The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials

We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.


Introduction
The quantum Knizhnik-Zamolodchikov (qKZ) equation, derived by Frenkel and Reshetikhin [9], is a system of difference equations satisfied by matrix elements of vertex operators in the representation theory of the quantum affine algebras. In this paper we consider the rational version of the qKZ equation. Let V = ⊕ N ǫ=1 Cv ǫ be the vector representation of the Lie algebra gl N . The rational qKZ equation is the following system of difference equations for an unknown function G(z 1 , . . . , z n ) taking values in V ⊗n : G(z 1 , . . . , z m − κ, . . . , z n ) = R m,m−1 (z m − z m−1 − κ) · · · R m,1 (z m − z 1 − κ) In [12] special solutions to the qKZ equation are constructed. Let us recall the construction briefly. Consider the coefficients in an expansion of the unknown function with respect to the standard basis of V ⊗n . In [12] a sufficient condition is given so that they form a solution to the qKZ equation. It is described as an eigenvalue problem for the q-Dunkl-Cherednik operators and the Demazure-Lusztig operators. The problem can be solved by using the non-symmetric Macdonald polynomials [1,13], which are by definition eigenfunctions of the q-Dunkl-Cherednik operators. Once one obtains a solution to the eigenvalue problem, the coefficients are generated from it through the action of the Demazure-Lusztig operators. Thus we obtain special solutions to the qKZ equation from the non-symmetric Macdonald polynomials. In this construction the polynomial representation of the affine Hecke algebra is a main tool.
In this article we apply the method of [12] to the rational qKZ equation. The main tool is the degenerate double affine Hecke algebra H κ . It is a trigonometric degeneration of the double affine Hecke algebra introduced by Cherednik [2]. Usually we consider the polynomial representation of H κ given by the differential operators called the Dunkl-Cherednik operators. They form a commuting family of differential operators and the polynomial eigenfunctions are the non-symmetric Jack polynomials [14]. In our construction of solutions to the rational qKZ equation, we consider another polynomial representation given in terms of commuting difference operators Y i (1 ≤ i ≤ n), which are difference analogues of the Dunkl operators where σ ij is the transposition of variables x i ↔ x j . The parameter κ will play the role of the difference step in the rational qKZ equation. The operator Y i has an expansion where Y i decreases the total degree by at least two. We can construct another family of commuting difference operators in the same way as the construction of the Dunkl-Cherednik operators from the Dunkl operators. Then the corresponding polynomial eigenfunctions are the shifted non-symmetric Jack polynomials, which are originally obtained by Knop [11] as a classical limit of the non-symmetric quantum Capelli polynomials. In our construction of special solutions to the rational qKZ equation we need a polynomial eigenfunction of the operators Y i (1 ≤ i ≤ n). Then, from the expansion Y i = 1 − D i + Y i , the eigenvalue should be one, and the leading degree part of a polynomial eigenfunction should belong to ∩ n i=1 ker D i (note that any eigenvalue of the Dunkl operator is equal to zero). A nonzero polynomial in the joint kernel is called a singular polynomial, which is deeply studied by Dunkl [3,4]. It is proved that constants are only singular polynomials for generic κ, and specific values of κ, called singular values, such that there exists a non-constant singular polynomial are completely determined by Dunkl, de Jeu and Opdam in more general setting associated with finite reflection groups [5]. When κ is equal to a singular value, the space of singular polynomials becomes a non-trivial CS n -module. In [3,4] Dunkl determined its irreducible components and proved that all the multiplicities are equal to one. Each irreducible component has a basis consisting of non-symmetric Jack polynomials which are well-defined at the singular value and singular. In this paper we consider the shifted case. We call a polynomial eigenfunction of Y i (1 ≤ i ≤ n) a shifted singular polynomial. In Theorem 4 below we prove that the space of shifted singular polynomials is isomorphic to that of singular polynomials as CS n -modules, and it has a basis consisting of shifted non-symmetric Jack polynomials. This fact follows from the inversion formula [11], which relates the non-symmetric Jack polynomials and the shifted ones. We can get a solution to the rational qKZ equation from a shifted singular polynomial which satisfies some symmetry conditions (see (10) below). Now we mention the result [6] by Di Francesco and Zinn-Justin. They constructed a class of polynomial solutions to the qKZ equation and observed that their scaling limit give "extended Joseph polynomials". As mentioned in [6], the scaling limit of their solutions solves the rational qKZ equation of level one. It had been used in the construction of a new integral formula for solutions to the rational qKZ equation of level zero in [16]. We prove that our solutions contain the special solutions above of level one. In our framework, the extended Joseph polynomials appearing in the solutions to the rational qKZ equation are shifted non-symmetric Jack polynomials which are singular. The leading degree part of the extended Joseph polynomials is nothing but the Joseph polynomial [10] which is a singular non-symmetric Jack polynomial. Thus the inversion formula for (shifted) non-symmetric Jack polynomials at a singular value may be regarded as a map which bridges some data of cohomology of orbital varieties (Joseph polynomials) and their equivariant counterpart (extended Joseph polynomials).
The rest of the paper is organized as follows. In Section 2 we summarize the definitions and properties of (shifted) non-symmetric Jack polynomials from a viewpoint of the representation theory of the degenerate double affine Hecke algebra. In Section 3 we give the rational qKZ equation and a corresponding eigenvalue problem. In Section 4 we construct polynomial solutions to the eigenvalue problem, and hence we obtain special solutions to the rational qKZ equation.
The key is the isomorphism as CS n -modules between the space of singular polynomials and that of shifted singular polynomials. At the very last we prove an explicit formula for some singular (shifted) non-symmetric Jack polynomials.
2 Shifted non-symmetric Jack polynomials We introduce some notation. We call an element of Z n ≥0 a composition. In the following we fix n, and denote by Λ the set of compositions. A composition λ = (λ 1 , . . . , λ n ) is called dominant if λ 1 ≥ · · · ≥ λ n . The symmetric group S n acts on Λ by σλ := (λ σ −1 (1) , . . . , λ σ −1 (n) ). For λ ∈ Λ, denote by λ + the unique dominant element in the orbit S n λ, and by w + λ the shortest element of S n such that w + λ λ + =λ. We define the dominance order ≤ on Λ by

Non-symmetric Jack polynomials
First let us recall the definition of the degenerate double affine Hecke algebra (or the trigonometric Cherednik algebra) following [15] 1 . Let κ be an indeterminate and denote the coefficient field by F = C(κ).
Definition 1. The degenerate double affine Hecke algebra H κ is the unital associative F-algebra generated by satisfying the following relations: Here s ij = (s i · · · s j−1 )(s j−2 · · · s i ) for i < j.
Note that H κ contains the group algebra of the symmetric group s 1 , . . . , s n−1 ≃ FS n . Denote by H + κ the subalgebra generated by x i , u i (1 ≤ i ≤ n) and s i (1 ≤ i < n). The algebra H + κ and its polynomial representation are crucial tools in the theory of Jack polynomials. Let F[x] = F[x 1 , . . . , x n ] be the polynomial ring. Denote by σ i,j the transposition of variables where D i is the Dunkl operator The The non-symmetric Jack polynomials ξ λ (λ ∈ Λ) are uniquely defined by the following conditions: Here where ρ := (n − 1, . . . , 1, 0).

Shifted non-symmetric Jack polynomials
The shifted non-symmetric Jack polynomials are defined by the vanishing conditions. Hereafter we use the variables z = (z 1 , . . . , z n ) for the shifted non-symmetric Jack polynomials, and denote by F[z] = F[z 1 , . . . , z n ] the ring of polynomials in z 1 , . . . , z n .
Remark 1. The shifted non-symmetric Jack polynomials defined in Theorem 1 are slightly different from the original ones given by Knop [11] as follows. In [11] the parameter is denoted by r. Let F := C(r) be the corresponding coefficient field. Define a C-algebra isomorphism Ψ : The shifted non-symmetric Jack polynomials are also characterized as joint eigenfunctions of some commuting difference operators. To construct the operators, we introduce another polynomial representation of H κ . Define the operators T i (1 ≤ i < n) and ω acting on F[z] by (ωf )(z 1 , . . . , z n ) := f (z n − κ, z 1 , . . . , z n−1 ).
Here σ i,i+1 is the transposition of variables z i ↔ z i+1 . Set Proof . It can be checked by direct calculation using Note that the generator x i acts as a difference operator and u i as the multiplication of the variable z i on F[z]. It is opposite to the action φ (1).
Remark 2. The usual polynomial representation φ is induced from the one-dimensional representation of the subalgebra generated by s i (1 ≤ i < n) and u i (1 ≤ i ≤ n). The representation ψ in Proposition 1 is constructed from an opposite side in the following sense. Consider the subalgebra generated by s i (1 ≤ i < n) and x i (1 ≤ i ≤ n). It is isomorphic to the group algebra of the extended affine Weyl group S n . Then the induced module of the trivial C S n -module is isomorphic to F[u 1 , . . . , u n ] as an F-vector space. Change the variables u i → z i and we obtain the polynomial representation ψ.
Then the following relations hold: Thus we get the first relation. The others can be proved by easy calculation.
Thus we obtain commuting difference operators ψ(v i ) (1 ≤ i ≤ n) acting on F[z]. The shifted non-symmetric Jack polynomials are joint eigenfunctions of them.
Theorem 2 ([11, Theorem 6.6]). The shifted non-symmetric Jack polynomials E λ (λ ∈ Λ) are uniquely characterized by the following properties: Here we briefly explain why the eigenvalues of ψ(v i ) (1 ≤ i ≤ n) are the same as those of the Dunkl-Cherednik operators φ(u i ) (1 ≤ i ≤ n). Since the operator T i − σ i decreases the total degree by one, we have an expansion Y i = 1− D i + Y i , where D i is the Dunkl operator (2), and Y i is an operator which decreases the total degree by at least two. Thus we see that ψ(v i ) − φ(u i ) is an operator which strictly decreases the total degree. Hence ψ(v i ) is triangular with respect to the order on monomials in F[z] induced from the dominance order ≤ and the total degree, and its eigenvalues are equal to those of φ(u i ).

Proposition 3. Any eigenvalue of the operators
Proof . From the argument above, the operator Y i − 1 = −D i + Y i decreases the total degree by at least one. This implies the proposition.

The inversion formula
The H κ -module F[z] is naturally an H + κ -module by restriction. However we consider another H + κ -module structure defined through the following homomorphism.
Then there exists an Hereafter is an isomorphism between H + κ -modules. The inverse Φ −1 is a map taking the leading degree part and changing the variables z i → x i .

Proof . The intertwining property follows from φ(s
. The operator ψ( u i ) − z i decreases the total degree of polynomials in z, and hence Φ(x λ ) = z λ + (lower degree terms) for any λ ∈ Λ. Therefore Φ is an isomorphism.
The equality Φ(ξ λ ) = E λ is called the inversion formula. The non-symmetric Jack polynomials ξ λ may have poles when κ is a non-positive rational number. From the inversion formula, we see that the shifted one E λ also has the same property. We say that ξ λ (or E λ ) is well-def ined at κ = κ 0 ∈ Q ≤0 if it does not have a pole at κ = κ 0 .
3 The rational qKZ equation

The rational qKZ equation
Let V = ⊕ N ǫ=1 Cv ǫ be the N -dimensional vector space. We regard V as the vector representation of the Lie algebra gl N . Denote by h i (1 ≤ i ≤ N ) a basis of the Cartan subalgebra, which acts The rational R-matrix R(z) ∈ End(V ⊗2 ) is defined by where P is the transposition: Fix a parameter κ ∈ C × . The rational quantum Knizhnik-Zamolodchikov (rational qKZ) equation is the following system of difference equations for an unknown function G(z 1 , . . . , z n ) taking values in V ⊗n :

Eigenvalue problem associated with the rational qKZ equation
Hereafter we assume that n ≥ N ≥ 2.
Fix positive integers d 1 , . . . , d N satisfying N j=1 d j = n, and define an n-tuple δ by δ = (1 d 1 , . . . , N d N ). Consider the following eigenvalue problem for a function F δ = F δ (z 1 , . . . , z n ): Here Y i and T i are given by (5) and (3), respectively. Note that the consistency of (9) and (10) implies that Once one solves the eigenvalue problem above, a solution to the rational qKZ equation can be constructed as follows. Suppose that F δ is a solution to the eigenvalue problem (9) and (10). Set I d 1 ,...,d N := S n δ ⊂ {1, . . . , N } n . For ǫ = (ǫ 1 , . . . , ǫ n ) ∈ I d 1 ,...,d N , take a shortest element w ǫ ∈ S n such that w ǫ ǫ = δ. Let w ǫ = s i 1 · · · s i l be a reduced expression, and set T wǫ := T i 1 · · · T i l . This does not depend on a choice of the reduced expression. Now define a function F ǫ (z 1 , . . . , z n ) by where ℓ(w) is the length of w ∈ S n . Set Let K(z 1 , . . . , z n ) be a function satisfying (11) for m = 1, . . . , n. For example, take Theorem 3. Set G(z 1 , . . . , z n ) := K(z 1 , . . . , z n )F (z 1 , . . . , z n ). Then G is a solution of the rational qKZ equation with Proof . In the same way as the proof of Theorem 3.6 in [12], we can see that the following relations hold: where ω is defined by (4). By setting p j = χ d 1 +···+d j , the two relations above are equivalent to Combining them and (11), we see that G = KF is a solution to the rational qKZ equation.

Special solutions to the rational qKZ equation
Now we construct a class of polynomial solutions to the eigenvalue problem (9) and (10). They create solutions to the rational qKZ equation as discussed in the previous subsection (see Theorem 3).

Shifted singular polynomials
First let us consider the equation (9). We want to obtain polynomial solutions. Then the eigenvalues χ i (1 ≤ i ≤ n) should be one from Proposition 3. As seen below the polynomial eigenfunctions can be regarded as a shifted version of singular polynomials. [3,4,5]. Denote by S the subspace of C[x] consisting of singular polynomials. Recall that C[x] has H κmodule structure defined by the map φ (1). From the commutation relations σ i D i = D i+1 σ i and σ i D j = D j σ i (j = i, i + 1), the maps φ(s i ) (1 ≤ i < n) preserve S. Thus S is a CS n -module. Now let us consider the shifted version.
Definition 2. We call g(z 1 , . . . , z n ) ∈ C[z] a shifted singular polynomial if it satisfies Y i g = g for all 1 ≤ i ≤ n.
Denote by S the subspace of C[z] consisting of shifted singular polynomials. From the relations T i Y i = Y i+1 T i and T i Y j = Y j T i (j = i, i + 1), which follow from Proposition 1, the maps T i (1 ≤ i < n) preserve S. Hence S is also a CS n -module. It is naturally obtained from the H + κ -module structure on τ C[z] (note that τ (s i ) = s i for all i).
Proof . Recall that Φ −1 is the map taking the leading degree part and changing the variables z i → x i . Then we easily see Φ −1 ( S) ⊂ S from the expansion Y i = 1 − D i + Y i given in the argument preceding to Proposition 3. Let us prove Φ(S) ⊂ S. Suppose that f ∈ C[x] be a singular polynomial. Then we have If κ is generic, we have S = C, that is, constants are only singular polynomials. A specific parameter value κ = κ 0 is called a singular value if there exists a non-constant singular polynomial at κ = κ 0 . In [5] it is proved that the set of singular values is equal to {−l/m | m ∈ Z >0 , l = 2, . . . , n and m ∈ lZ} 2 . From Corollary 1, Corollary 2 and Theorem 4, we find Corollary 3. Let κ 0 be a singular value. Suppose that λ is a composition such that ξ λ is well-defined and singular at κ = κ 0 . Then E λ is also well-defined and shifted singular.
In [3,4] Dunkl obtained an explicit description of S as a CS n -module. At a singular value κ = κ 0 , the module S is multiplicity-free and the set of the irreducible components is completely determined from κ 0 and n. Each irreducible component has a basis consisting of non-symmetric Jack polynomials which are well-defined and singular. From Theorem 4 and Corollary 3, the CS n -module structure of S is the same as S. In the shifted case, S has a basis consisting of shifted non-symmetric Jack polynomials. Now we give a limiting procedure from non-symmetric Macdonald polynomials to shifted singular ones. Let us recall the definition of non-symmetric Macdonald polynomials. Let q and t 1/2 be indeterminates and denote the coefficient field by K = C(q, t 1/2 ). Consider the following operators T i (1 ≤ i < n) and ω acting on the polynomial ring K[X 1 , . . . , X n ]: The non-symmetric Macdonald polynomials E q,t λ (λ ∈ Λ) are defined by the following properties: Here X λ := X λ 1 1 · · · X λn n .
Introduce a small parameter ǫ and set Then we find T ±1 i → T i and ω → ω in the limit ǫ → 0, and hence Y i → Y i . Suppose that E q,t λ has the Laurent expansion at ǫ = 0 in the form for some α ∈ Z and nonzero g λ (z) ∈ F[z]. Then we obtain Y i g λ = g λ for 1 ≤ i ≤ n, that is, g λ is shifted singular. From the discussion preceding to Corollary 3, g λ can become non-constant only if κ is a singular value. In Proposition 7 below, we give an example of λ and a singular value κ 0 such that g λ is non-constant at κ = κ 0 .

4.3
Factorized formula for the case of (k, r, s) = (N, 2, 0) In [16] a new integral formula for solutions of the rational qKZ equation of level zero is constructed. In the construction we use as an ingredient a special solution of level one given as follows. We consider the case of n = N m for some m ≥ 1, and d 1 = · · · = d N = m. Then the special solution is given in the form G = KF , where K = K(z 1 , . . . , z n ) is the function defined by (12), and It is obtained as a limit of a matrix element of the vertex operators associated with the quantum affine algebra U q ( sl N ). It is also found by Di Francesco and Zinn-Justin in [6]. The special solution above is contained in the polynomial solutions given in Proposition 6 as follows.