in his MathSciNet review of the paper [22]). Let Define the corresponding Jack-Laurent symmetric polynomial P (k) where P are the usual Jack polynomials [8]. It is well-defined because of the well-known property of Jack polynomials for all b 0 (see, for example, [25]). There exists a natural involution * on the algebra Λ ± The following lemma shows how this involution acts on the Jack-Laurent symmetric polynomials.

Lemma 3.1. For any non-increasing sequence of integers
where w is the following involution: Proof. It is enough to consider the case when χ = λ is a partition with l(λ) N. In that case we have to show that where ν = (a − λ N , . . . , a − λ 1 ) and a λ 1 . Recall that the Jack polynomial P (k) λ (x 1 , . . . , x N ) can be uniquely characterized by the following properties: it is an eigenfunction of the CMS operator L k,N given by (1) and has an expansion where m μ are the standard monomial polynomials [8] and μ λ means dominance order: so we only need to show that a − μ < a − λ. But the inequalities and thus This proves the lemma.
Now we are going to present the Laurent version of the Harish-Chandra homomorphism. Let D N (k) be the algebra of quantum integrals of the CMS generated by the integrals L , where ν is a partition and P (k) ν is the usual Jack polynomial. The * -involution on Λ ± N gives rise to the involution on the algebra D N (k), which we will denote by the same symbol:

Theorem 3.2. For any integral L ∈ D N (k) and any non-increasing sequence of integers
and Proof. Let us prove first (25). It is enough to prove this for the integrals L for all a. For positive a from formula (22) it follows that Since both sides are polynomial, this is true for negative a as well, which implies the claim.
The second statement follows from the relations Thus we see that the involution * on the integrals goes under the Harish-Chandra homomorphism to the involution w : This involution can be described also in the following way.
Proof. We have Let us present now a Laurent version of Pieri formula. Define the following functions for positive integers r, i and any b: and Let ε i be the sequence of length N with all zeroes except 1 at the ith place.
Theorem 3.4. The Jack-Laurent polynomials satisfy the following Pieri formula: where the sum is taken over 1 i N such that χ + ε i is a non-increasing sequence of integers.
Proof. If χ is a partition, then the result is well-known [8]. In the general case choose integer a such that ν = χ + a is a partition, multiply both sides of the Pieri formula for ν by (x 1 · · · x N ) −a and take into account that V i (χ) = V i (χ + a).
We will need also the following corollary of the Pieri formula. Let χ be a non-increasing sequence of N integers and set which is the eigenvalue of the CMS operator. Define the following polynomial in variable t depending on a complex number s where the product is taken over all j such that χ + ε j is a non-increasing sequence of integers and e N (χ + ε j ) = s.
Proof. Since k is not positive rational or zero, then, for i = j, In other words, all these quantities are pairwise distinct. One can check also that, for these k, the quantities V i (χ) are well-defined and non-zero. Now the result directly follows from the Pieri formula.
, then, for any bipartition α, there exists a unique element P (k,p0) α ∈ Λ ± (called Jack-Laurent symmetric function) such that, for every N ∈ N, Proof. Let us prove the existence first. We will prove it by induction in |λ|. If |λ| = 0, then we set is the usual Jack symmetric function [8]. We see that P (k) μ does not depend on p 0 and, for l(μ) N, μ )) * = 0, so we proved the theorem when |λ| = 0.
Let α be a bipartition. Denote by X(α) and Y (α) the sets of bipartitions, which can be obtained from α by adding one box to λ and by deleting one box from μ, respectively, and define Similarly to the previous section, define for any bipartition α and consider the following polynomial in t, depending rationally on p 0 and on an additional parameter s, where the product is over all bipartitions γ ∈ Z(α) such that e p0 (γ) = s. Now suppose that the theorem is true for all α = (λ, μ) with |λ| M . Let β = (ν, τ ) be a bipartition such that |ν| = M + 1. Let α be a bipartition obtained from β by removing one box (30) with i corresponding to the removed box. One can check that if k is not rational and p 0 = n + k −1 m with natural m, n, the coefficient V (α, β) is well-defined and non-zero. Therefore, we can define From the previous formula, we see that P β is well-defined if p 0 = n + k −1 m, m, n ∈ Z >0 and ). Now we are going to compare two polynomials R α (N, t, s) and R χN (α) (t, s). If l(α) < N, then l(β) N and By induction assumption and Proposition 3.5, we have x 1 , . . . , x N ). If l(α) = l(β) = N, then there exists γ ∈ X(α) with the added box (l(λ) + 1, 1) and it is easy to check that which, by Proposition 3.5, imply that ϕ N (P . . . , x N ).
Suppose now that l(β) > N and consider two cases: l(α) > N and l(α) = N . In the first case, by induction ϕ N (P α ) = 0, therefore ϕ N (P (k,p0) β ) = 0. In the second case, we have again equality R α (N, t, s) = R χN (α) (t, s), but this time s = e N (β) = e N (χ(α) + ε j ), 1 j N and, according to Proposition 3.5, This proves the existence. The uniqueness follows from the same arguments as in the proof of Lemma 2.3.
We will show in the next section that the Jack-Laurent symmetric functions P  Here is the explicit form of the Jack-Laurent symmetric functions in the simplest cases: Proof. Choose N l(α); then we have w(α) ), which implies the claim.

Harish-Chandra homomorphism and Polychronakos operator
In this section, it will be convenient for us to think of p 0 as a variable, while k should be still considered as a fixed parameter. The difference between variable and parameter is only in the point of view, which we will continue to change, hopefully without much problems for the reader.
Recall that the usual Harish-Chandra homomorphism maps the algebra generated by the quantum CMS integrals (8) onto the algebra of the shifted symmetric polynomials Λ N (k). The algebra D N (k) acts on the algebra of symmetric polynomials Λ N and there is a natural homomorphism induced by the homomorphism Λ N → Λ N −1 sending x N to zero. Consider the inverse limit which we will call the algebra of stable CMS integrals, and the inverse limit of Harish-Chandra where Λ(k) = lim ← Λ N (k) is the algebra of the shifted symmetric functions [15].
The algebra D(k) can be naturally considered as a subalgebra of the algebra of the differential operators acting on Λ, which depends on the parameter k. It has a natural extension D(k)[p 0 ] = D(k) ⊗ C[p 0 ], which also acts on Λ if we specialize p 0 .
Let D L (k, p 0 ) be the algebra of differential operators on Λ generated over C[p 0 ] by the CMS integrals (16).
We claim that any operator from this algebra can be represented as a polynomial in p 0 with coefficients, which are stable CMS integrals: To construct the stable CMS integrals, we will use the following version of the Dunkl operator, which was introduced by Polychronakos [16]: Note the difference with Dunkl-Heckman operator: which implies in particular thatΔ i,N (and hence π i,N ) do not have symmetry (11) with respect to * -involution. This makes the extension of the operators from Λ to Λ ± a bit more tricky, so we will consider the Laurent version of the operators. The reduction to the usual polynomial case is obvious.
The action of the operator π i,N on Λ ± N [x ± i ] is described by Polychronakos used these operators to give an alternative way to construct the CMS integrals.
Theorem 5.1 [16]. The operators I are the commuting quantum integrals of the CMS system (9).
The proof is simple and based on the commutation relations Now we would like to express Heckman's CMS integrals L where the operator coefficients f (r) a are defined by the following recurrent relations: with initial data f Proof. We claim that where Res, as before, is the restriction on symmetric polynomials Λ ± N . Indeed, introduce S i = j =i (1 − s ij ) and assume (46). Then

Let us define now the infinite-dimensional analogue of the Polychronakos operator on Λ ± [x] by the formulas
be the operator defined by formula (15) above and consider the following set of infinite-dimensional CMS integrals where again the action of the right-hand side is restricted to Λ ± . Their commutativity follows from the same arguments as in Theorem 2.2. Let D I (k, p 0 ) be the algebra of differential operators on Λ generated over C[p 0 ] by the CMS integrals I (j) k,p0 , j ∈ N. k (see formula (53)), which do not depend on p 0 and generate the algebra D(k). It turns out that all these three sets of CMS integrals generate the same algebra if we allow the coefficients to depend polynomially on p 0 .

Proof. It is enough to show that the integrals L
Theorem 5.4. The following two algebras coincide: Proof. To define the stable quantum CMS integrals, we note first that on the algebra Λ[x] we have Indeed, the operator ∂ i maps algebra Λ N [x i ] into the ideal J generated by x i , while the operator π i,N + kN maps the ideal J into itself (see formula (40)). Therefore, the operator (π i,N + kN ) r−1 ∂ i maps the algebra Λ N [x i ] into the ideal J and can be checked to be stable. (52), it follows that these operators are stable and their inverse limit can be naturally identified with

Consider the operators
do not depend on p 0 and belong to D(k). In particular, for r = 2 we have H k,p0 , which is the stable version of the CMS operator (3) on Λ (see [1,25]): From (53), one can also express I More precisely, the following diagram is commutative: Proof. Let ϕ N be defined by (20) and use the same notation for its action on D(k)[p 0 ]. Define also φ N : Λ(k)[p 0 ] → Λ N (k) by setting p 0 = N and φ N (p r,0 ) = p r,0,N .
We have the following commutative diagram: Therefore, we only need to prove that φ N (w(p)) = w N (φ N (p)) for any p ∈ Λ(k)[p 0 ]. It is enough to show this for shifted power sums, when this reduces to the identity p r,a,N (w(χ)) = (−1) r p r,k−kN −a,N (χ), which is easy to check. Thus the theorem is proved.
Since N l(λ) + l(μ), we have It is easy to see that since, for any 1 i N , λ i w N (μ) i = 0, we have p r,N (χ N (λ)) + p r,N (w N (χ N (μ))) = p r,N (χ N (α)) = φ N (p r )(χ N (α)), which completes the proof. Now we can use this to prove the following important result. Let us consider again p 0 as a parameter.
Theorem 5.7. If k / ∈ Q and kp 0 = m + nk for all m, n ∈ Z >0 , then the spectrum of the algebra D(k)[p 0 ] of quantum CMS integrals on Λ ± is simple.
Proof. Consider the following shifted symmetric functions: where B l (z) are the classical Bernoulli polynomials [28] and a is a parameter. They generate the algebra of shifted symmetric functions Λ(k).
Proof. Indeed, because of the symmetry property (17) l (λ , k −1 , k −1 a) and the simplicity of the spectrum for generic k and p 0 . An explicit form of the constants d α (k, p 0 ) is given below by (73).

Pieri formula for Jack-Laurent symmetric functions
Let α be a bipartition represented by a pair of Young diagrams λ and μ. Define, for any positive integers i, j, the following functions: where λ , as before, is the Young diagram conjugated (transposed) to λ.
Let x = (ij) be a box such that the union λ + x := λ ∪ x ∈ P is also a Young diagram, and, similarly to the Pieri formula for Jack polynomials [8], define If x cannot be added to λ, we assume that the corresponding V (x, α) = 0. Similarly, if the box y = (ij) can be removed from the Young diagram μ in the sense that μ − y := μ \ y ∈ P, then we define where l(λ) is the length, which is the number of non-zero parts in partition λ. If y = (ij) cannot be removed from μ, then we define U (y, α) = 0.
The following theorem follows from the Pieri formula for Jack-Laurent polynomials (31).
Define the following analogues of rows: and columns with all other y i , y j being zero. For every box with integer coordinates (j, i) define the function Define, for the added box = (j, i), the following subset in Y λ : and, for deleted box = (ji), the subsets in Y : The meaning of these subsets is clear from Figure 2, where the deleted box is black and the added box is cross-hatched.
In these terms the Pieri formula (63) can be written as with (j + 1 + k(y j − l(λ) + p 0 + 1))(j + k(y j + l(μ)) (j + k(y j − l(λ) + p 0 ))(j + 1 + k(y j + l(μ) + 1) with the convention that the product over empty set is equal to 1. A non-symmetry between λ and μ is due to the choice of p 1 in the left-hand side of the Pieri formula (63). By applying * -involution to formula (63), one has the corresponding formula for p −1 , where the roles of λ and μ are interchanged.
Another remark is that in the Pieri formula (64) one can replace the rectangle containing the figure Y by any bigger rectangle with −M i L by changing in formula (66) the lengths l(λ) and l(μ) to L and M, respectively.

Evaluation theorem
Consider a pair of Young diagrams λ and μ which can be jointed together to form an a × b rectangle (see Figure 3): We will call such two diagrams complementary. Define the following function on Young diagrams depending on two parameters p and x: with the assumption that, for empty Young diagram, ϕ p (∅, x) = 1. Such a function was first introduced by Stanley [25] in the theory of Jack polynomials.
Proof. The proof is by induction in a + b. If a + b = 2, then we have λ = (1), μ = ∅ or the other way around. Therefore, Now let a + b > 2. There are two cases: first when λ 1 = a or μ 1 = a and the second when Let us consider the first case. By symmetry we can assume that λ 1 = a and set ν = λ \ λ 1 . Then we have where in the last row we have made the change j → a − j + 1 and use the equality λ j + μ a−j+1 = b in the denominator. Thus we see that Consider now the second case. Set ν = λ \ λ 1 . As before, we can assume that Lemma 7.2. Let λ, μ be two partitions, a μ 1 and N l(λ) + l(μ) and ν = (λ 1 + a, . . . , λ r + a, a, . . . , a, a − μ s , . . . , a − μ 1 ) ∈ P. Then where ϕ p (λ, μ, x) is given by the formula Proof. Let τ ⊂ λ be some subset of λ ∈ P. Define ψ p (λ, τ, x) similarly to ϕ p (λ, x) as Split the Young diagram ν into three parts as follows: Figure 4). We have (using the second formula (68) for ϕ p (λ, x)) It is easy to see that where we made the change j → a − j + 1. Now we only need to compute ψ N (ν 2 ∪ ν 3 , ν 2 , x). But we can get this product from the previous one by setting λ i = 0 to have Taking a = l(μ ), we have the claim. Now we are ready to prove the main result of this section.
Corollary 7.4. Jack-Laurent symmetric functions P (k,p0) α satisfy the θ-duality where as before θ is defined by θ(p a ) = kp a and Proof. We know from Corollary 5.9 that θ −1 (P for some constants d λ,μ . Applying to both sides the evaluation homomorphism ε kp0 and using and thus which implies (73).

Symmetric bilinear form
Let us fix the parameter k, which we assume in this section to be negative real.
We start with the finite-dimensional case. The original CMS operator is clearly formally self-adjoint with respect to the standard scalar product (ψ 1 , ψ 2 ) = ψ 1 (z)ψ 2 (z) dz with the standard Lebesgue measure dz = dz 1 · · · dz N on R N . After the gauge ψ = f Ψ 0 and change x j = e 2zj , we naturally come to the following symmetric bilinear form for the Laurent polynomials f, g ∈ Λ ± N : where T N is the complex torus with |x j | = 1, i = 1, . . . , N, dx is the Haar measure on (cf. Macdonald [8, p. 383], who uses parameter α = −1/k). The normalization constant c N (k) is chosen in such a way that (1, 1) N = 1: Note that for negative real k the integral (74) is clearly convergent for all Laurent polynomials f, g and that on the Laurent polynomials with real coefficients (in particular, for Jack-Laurent polynomials with real k) the product (74) coincides with the Hermitian scalar product Since the eigenfunctions of a self-adjoint operator are orthogonal the Jack polynomials P (k) λ (x 1 , . . . , x N ) are orthogonal with respect to the product (74). Using formulae (10.37), (10.22) from [8] we have which can be rewritten in our notations as We can extend now this formula to the Jack-Laurent polynomials P (k) χ for any integer nondecreasing sequence χ = (χ 1 , . . . , χ N ) by adding a large a to all its parts to make them positive. Note that both ϕ N (λ, 0) and ϕ N (λ, 1 + k) do not change under this operation and that the integral for all sufficiently large N, where the product in the left-hand side is defined by (74). The corresponding square norm of the Jack-Laurent symmetric function P (k,p0) α with bipartition α = (λ, μ) is equal to Proof. The uniqueness is obvious since the rational function is determined by its values at sufficiently large integers.
To prove the existence, we simply check that the formula (78) defines the symmetric bilinear form satisfying (77). We have according to (34) that which by formula (69) from Lemma 7.2 coincides with the right-hand side of (78) for p 0 = N.
Note that, in contrast to the usual Jack case [8], the bilinear form ( , ) p0 is not positive definite on real Laurent symmetric functions, as it follows from (78). In order to have positive definite form, one should send p 0 to infinity; see the last section.

Special case k = −1 : Schur-Laurent symmetric functions
The case k = −1 is very important for representation theory of Lie superalgebra gl(n, m) (see [3,11]). In this case, the corresponding Jack-Laurent symmetric functions (whose existence is not obvious) do not depend on p 0 , as one can see already in the simplest case Proposition 9.1. The limit S λ,μ := lim does exist for generic p 0 and does not depend on p 0 .
We call S λ,μ the Schur-Laurent symmetric functions. The image of these functions under the homomorphism ϕ N coincide with the symmetric Schur polynomials sμ ,λ (x) indexed by a composite partitionμ; λ (see [9] for a brief history of these polynomials and their role in representation theory). Here are the two simplest examples of Schur-Laurent symmetric functions Proof. Use the induction on |λ|. When λ = ∅, then P is well-defined (recall that k = −1 corresponds to α = 1 in Jack's notations) and coincide with Schur symmetric function S μ (see, for example, [8]).
To prove the induction step, one can use the Pieri formula (63). The left-hand side is welldefined at k = −1 by induction assumption. Restrict the CMS operator L k,p0 onto the invariant subspace generated by the linear combinations of the Jack-Laurent symmetric functions in the right-hand side of Pieri formula for generic values of the parameters. One can check analysing the proof of Theorem 3.1 that, for k = −1 and generic p 0 , the corresponding eigenvalues E 1 , . . . , E k are distinct. This means that the component V (x, α)P λ+x,μ = Q(L k,p0 )(p 1 P λ,μ ) with polynomial where E 1 is the eigenvalue corresponding to P λ+x,μ . Since L k,p0 is polynomial in parameters and E 1 = E j the product V (x, α)P λ+x,μ is well-defined for k = −1 and generic p 0 . Since the coefficients V (x, α) tend to 1 when k → −1, this means that P λ+x,μ is well-defined as well. This proves the existence of Schur-Laurent symmetric functions for generic p 0 ; their independence on p 0 follows from the Laurent version of the Jacobi-Trudi formula below.

Some conjectures and open questions
The usual Jack symmetric functions can be defined using the following scalar product in Λ defined in the standard basis p λ = p λ1 p λ2 · · · by < p λ , p μ >= (−k) −l(λ) where m j is the number of parts of λ equal to j (see [8, p. 305]). It is known (see, for example, [8, p. 383]) that this scalar product is the limit of the scalar product (78) restricted on Λ when p 0 → ∞. An interesting question is what happens on Λ ± .
We believe that the limit ( , ) ∞ of the indefinite bilinear form (78) does exist and is positive definite on real Laurent symmetric functions for real negative k. More precisely, we conjecture that the limits of the Jack-Laurent symmetric functions P (k,∞) α := lim p0→∞ P (k,p0) α exist for all k / ∈ Q. Then, by (78) they would provide an orthogonal basis in Λ ± with where Φ(λ, k) = (i,j)∈λ which can be checked to be positive for k < 0 and all λ. Note that in contrast to the Jack polynomial case, the Laurent polynomials p λ,μ = p λ p μ , as well as the products of Jack symmetric functions P (k) λ P (k) * μ , are not orthogonal with respect to ( , ) ∞ . What are the transition matrices between these bases and the Jack-Laurent basis P (k,∞) λ,μ ?
In the theory of Jack symmetric functions [8] it is known that the product A(λ)P (k) λ with A(λ) = (ij)∈λ (λ i − j + k(i − 1 − λ j )) depends on k polynomially. We conjecture that a similar fact is true for Jack-Laurent symmetric functions, namely that the product J (k,p0) λ,μ

:= A(λ, μ)A(λ)A(μ)P
is polynomial in k and p 0 , where A weaker version of this conjecture is that the product A(λ, μ)P (k,p0) λ,μ is polynomial in p 0 . The case of special parameters k and p 0 with p 0 = n + k −1 m is very important for the representation theory of Lie superalgebras and is discussed in our paper [21].