Representation Theory and Quantum Inverse

Scattering Method:

Open Toda Chain and Hyperbolic Sutherland Model

A. Gerasimov^{1}, S. Kharchev^{2}, D. Lebedev^{3}

Max-Planck-Insitut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

and

Institute of Theoretical & Experimental Physics, 117259, Moscow, Russia

Abstract

Using the representation theory of 𝔤𝔩(N, ℝ) we explain the expression through integrals for GL(N, ℝ) Toda chain wave function obtained recently by Quantum Inverse Scattering Method. The main tool is the generalization of the Gelfand-Tsetlin method to the case of the infinite-dimensional representations of 𝔤𝔩(N, ℝ) . The interpretation of this generalized construction in terms of the coadjoint orbits is given and the connection with Yangian Y (𝔤𝔩(N)) is discussed. We also provide the representation by integrals for the hyperbolic Sutherland model eigenfunctions in Gelfand-Tsetlin representation. Using the example of the open Toda chain we discuss the connection between the Quantum Inverse Scattering Method and Representation Theory.

1E-mail: gerasimov@vitep5.itep.ru 2E-mail: kharchev@gate.itep.ru 3E-mail: dlebedev@vitep5.itep.ru

Contents

1 Introduction 2

2 Whittaker modules in Gelfand-Tsetlin representation 4

2.1 Generalization of Gelfand-Tsetlin construction to the case of infinite-dimen-sional representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Whittaker modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Orbit method for GL(N, ℝ) and Gelfand-Tsetlin representation 11

4 Yangian Y (𝔤𝔩(N)) and Gelfand-Tsetlin representation 15

5 Application to the quantum integrable systems 18

5.1 The open Toda chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 The hyperbolic Sutherland model . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Connection with QISM-based approach 22

1 Introduction

In the late seventies the two basic approaches to study the quantum integrable systems have been formulated.

M.Olshanetsky and A.Perelomov discovered the direct relation between the representation theory of non-compact semi-simple Lie groups and Sutherland-Calogero-Moser (SCM) systems [1]-[3]. Methods of Representation Theory provide the complete solution of the quantum SCM system. Soon B.Kostant and D.Kazhdan realized that the more general class of integrable systems, including open Toda chains could be solved by this method [4], [5]. The wave function of the SCM systems and Toda chains appears to be zonal spherical functions [6], [7] and Whittaker functions [8]-[11] respectively. There is an obvious generalization of the approach to a more general class of integrable systems corresponding formally to the affine Lie algebras. But up to now there was no progress in derivation of the explicit representation through integrals for the wave functions along this line due to analytic difficulties in the corresponding part of Representation theory.

At the same time another approach to the quantum integrability emerged. In the work of L.Faddeev, E.Sklyanin, P.Kulish and L.Takhtajan [12]-[15]^{4} the Quantum Inverse Scattering Method (QISM) was formulated and successively applied to numerous quantum integrable systems [17]. The recent development of this approach is connected with the Separation of Variables (SoV) method [18], [19]. However, the understanding of the connection of QISM with the Representation Theory remains unclear up to now.

4See the summary of the early period of the development in [16]

The starting point of this paper was the new method of deriving the wave function for GL(N, ℝ) Toda chain given in [20]. It had no obvious relation with the Representation Theory approach, but instead the QISM was used to construct the wave function. Understanding of this construction in terms of the Representation Theory approach and comparing the two basic approaches to quantum integrability for this simple integrable system may give additional insight into the nature of QISM. In this paper we give interpretation of the representation by integrals of the wave function in the open Toda chain [20] in terms of the Representation Theory approach to quantum integrability. The basic tool is a generalization of the Gelfand-Tsetlin method [21], [22] to the case of infinite-dimensional representations of 𝔤𝔩(N, ℝ) . This gives a construction of the representation of U(𝔤𝔩(N, ℝ)) in terms of difference operators ^{5} which one can use for finding explicit expressions for Whittaker vectors. Thus, we reproduce the representation through integrals for Whittaker functions obtained previously in [20] by QISM.

Comparing the methods of [20] with those proposed in this paper reveals a deep connection between SoV [18] and Gelfand-Tsetlin’s type representations. In [24], [18] the separated variables for the classical Toda chain were found. They could be successfully quantized to give an explicit solution of the corresponding quantum theory. In this paper we use the generalization of Gelfand-Tsetlin representation for 𝔤𝔩(N, ℝ) which gives rise to a much larger set of the separated variables ( \frac{1}{2}N(N − 1) instead of (N − 1) ). The idea of introducing the enlarged set of separated variables could be traced back to the work of I.M.Gelfand and A.A.Kirillov on the field of fractions of the universal enveloping algebras [25]. On the other hand, the representation we have constructed appears to be closely connected with the representation theory of the Yangian Y (𝔤𝔩(N)) . The enlargement of the set of variables for solving the quantum problem is a crucial part of the Representation Theory approach to quantum integrability. One may hope that understanding of the correct notion of the separated variables in Representation Theory will turn out to be fruitful for both the theory of Quantum Integrable Systems and Representation Theory.

The plan of the paper is as follows. In section 2 we give a direct generalization of the Gelfand-Tsetlin method to the case of infinite-dimensional representations of U(𝔤𝔩(N)) . The crucial point is to use continuous unconstrained variables instead of integer valued Gelfand-Tsetlin parameters. We define Whittaker modules W and W′ in this realization of the representation and show that there exists the hermitian pairing between W and W′.

In the section 3 we give interpretation of the representation through Kirillov-Kostant orbit method [26]. We give an explicit parameterization of the finite cover of the open part of the coadjoint orbit of GL(N, ℝ) with the stabilizer H_{N} = GL(1, ℝ)^{⊗N} . We show that the parameterization provides Darboux coordinates with respect to the canonical Kirillov-Kostant symplectic structure on the orbit. The above construction is a version of those by A.Alekseev, L.Faddeev and S.Shatashvilli [27], [28] adopted for non-compact groups. Our parameterization leads to the classical counterpart for generators of 𝔤𝔩(N) given in section 2. A subtle point of the orbit method interpretation of our representation is that we use only (the finite cover of ) the open part of the orbit. This leads to a representation of 𝔤𝔩(N, ℝ) which can not be integrated to a representation of the whole group. However it is obvious that the action of the Cartan subalgebra integrates to the action of a Cartan torus and this is enough for constructing solutions of the open Toda chain. We end this section giving contour integral formulas for the generators of 𝔤𝔩(N, ℝ) .

5Accidentally the version of this construction for the affine algebra 𝔰̂𝔩(2)_{k} was discovered previously in [23]. The generalization to the case { \atop 𝔰𝔩(N)}_{k} will be given elsewhere.

In section 4 we re-derive the Gelfand-Tsetlin parameterization using the representation theory of Yangian [29] (closely connected ideas for the classical Gelfand-Tsetlin construction were developed in [30]-[32]). Moreover, it reveals a deep connection of Gelfand-Tsetlin representation with QISM based on representation theory of Yangian.

In section 5 we apply the representation constructed in section 2 to find the explicit solution of the quantum open Toda chain and quantum hyperbolic Sutherland model. For Toda chain we reproduce the results of [20]. The corresponding representation of the wave functions of quantum hyperbolic Sutherland model appears to be new.

In the last section we compare QISM-based approach and those developed in this paper to the integrability of the open Toda chain.

Acknowledgments: We would like to to thank the Max Planck Institut für Mathematik (Bonn, Germany), where the paper was finished, for financial support and stimulating atmosphere.

The authors are grateful to V.Gorbunov, A.Levin, Y.Neretin, M.Olshanetsky, and M.Semenov-Tian-Shansky for useful discussions.

The research was partly supported by grants RFBR 01-01-00548 (A.Gerasimov); INTAS OPEN 00-00055; RFBR 00-02-16477, (S.Kharchev); INTAS OPEN 00-00055; RFBR 01-01-00548 (D.Lebedev) and by grant 00-15-96557 for Support of Scientific Schools.

2 Whittaker modules in Gelfand-Tsetlin representation

2.1 Generalization of Gelfand-Tsetlin construction to the case of infinite-dimensional representations

We start recalling the original Gelfand-Tsetlin construction of the canonical basis in the space of an irreducible finite-dimensional representation of GL(N, ℂ) [21], [22]. It is based on the two well known facts:

Any irreducible finite-dimensional representation is uniquely determined by a n -dimen-sional vector m_{N} = (m_{N1}, …, m_{_{N}_{N}}) with integer entries such that: m_{N1} ≥ … ≥ m_{_{N}_{N}} . The vector m_{N} is called the label or the highest weight of the representation.

Consider GL(N − 1, ℂ) as an upper left corner subgroup of GL(N, ℂ) . Then in the decomposition of a finite-dimensional irreducible representation of GL(N, ℂ) each irreducible representations of the subgroup GL(N − 1, ℂ) enters with multiplicity not greater than one. If we denote by m_{N−1} the labels of representations involved into the decomposition, then their entries satisfy the additional conditions: m_{Nj} ≥ m_{N−1,j} ≥ m_{N,j+1} for all j = 1, …, N − 1 .

If we continue the decomposition step by step, we obtain a decomposition of an irreducible finite-dimensional representation of GL(N, ℂ) into the direct sum of the one dimensional subspaces. Fixing an element in each one-dimensional summand one obtains the Gelfand-Tsetlin basis. The vectors of this basis are labeled by the triangular arrays, consisting of all m_{n} ( n = 1, …, N ) with entries satisfying the above mentioned constraints.

Now we wish to generalize the Gelfand-Tsetlin construction to the case of infinitedimensional representations of the universal enveloping algebra U(𝔤𝔩(N)) . We start with the following generalization of the Gelfand-Tsetlin (GT) representation of 𝔤𝔩(N) . Namely, let (γ_{1} … γ_{N}) be a triangular array consisting of \frac{1}{2}N(N+1) variables γ_{n} = (γ_{n1}, …, γ_{nn}); n = 1, …, N .

Proposition 2.1 Let M be the space of meromorphic functions in \frac{1}{2}N(N − 1) variables γ_{1}, …, γ_{N−1} . Then the operators ^{6}

E_{nn} = \frac{1}{iℏ}( ∑_{j=1}^{n} γ_{jn} − ∑_{j=1}^{n−1} γ_{j,n−1});  (n = 1, …, N),  (2.1a)

E_{n,n+1} = −\frac{1}{iℏ} ∑_{j=1}^{n} \frac{∏_{r=1}^{n+1}(γ_{nj} − γ_{n+1,r} − \frac{iℏ}{2})}{∏_{s≠j}(γ_{nj} − γ_{ns})} e^{−iℏ∂_{γ_{nj}}}; (n = 1, …, N − 1), (2.1b)

E_{n+1,n} = \frac{1}{iℏ} ∑_{j=1}^{n} \frac{∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r} + \frac{iℏ}{2})}{∏_{s≠j}(γ_{nj} − γ_{ns})} e^{iℏ∂_{γ_{nj}}}; (n = 1, …, N − 1) (2.1c)

realize a representation of 𝔤𝔩(N) in M .

Proof. It is sufficient to show that the generators (2.1) satisfy the standard commutation relations

\begin{matrix}[E_{nn}, E_{k,k+1}] = (δ_{nk} − δ_{n,k+1})E_{k,k+1}, \\ [E_{nn}, E_{k+1,k}] = −(δ_{nk} − δ_{n,k+1})E_{k+1,k}, \\ [E_{n,n+1}, E_{m+1,m}] = (E_{nn} − E_{n+1,n+1})δ_{nm} \end{matrix}  (2.2)

and obey the Serre identities

\begin{matrix}[E_{n,n+1}, [E_{n,n+1}, E_{n+1,n+2}]] = 0, \\ [E_{n+1,n}, [E_{n+1,n}, E_{n+2,n+1}]] = 0. \end{matrix}  (2.3)

6Actually, we construct the family of representations depending on the auxiliary parameter ℏ. One could rescale the variables γ_{nj} → ℏγ_{nj} to get rid of this parameter. However, we will retain ℏ in the formulae to make more explicit the connection with standard notations in the theory of quantum integrable systems.

This follows from the direct verification of above relations. Defining the composite root generators recursively as E_{jk} = [E_{jm}, E_{mk}] for j < m < k and j > m > k, one obtains the complete set of commutation relations

[E_{jk}, E_{lm}] = E_{jm}δ_{lk} − E_{lk}δ_{jm}.  (2.4)

The representation described above is parameterized by label γ_{N} = (γ_{N1}, …, γ_{_{N}_{N}}) and extends to a representation of the universal enveloping algebra U(𝔤𝔩(N)) . Obviously, there is a natural representation of any subalgebra U(𝔤𝔩(n)) ⊂ U(𝔤𝔩(N)), (n = 1, …, N − 1) on the same space M with the label γ_{n} = (γ_{n1}, …, γ_{nn}) . Let Z(𝔤𝔩(n)) be the center of U(𝔤𝔩(n)) .

Definition 2.1 We say that an U(𝔤𝔩(n)) -module V admits an infinitesimal character ξ if there is a homomorphism ξ : Z(𝔤𝔩(n)) → ℂ such that zv = ξ(z)v for all z ∈Z(𝔤𝔩(n)), v ∈ V.

It is possible to show that U(𝔤𝔩(N)) -module M defined above admits an infinitesimal character. Actually, more general statement holds:

Lemma 2.1 Each central element of U(𝔤𝔩(n)) acts on M by the multiplication on symmetric polynomials of variables γ_{nj}.

Proof. By construction, an operator u ∈ U(𝔤𝔩(n)) acting on M is invariant under permutation of variables entering in each row γ_{m}, (m = 1, …, n) . It is easy to prove that any symmetric operator commuting with the Cartan subalgebra is just a function in γ_{m}. If this function commutes with all root generators it should be iℏ periodic with respect to any variable γ_{mj}, (m = 1, …, n − 1) . The polynomial generators of the center of U(𝔤𝔩(n)) may act only as a rational functions in this representation and thus we conclude that they do not depend on the variables γ_{mj}, (m = 1, …, n − 1) . Taking into account that the generators of U(𝔤𝔩(n)) depend polynomially on γ_{n} we conclude that the generators of the center Z(𝔤𝔩(n)) act on M as symmetric polynomials in γ_{n}.

Therefore, the elements z ∈ Z(𝔤𝔩(n)) act on U(𝔤𝔩(n)) -module M as an operators of multiplication by scalar and the U(𝔤𝔩(n)) -module M (n = 1, …, N) admits an infinitesimal character.

In the next subsection we calculate the explicit form of central elements acting on M using the notion of Whittaker vectors.

2.2 Whittaker modules

Let us now give a construction of Whittaker module using the representation of U(𝔤𝔩(N)) described above. We recall some facts from [5] which we use in our construction.

Let n_{+} and n_{−} be the subalgebras of 𝔤𝔩(N) generated by positive and negative root generators respectively. The characters χ_{+} : n_{+} → ℂ, χ_{−} : n_{−} → ℂ are uniquely determined by their values on the simple root generators; they are called non − degenerate if the (complex) numbers χ_{+}(E_{n,n+1}) and χ_{−}(E_{n+1,n}) are non-zero for all n = 1, …, N − 1 .

Let V be any U = U(𝔤𝔩(N)) -module. Denote the action of u ∈ U on v ∈ V by uv. A vector w ∈ V is called the Whittaker vector with respect to character χ_{+} if

E_{n,n+1}w = χ_{+}(E_{n,n+1})w ,  (n = 1, …, N − 1),  (2.5)

and an element w′ ∈ V′ is called the Whittaker vector with respect to character χ_{−} if

E_{n+1,n}w′ = χ_{−}(E_{n+1,n})w′ ,  (n = 1, …, N − 1).  (2.6)

A Whittaker vector is called the cyclic for V if Uw = V and an U -module is called a Whittaker module if it contains a cyclic Whittaker vector. The U -modules V and V′ are called dual if there exists a non-degenerate pairing of V′ and V such that ⟨Xv′, v⟩ = −⟨v′, Xv⟩ for all v ∈ V , v′ ∈ V′ and X ∈ 𝔤𝔩(N) .

We proceed with explicit formulae for Whittaker vectors corresponding to representation given by (2.1).

Proposition 2.2 Whittaker vectors satisfying the equations

E_{n+1,n}w′_{N} = −iℏ^{−1}w′_{N},  (2.7)

E_{n,n+1}w_{N} = −iℏ^{−1}w_{N}  (2.8)

for all n = 1, …, N − 1 , are given by the following formulae:

\begin{matrix}w′_{N} = 1, \\ w_{N} = e^{− \frac{π}{ℏ} ∑_{n=1}^{N−1}(n−1) ∑_{j=1}^{n} γ_{nj}} ∏_{n=1}^{N−1} g_{n}(γ_{n}, γ_{n+1}), \end{matrix}  (2.9)

where

g_{n}(γ_{n}, γ_{n+1}) = ∏_{k=1}^{n} ∏_{m=1}^{n+1} ℏ^{\frac{γ_{nk}−γ_{n+1,m}}{iℏ} + \frac{1}{2}} Γ(\frac{γ_{nk} − γ_{n+1,m}}{iℏ} + \frac{1}{2}).  (2.10)

Proof.

E_{n,n+1}w_{N} = −iℏ^{−1}w_{N} ∑_{j=1}^{n} \frac{∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r} − \frac{iℏ}{2})}{∏_{s≠j}(γ_{nj} − γ_{ns})} .  (2.11)

Using the identity

∑_{j=1}^{n} \frac{x_{j}^{m}}{∏_{s≠j}(x_{j} − x_{s})} = ∑_{{k_{r}≥0 \atop k_{1}+...+k_{n}=m−n+1}} x_{1}^{k_{1}} … x_{n}^{k_{n}}  (2.12)

one arrives to (2.8). The equation (2.7) is proved similarly.

The set of Whittaker vectors (2.9), (2.10) is closed under multiplication by an arbitrary iℏ -periodic function of variables γ_{nj}. Hence, there are infinitely many invariant subspaces in M corresponding to infinitely many Whittaker vectors. To construct irreducible submodules, let us introduce the Whittaker modules W and W′ cyclically generated by Whittaker vectors w_{N} and w′_{N} respectively.

Theorem 2.1 The Whittaker module W = Uw_{N} is spanned by the elements of the form

w_{m_{1},... ,m_{N−1}} = ∏_{n=1}^{N−1} ∏_{k=1}^{n} σ_{k}^{m_{nk}}(γ_{n})w_{N},  (2.13)

where σ_{k}(γ_{n}) is the elementary symmetric function of the variables γ_{n1}, …, γ_{nn} of order k:

σ^{k}(γ^{n}) = ∑_{j_{1}<...<j_{k}} γ^{nj_{1}} … γ^{nj_{k}}.  (2.14)

Similarly, the Whittaker module W′ = Uw′_{N} is spanned by the polynomials

w′_{m_{1},... ,m_{N−1}} = ∏_{n=1}^{N−1} ∏_{k=1}^{n} σ_{k}^{m_{nk}}(γ_{n}).  (2.15)

The Whittaker modules W and W′ are irreducible.

Proof. Let us prove the statement for the module W . Using the identity e^{iℏ∂_{γ_{nj}}}σ_{k}(γ_{n}) =

σ_{k}(γ_{n}) + iℏ ∑_{r=1}^{k−1}(−1)^{r}σ_{k−r−1}(γ_{n})γ_{nj}^{r} and the explicit formula for the action of E_{n+1,n} on the

Whittaker vector:

E_{n+1,n}w_{N} = iℏ^{−1}w_{N} ∑_{j=1}^{n} \frac{∏_{r=0}^{n+1}(γ_{nj} − γ_{n+1,r} + \frac{iℏ}{2})}{∏_{s≠j}(γ_{nj} − γ_{ns})} ,  (2.16)

one shows that the Whittaker module is spanned by the elements of the form (2.13).

To prove the irreducibility of Whittaker modules we need the following fact ([5], theorem 3.6.2): Let M be any U -module which admits an infinitesimal character. Assume w ∈ M is a Whittaker vector. Then the submodule Uw ⊂ M is irreducible.

Hence, the modules W and W′ are irreducible according to Lemma 2.1 and the above mentioned theorem.

Let us note that for any subalgebra U(𝔤𝔩(n)) ⊂ U(𝔤𝔩(N)) , 2 ≤ n < N the vector space

with the basis ∏_{s=1}^{n−1} ∏_{k=1}^{n} σ_{k}^{m_{sk}}(γ_{s})w_{N} is a U(𝔤𝔩(n)) Whittaker module.

Now one can calculate the explicit form of the action of the central elements of U(𝔤𝔩(n)) on the space M . It is well known that the generating function A_{n}(λ) of the central elements of U(𝔤𝔩(n)) (the Casimir operators) can be represented as follows [33]:

\begin{matrix}A_{n}(λ) \\ = ∑_{p∈P_{n}} sign p[(λ − iℏρ_{1}^{(n)})δ_{p(1),1} − iℏE_{p(1),1}] … [(λ − iℏρ_{n}^{(n)})δ_{p(n),n} − iℏE_{p(n),n}], \end{matrix} (2.17)

where ρ_{k}^{(n)} = \frac{1}{2}(n − 2k + 1), k = 1, …, n and summation is given over elements of the permutation group P_{n}.

Proposition 2.3 The operators (2.17) has the following representation on M :

A_{n}(λ) = ∏_{j=1}^{n}(λ − γ_{nj}),  (n = 1, …, N).  (2.18)

Proof. Due to Lemma 2.1 it is sufficient to calculate the action of the operators A_{n}(λ) on the Whittaker vector w_{N} (or w′_{N} ) only. Due to (2.8) one has

\begin{matrix}A_{n}(λ)w_{N} = A_{n−1}(λ − \frac{iℏ}{2})[λ − iℏρ_{n}^{(n)} − iℏE_{nn}]w_{N} \\ − iℏ ∑_{k=2}^{n} A_{n−k}(λ − \frac{ikℏ}{2})E_{n,n−k+1}w_{N} . \end{matrix}  (2.19)

Further, using the relation (2.16), it is easy to calculate the action of the composite generators:

E_{n,n−k+1}w_{N} = iℏ^{−1}w_{N} ∑_{j_{1}=1}^{n−1} … ∑_{j_{k}=1}^{n−k+1} ∏_{p=1}^{k−1} \frac{∏_{r_{p}≠j_{p−1}}(γ_{n−p.j_{p}} − γ_{n−p+1,r_{p}} + \frac{iℏ}{2})}{∏_{s_{p}≠j_{p}}(γ_{n−p,j_{p}} − γ_{n−p,s_{p}})} .  (2.20)

Substituting (2.20) to (2.19) one arrives to (2.18) after some combinatorics. The similar calculation is obviously valid for the Whittaker module W′ .

One thing that remains is to construct the hermitian pairing between W and W′ and proving that the Whittaker modules W and W′ are dual with respect to this pairing.

Definition 2.2 Let ϕ ∈ W′ and ψ ∈ W . Define the hermitian pairing ⟨ , ⟩: W′ ⊗ W → ℂ by

⟨ϕ, ψ⟩ = ∫_{ℝ^{\frac{N(N−1)}{2}}} μ^{0}(γ)\overline{ϕ}(γ) ψ(γ) ∏_{{n=1 \atop j≤n}}^{N−1} dγ^{nj},  (2.21)

where

μ_{0}(γ) = ∏_{n=2}^{N−1} ∏_{s<p}(γ_{ns} − γ_{np})(e^{\frac{2πγnp}{ℏ}} − e^{\frac{2πγns}{ℏ}}).  (2.22)

Then the integral (2.21) converges absolutely. Actually, this is a simple corollary of more general statement. Namely, denote the integrand in the r.h.s. of (2.21) by the I_{N}(γ) .

Lemma 2.2 For the set of Im γ_{nj} such that min_{j}\{Im γ_{nj}\} > max_{k}\{Im γ_{n+1,k}\} − \frac{1}{2} the following inequality holds:

|I_{N}(γ)| ≤ |P (γ)| exp \{ − \frac{1}{(2N − 3)!!} ∑_{n=1}^{N−1} ∑_{j=1}^{n} |Re γ_{nj}|\}.  (2.23)

where P (γ) is a polynomial in γ_{1}, …, γ_{N−1} .

Proof. For any x > 0 the inequality |Γ(x + iy)| ≤ Γ(x)p_{x}(y) cosh^{− \frac{1}{2}} (y) is valid for some polynomial p_{x}(y) with degree linearly depending on x. Hence, the integrand satisfies:

|I_{N}(γ)| ≤ |P (γ)| e^{\frac{π}{ℏ}S_{N} (γ)},  (2.24)

where

S_{N}(γ) = ∑_{n=2}^{N−1} ∑_{s<p} |Re γ_{ns} − Re γ_{sp}|

− \frac{1}{2} ∑_{n=1}^{N−2} ∑_{j,k} |Re γ_{nj} − Re γ_{n+1,k}| − \frac{N}{2} ∑_{j=1}^{N−1} |Re γ_{N−1,j}|.

(2.25)

Using inequality

∑_{j,k=1}^{m} |a_{j} − b_{k}| − ∑_{j<k}^{m} |a_{j} − a_{k}| − ∑_{j<k}^{m} |b_{j} − b_{k}| ≥ 0  (2.26)

which holds for any set of real parameters {a_{j}, b_{j}}; j = 1, …, m (see (11.118) in [34]), one proves the recursive relation S_{N} ≤ \frac{1}{2N−3}(S_{N−1} − ∑_{j} |Re γ_{N−1,j}|) thus arriving to (2.23).

Proposition 2.4 Let γ_{N} ∈ ℝ^{N} . Whittaker modules W and W′ are dual with respect to the pairing defined by (2.21), which means that for any ϕ ∈ W′ and ψ ∈ W , the generators X ∈ 𝔤𝔩(N, ℝ) possess the property

⟨ϕ, Xψ⟩ = − ⟨Xϕ, ψ⟩.  (2.27)

Proof. For example, consider ⟨ϕ, E_{n,n+1}ψ⟩, where E_{n,n+1} is defined by (2.1b). It is easy to see that expression \overline{E_{n,n+1}ϕ} ψ does not have the poles in the upper hyper-plane and it is possible to deform the integration contour. Consider the shifts γ_{nj} → γ_{nj} + iℏ. Note that there is a difference equation

e^{iℏ∂_{γ_{nj}}}μ^{0}(γ) = μ^{0}(γ) ∏_{s≠j} \frac{γ_{nj} − γ_{ns} + iℏ}{γ_{nj} − γ_{ns}} .  (2.28)

Then using the deformation of the contour and the estimates (2.23), one obtains (2.27).

3 Orbit method for GL(N, ℝ) and Gelfand-Tsetlin representation

In this section we provide the orbit method interpretation for the explicit algebraic construction of the infinite-dimensional representation of U(𝔤𝔩(N)) given in the previous section. Similar results for the case of the compact group were obtained previously in [27],[28].

Let G be a Lie group, 𝔤 corresponding Lie algebra and 𝔤^{∗} its dual. Coadjoint orbits O of G are supplied with the canonical symplectic structure, given by Kirillov-Kostant two-form [26]. The space of linear functions on 𝔤^{∗} is closed under Kirillov-Kostant Poisson bracket and coincides with 𝔤 as a Lie algebra.

It reproduces the relations in the algebra 𝔤. Consider coadjoint orbits of the group GL(N, ℝ) with stabilizer H_{N} = GL(1, ℝ)^{⊗N} . Below we construct Darboux coordinates on the finite cover Õ^{(0)} of the open part O^{(0)} of such an orbit O of G = GL(N, ℝ) . We also give expressions for the restriction onto O^{(0)} of the linear functions on 𝔤^{∗} in these coordinates. The corresponding Hamiltonian vector fields act on O^{(0)} . The explicit realization of U(𝔤𝔩(N)) by difference operators given in the previous section may be considered as a “quantization” of this Poisson algebra. Let us stress that this construction gives the representation of the U(𝔤𝔩(N)) that can not be integrated to the representation of the whole group. From the picture that will be discussed in this section it is quite natural because the part of the orbit O^{(0)} is not stable with respect to the action of the group on the whole orbit O.

Let G = GL(N, ℝ) , 𝔤 = 𝔤𝔩(N, ℝ)) and we identify the algebra 𝔤 and its dual 𝔤^{∗} with the help of the Killing form. Consider the coadjoint orbit O of the diagonal element u_{N} ∈ 𝔤^{∗} with pairwise different diagonal entries: γ_{N} = (γ_{N1}, γ_{N2}, ⋯ γ_{_{N}_{N}}) :

u_{N} = (\begin{matrix}γ_{N1} & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & γ_{_{N}_{N}} \end{matrix} ) .  (3.1)

The stabilizer of u_{N} is H_{N} = GL(1, ℝ)^{⊗N} and the general element of the orbit may be parameterized as u = g^{−1}u_{N}g, where g ∈ GL(N, ℝ) is defined up to the left action of H_{N} . There is a canonical symplectic two-form on O which in this parameterization is given by:

Ω = T r u_{N}(dg g^{−1})^{2} .  (3.2)

Consider the subspace in GL(N, ℝ) which consists of the elements of the form:

g = g_{N}g_{N−1} ⋯ g_{2} .  (3.3)

The matrices g_{n} differ from the unit matrix only in the upper-left n × n square and recursively defined as follows. Denote the upper-left (n × n) sub-matrix of g_{n} by f_{n}. There should exist such γ_{n−1,j} , (j = 1, …, n − 1) that the following condition holds:

f_{n}^{−1} (\begin{matrix}γ_{n1} & 0 & ⋯ & 0 \\ 0 & ... & & ⋮ \\ ⋮ & & γ_{n,n−1} & 0 \\ 0 & ⋯ & 0 & γ_{nn} \end{matrix}) f_{n} = (\begin{matrix}γ_{n−1,1} & 0 & ⋯ & ∗ \\ ⋮ & ... & 0 & ⋮ \\ 0 & ⋯ & γ_{n−1,n−1} & ∗ \\ ∗ & ⋯ & ∗ & ∗ \end{matrix}) .  (3.4)

This means that the diagonal matrix conjugated by f_{n} may have non-zero non-diagonal entries only in the last column and the last row. The matrix f_{n} is defined uniquely by γ_{nj}, γ_{n−1,k} up to the left action by the diagonal matrix and the right action by the diagonal matrix with the last diagonal element being equal to one. Let us choose the following representatives for f_{n} in the coset of G by the left action of the diagonal matrices:

\begin{matrix}(f_{n})_{jk} = \frac{Q_{n−1,k}}{γ_{nj} − γ_{n−1,k}} \frac{∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r})}{∏_{s≠j}(γ_{nj} − γ_{ns})}  (k = 1, …, n − 1), \\ (f_{n})_{jn} = \frac{∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r})}{∏_{s≠j}(γ_{nj} − γ_{ns})}, \end{matrix}  (3.5)

where Q_{n−1,k}, k = 1...(n − 1) are additional coordinates on the orbit. This gives the map from ℝ^{N(N−1)}/Δ to the open part O^{(0)} of the orbit. Here Δ is a union of three subspaces of ℝ^{N(N−1)} : Δ = D_{1} ∪ D_{2} ∪ D_{3} . The subspace D_{1} is a subspace where at least two γ -coordinates of the same level coincide: γ_{nj} = γ_{nk}. The subspace D_{2} is defined as a subspace where at least two of γ -coordinates from the consecutive levels coincide γ_{nj} = γ_{n−1,k}. Finally, the subspace D_{3} is a subspace where at least one of Q_{nj} is zero. Now let us construct the inverse map. Consider the element of the orbit u = g^{−1}u_{N}g shifted by the unit matrix multiplied by a formal variable λ:

T (λ) = 1λ − u .  (3.6)

The n -th principal minor a_{n}(λ) of T (λ) does not depend on g_{m}, m < n+1 . For the general point of the orbit the roots of the minors may be complex. Consider the open part O^{(0)} of the orbit O such that all upper-left sub-matrices are diagonalizable and their determinants have distinct real roots. We also impose the condition that there are no identical roots of the minors of the consecutive ranks. Then define γ -variables as roots of the minors:

a_{n}(λ) = ∏_{j=1}^{n}(λ − γ_{nj}) .  (3.7)

Note that by this condition the roots are defined only up to the action of symmetric group S_{n}. As a result we get the coordinates on the finite cover of the part of the orbit.

Now let us give the explicit expressions for the Q -coordinates. Let b_{n}(λ) be the determinant of n × n corner sub-matrix of the matrix which is obtained from T (λ) by the interchanging the n -th and the (n + 1) -th columns, i.e. b_{n}(λ) = det_{n×n} ||T (λ)S_{n,n+1}|| where N × N matrix S_{n,n+1} is a unit matrix with the diagonal 2 × 2 sub-matrix in the n, n + 1

columns and n, n + 1 rows being replaced by the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Similarly, let us

define c_{n}(λ) as the determinant of the n × n corner sub-matrix obtained from T (λ) by interchanging the n -th and the (n + 1) -th rows, i.e. c_{n}(λ) = det_{n×n} ||S_{n,n+1}T (λ)||. Then the other part of the variables may be introduced as follows:

Q_{nj} = b_{n}^{−1}(γ_{nj})a_{n+1}(γ_{nj}) = −c_{n}(γ_{nj})a_{n−1}^{−1}(γ_{nj}) ,  (3.8)

where the last equality can be proved by straightforward check. It is not difficult to verify that this definition is compatible with the explicit parameterization in terms of (γ_{nj}, Q_{nj}) given above. Due to the condition (3.4) the n -th left-upper corner sub-matrix of the matrix u_{n} = g_{n+1}^{−1} ⋯ g−1_{N}u_{N}g_{N} ⋯ g_{n+1} is the diagonal matrix, a_{n}(λ) is given by (3.7). The determinant b_{n}(λ) is invariant under conjugation by the matrix which differs from the unit matrix only in the upper-left (n − 1) × (n − 1) corner. Thus, b_{n}(λ) may be represented as:

b^{n}(λ) = det(g_{n}^{−1}) det_{n×n} ||(λ − u^{n})g^{n}S^{n,n+1}|| .  (3.9)

Straightforward calculation gives

b_{n}(λ) = ∑_{j=1}^{n} Q_{nj}^{−1} ∏_{r=1}^{n+1}(γ_{nj} − γ_{n+1,r}) ∏_{s≠j} \frac{λ − γ_{ns}}{γ_{nj} − γ_{ns}} .  (3.10)

Similarly, for c_{n}(λ) one has

c_{n}(λ) = − ∑_{j=1}^{n} Q_{nj} ∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r}) ∏_{s≠j} \frac{λ − γ_{ns}}{γ_{nj} − γ_{ns}} .  (3.11)

This gives the inverse transformation from the finite cover of the open part of the orbit Õ^{(0)} to ℝ^{\frac{N(N−1)}{2}} /Δ .

Simple calculation of the Kirillov-Kostant simplectic form in the coordinates (γ_{nj}, Q_{nj}) gives the following

Proposition 3.1 1. The coordinates (γ_{nj}, Q_{nj}) parameterize the finite cover Õ^{(0)} of the open part of the orbit O^{(0)} . The product of the symmetric groups S = ∏n=2^{N} S_{n} acts freely on the fibers of the projection: π : Õ^{(0)} → O^{(0)} and the factor is O^{(0)} .

2. The lift of 2-form Ω on the Õ^{(0)} has the canonical form

π^{∗}Ω = ∑_{n=1}^{N−1} ∑_{j=1}^{n} dγ_{nj} ∧ Q_{nj}^{−1}dQ_{nj}.  (3.12)

Coordinates (γ_{nj}, Q_{nj}) give Darboux coordinates on the finite cover of the open part O^{(0)} of the coadjoint orbit of GL(N, ℝ) with the stabilizer conjugated to H_{N} = GL(1, ℝ)^{⊗N} .

Now let us give explicit expressions for the elements on the sub-diagonal, diagonal and over-diagonal of the matrix u = g^{−1}u_{N}g in the GT-coordinates. This gives the classical counterparts of the generators (2.1).

Proposition 3.2 The linear functions u_{nn}, u_{n,n+1} , u_{n+1,n} on 𝔤𝔩(N, ℝ)^{∗}, being restricted to O^{(0)} and lifted by the projection map π : Õ^{(0)} → O^{(0)} to Õ^{(0)} , have the following form in the GT-coordinates ^{7} :

\begin{matrix}u_{nn} = ∑_{j=1}^{n} γ_{nj} − ∑_{j=1}^{n−1} γ_{n−1,j}, \\ u_{n,n+1} = − ∑_{j=1}^{n} \frac{∏_{r=1}^{n+1}(γ_{nj} − γ_{n+1,r})}{∏_{s≠j}(γ_{nj} − γ_{ns})} Q_{nj}^{−1}, \\ u_{n+1,n} = ∑_{j=1}^{n} \frac{∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r})}{∏_{s≠j}(γ_{nj} − γ_{ns})} Q_{nj}. \end{matrix}  (3.13)

Proof. Let us start with the diagonal elements. Coadjoint action of the diagonal matrix Φ = diag(e^{ϕ_{1}}, ⋯ e^{ϕ_{N}}) on u may be represented as the right action on the element g (3.3). It is easy to see that this is equivalent to the shift of the variables:

\begin{matrix}Q_{nj} → Q_{nj}e^{ϕ_{n}−ϕ_{n+1}}, \\ Q_{Nj} → Q_{Nj}e^{ϕ_{N}} . \end{matrix}  (3.14)

7There is another representation with “square roots” which is more close to the one given in [21]. It may be obtained by a different choice of the parameterization of the solution of (3.4).

Thus, the corresponding Hamiltonian functions u_{nn} are:

u_{nn} = ∑_{j=1}^{n} γ_{nj} − ∑_{j=1}^{n−1} γ_{n−1,j} .  (3.15)

This result could also be derived from the explicit representation of the matrix u. Moreover, from the structure of the matrices u_{n}, f_{n} one could easily obtain the expressions for the other generators. For example,

u_{n+1,n} = ∑_{j=1}^{n}(u_{n})_{n+1,j}(f_{n})_{jn} = ∑_{j=1}^{n} \frac{∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r})}{∏_{s≠j}(γ_{nj} − γ_{ns})} Q_{nj} .  (3.16)

Quite similarly,

u_{n,n+1} = − ∑_{j=1}^{n} \frac{∏_{r=1}^{n+1}(γ_{nj} − γ_{n+1,r})}{∏_{s≠j}(γ_{nj} − γ_{ns})} Q_{nj}^{−1} .  (3.17)

There is a representation for u_{n,n}, u_{n,n+1}, and u_{n+1,n} by the contour integrals in terms of the polynomials a_{n}(λ), b_{n}(λ), and c_{n}(λ) :

\begin{matrix}u_{n,n}(λ) = − ∮_{∞} \frac{a_{n}(λ)}{a_{n−1}(λ)} \frac{dλ}{λ} , \\ u_{n,n+1}(λ) = − ∮_{∞} a_{n}^{−1}(λ)b_{n}(λ)dλ , \\ u_{n+1,n}(λ) = − ∮_{∞} c_{n}(λ)a_{n}^{−1}(λ)dλ . \end{matrix}  (3.18)

In the next section we discuss the connection of this representation with the Drinfeld’s “new realization” of the Yangian [29].

4 Yangian Y (𝔤𝔩(N)) and Gelfand-Tsetlin representation

In the previous section we outline the construction of the Gelfand-Tsetlin parameterization of the coadjoint orbit of GL(N, ℝ) . The polynomials a_{n}(λ), b_{n}(λ), and c_{n}(λ) that appear in the invariant formulation of this parameterization bear obvious similarity to the basic ingredients of Drinfeld’s “new realization” of Yangian. In this section we systematically use the Yangian relations to derive Gelfand-Tsetlin representation starting with the representation theory of Yangian Y (𝔤𝔩(N)) . ^{8} It appears that this approach provides us with a new interpretation of the QISM to the solution of the open Toda chain [20].

We recall some well known facts about the Yangian Y (𝔤𝔩(N)) [29]. The Yangian is an associative Hopf algebra generated by the elements T_{ij}^{(r)} where i, j = 1, …, N and r = 0, …, ∞ subject the following relations. Consider an N × N matrix ||T_{ij}(λ)||ij=1^{N} with the operator valued entries

T_{ij}(λ) = δ_{ij}λ + ∑_{r=0}^{∞} T_{ij}^{(r)}λ^{−r}.  (4.1)

Let

R_{N}(λ) = I ⊗ I + iℏP/λ ;  P_{ik,jl} = δ_{il}δ_{kj}  (4.2)

be an N^{2} × N^{2} numerical matrix (the Yang R -matrix). Then the relations between the generators T_{ij}^{(r)} can be written in standard form:

R_{N}(λ − μ)(T (λ) ⊗ I)(I ⊗ T (μ)) = (T (μ) ⊗ I)(T (λ) ⊗ I)R_{N}(λ − μ).  (4.3)

Let ||X_{ij}(λ)||_{i,j=1}^{n} be a n × n sub-matrix of the matrix ||T_{ij}(λ)||i,j=1^{N}. It is obvious from the explicit form of R_{N}(λ) that this sub-matrix satisfies RT T relations with the corresponding n^{2} × n^{2} matrix R_{n}(λ) . The center of the Yangian is generated by the coefficients of the following polynomial (quantum determinant of T_{ij}(λ) ):

\begin{matrix}det_{q}T (λ) \\ = ∑_{p∈P_{n}} sign p T_{p(1),1}(λ − iℏρ_{1}^{(n)}) … T_{p(k),k}(λ − iℏρ_{k}^{(n)}) … T_{p(n),n}(λ − iℏρ_{n}^{(n)}) . \end{matrix}  (4.4)

The following way to describe the Yangian Y (𝔤𝔩(N))) was introduced in [29]. Let A_{n}(λ) ( n = 1, …, N ) be the quantum determinants of sub-matrices, consisting of the first n rows and columns, while the operators B_{n}(λ), C_{n}(λ) be the quantum determinants of submatrices with the elements T_{ij}(λ) , where i = 1, …, n; j = 1, …, n − 1, n + 1 and i = 1, …, n−1, n+1 ; j = 1, …, n respectively. Finally, let D_{n}(λ) be the quantum determinant of sub-matrix with the elements T_{ij}(λ) with i = 1, …, n − 1, n + 1; j = 1, …, n − 1, n + 1 . The expansion coefficients of A_{n}(λ), B_{n}(λ), C_{n}(λ), D_{n}(λ) ( n = 1, …, N −1 ) with respect to λ together with those of A_{N}(λ) generate the algebra Y (𝔤𝔩(N)) . The whole set of relations between A_{n}(λ) , B_{n}(λ) , C_{n}(λ) and D_{n}(λ) could be found in [32]. The part which is relevant for what follows is given by the relations

\begin{matrix}[A_{n}(λ), A_{m}(μ)] = 0 ;  (n, m = 1, …, N), \\ [B_{n}(λ), B_{n}(μ)] = 0 ; [C_{n}(λ), C_{m}(μ)] = 0 ;  (m ≠ n ± 1), \\ (λ − μ + iℏ)A_{n}(λ)B_{n}(μ) = (λ − μ)B_{n}(μ)A_{n}(λ) + iℏA_{n}(μ)B_{n}(λ), \\ (λ − μ + iℏ)A_{n}(μ)C_{n}(λ) = (λ − μ)C_{n}(λ)A_{n}(μ) + iℏA_{n}(λ)C_{n}(μ). \end{matrix}  (4.5)

8For the connection of finite-dimensional Gelfand-Tsetlin representation with Yangian see [30], [32].

Drinfeld’s “new realization” emerges if we consider the coefficients of the expansions with respect to λ of some simple combinations of the generators A_{n}(λ), B_{n}(λ), C_{n}(λ) [29].

There is a natural homomorphism π_{N} : Y (𝔤𝔩(N)) → U(𝔤𝔩(N)) defined by

T_{jk}(λ) = λδ_{jk} − iℏE_{jk} ,  (j, k = 1, …, N).  (4.6)

Let us describe the images of the generators A_{n}(λ), B_{n}(λ), and C_{n}(λ) explicitly. The image of A_{N}(λ) which is a polynomial of order N gives a generating function for the generators of the center Z ⊂ U(𝔤𝔩(N)) . Consider the chain of Yangians

Y (𝔤𝔩(1)) ⊂ … ⊂ Y (𝔤𝔩(N))  (4.7)

defined by the standard embeddings. Let A be the maximal commutative sub-algebra of Y (𝔤𝔩(N))) . The images of the generators of A under the homomorphism (4.6) have the form (2.17). The other generators B_{n}(λ) , C_{n}(λ) can be written as

Lemma 4.1

B_{n}(λ) = [A_{n}(λ), E_{n,n+1}],  (4.8)
C_{n}(λ) = [E_{n+1,n}, A_{n}(λ)],  (4.9)

where n = 1, …, N − 1 .

The homomorphism π_{N} : Y (𝔤𝔩(N)) → U(𝔤𝔩(N)) is really an epimorphism. The following integral formula expresses the generators of U(𝔤𝔩(N)) in terms of A_{n}(λ), B_{n}(λ), C_{n}(λ) :

\begin{matrix}E_{n,n+1} = −\frac{1}{iℏ} ∮_{∞} A_{n}^{−1}(λ)B_{n}(λ)dλ , \\ E_{n+1,n} = −\frac{1}{iℏ} ∮_{∞} C_{n}(λ)A_{n}^{−1}(λ)dλ , \\ E_{nn} = −\frac{1}{iℏ} ∮_{∞} \frac{A_{n}(λ)}{A_{n−1}(λ − \frac{iℏ}{2})} \frac{dλ}{λ} − \frac{1}{2}(n − 1) , \end{matrix}  (4.10)

where the contours of integrations are taken around ∞. These expressions may be considered as a “quantization” of the classical relations (3.18). Thus it is not surprising that Drinfeld’s construction of the “new realization” of the Yangian leads naturally to the the generalized Gelfand-Tsetlin representation discussed in sections 2,3. Consider the model for representation of the U(𝔤𝔩(N)) based on the commutative subalgebra A. In this realization all polynomial generators A_{n}(λ) act by multiplication by the polynomials depending on λ and some auxiliary variables. We restrict the representation by the condition that all roots of these polynomials are real (this is compatible with commutation relations (4.5)). Thus, in this realization we have

A_{n}(λ) = ∏_{j=1}^{n}(λ − γ_{nj}).  (4.11)

As auxiliary variables one could choose γ_{nj}. Using the commutation relations between the generators A_{n}(λ), B_{n}(λ), and C_{n}(λ) one could derive the expression for these generators in γ_{nj} - parameterization.

Proposition 4.1 The operators

\begin{matrix}A_{n}(λ) = ∏_{j=1}^{n}(λ − γ_{nj}), \\ B_{n}(λ) = ∑_{j=1}^{n} ∏_{s≠j} \frac{λ − γ_{ns}}{γ_{nj} − γ_{ns}} ∏_{r=1}^{n+1}(γ_{nj} − γ_{n+1,r} − \frac{iℏ}{2}) e^{−iℏ∂_{γ_{nj}}}, \\ C_{n}(λ) = − ∑_{j=1}^{n} ∏_{s≠j} \frac{λ − γ_{ns}}{γ_{nj} − γ_{ns}} ∏_{r=1}^{n−1}(γ_{nj} − γ_{n−1,r} + \frac{iℏ}{2}) e^{iℏ∂_{γ_{nj}}} \end{matrix}  (4.12)

satisfy the full set of the Yangian relations [32]. In particular, (4.5) hold.

The formulae (4.12) have also the “classical” counterpart (3.7), (3.10), (3.11) derived in section 2. The easy check shows that the combination of Proposition 4.1 and the integral representation (4.10) leads to the representation (2.1) of 𝔤𝔩(N) derived in section 2.

5 Application to the quantum integrable systems

5.1 The open Toda chain

The open Toda chain corresponding to 𝔤𝔩(N, ℝ)) is one of the simplest examples of the integrable quantum mechanical system with N degrees of freedom (see [35] and references therein). It has N mutually commuting Hamiltonians, the first two of which are given by

\begin{matrix}h_{1} = ∑_{j=1}^{N} p_{j} , \\ h_{2} = ∑_{j<k} p_{j}p_{k} − ∑_{j=1}^{N−1} e^{x_{j}−x_{j+1}} , \end{matrix}  (5.1)

where [x_{n}, p_{m}] = iℏδ_{nm}.

Recall the general idea behind the Representation Theory approach to the quantum integrability in the case of the open Toda chain. Let V and V′ be any dual irreducible Whittaker U(𝔤𝔩(N, ℝ)) - modules and w ∈ V and w′ ∈ V′ be the corresponding cyclic Whittaker vectors with respective characters χ_{+}(E_{n,n+1}) = −iℏ^{−1} and χ_{−}(E_{n+1,n}) = −iℏ^{−1}. We assume that the action of the Cartan subalgebra is integrated to the action of the Cartan torus so the following function is well defined

ψ_{γ_{N1},... ,γ_{NN}} = e^{−x·ρ^{(N)}}⟨w′_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}w_{N}⟩ .  (5.2)

In each irreducible representation the action of the elements of the center Z of the universal enveloping algebra U(𝔤𝔩(N, ℝ)) is proportional to the action of the unit operator. Consider the function

ψ̃_{γ_{N1},... ,γ_{NN}} = e^{−x·ρ^{(N)}}⟨w′_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}z w_{N}⟩ ,  (5.3)

where z belongs to the center Z. Using the properties of the Whittaker vectors one could show that there is a differential operator D_{z} in the variables x_{k} such that ψ̃_{γ_{N1},... ,γ_{NN}} = D_{z}ψ_{γ_{N1},... ,γ_{NN}} Thus, taking the first two Casimir operators, one gets the first two Hamiltonians (5.1). On the other hand, z is an element of the center Z and ψ̃_{γ_{N1},... ,γ_{NN}} is proportional to ψ_{γ_{N1},... ,γ_{NN}} with some numerical coefficient. Hence, we get the common eigenfunction for the set of the differential operators corresponding to the elements of the center Z.

Let us given an explicit realization of the representation of U(𝔤𝔩(N, ℝ)) in terms of some difference/differential operators which integrates to the representation of the Cartan group. This leads to the integral formula for the wave function of the open Toda chain. To obtain such a formula for GT-representation, we substitute the expressions (2.9), (2.10), and (2.1a) into (5.2) thus getting

\begin{matrix}ψ_{γ_{N1},... ,γ_{NN}} = e^{−x·ρ^{(N)}} \\ × ∫_{ℝ^{\frac{N(N−1)}{2}}} ∏_{n=1}^{N−1} \frac{∏_{k=1}^{n} ∏_{m=1}^{n+1} ℏ^{\frac{γ_{nk}−γ_{n+1,m}}{iℏ} + \frac{1}{2}} Γ(\frac{γ_{nk}−γ_{n+1,m}}{iℏ} + \frac{1}{2})}{∏_{s<p} |Γ(\frac{γ_{ns}−γ_{np}}{iℏ})|^{2}} e^{\frac{i}{ℏ} ∑_{n,j=1}^{N}(γ_{nj}−γ_{n−1,j})x_{n}} ∏_{{n=1 \atop j≤n}}^{N−1} dγ_{nj} . \end{matrix} (5.4)

In the study of the analytic properties over γ_{N} of this solution, the following reformulation is very usefull:

Proposition 5.1 An analytical continuation of the eigenfunction (5.4) as a function of γ_{N} can be expressed in the form

\begin{matrix}ψ_{γ_{N}} (x_{1}, …, x_{N}) \\ = ∫_{S} ∏_{n=1}^{N−1} \frac{∏_{k=1}^{n} ∏_{m=1}^{n+1} ℏ^{\frac{γ_{nk}−γ_{n+1,m}}{iℏ}} Γ(\frac{γ_{nk}−γ_{n+1,m}}{iℏ})}{∏_{s≠p} Γ(\frac{γ_{ns}−γ_{np}}{iℏ})} e^{\frac{i}{ℏ} ∑_{n,j=1}^{N}(γ_{nj}−γ_{n−1,j})x_{n}} ∏_{{n=1 \atop j≤n}}^{N−1} dγ_{nj} , \end{matrix}  (5.5)

where the domain of integration S is defined by conditions min_{j}{Im γ_{kj}} > max_{m}{Im γ_{k+1,m}} for all k = 1, …, N − 1 . Recall that we assume that γ_{nj} = 0 for j > n. The integral (5.5) converges absolutely.

Proof. Let us change the variables of integration in (5.4):

γ_{nj} → γ_{nj} − \frac{iℏ}{n} ∑_{s=1}^{n} ρ_{s}^{(N)} ;  (n = 1, …, N − 1) .  (5.6)

We recall that ρ_{s}^{(N)} = \frac{1}{2}(N − 2s + 1) . After elementary calculations, the integral in (5.4) acquires the form (5.5). It is worth mentioning that after the change of variables (5.6) we shift the domain of integration ℝ^{\frac{N(N−1)}{2}} to the complex plain in such a way that the domain of integration over the variables γ_{n−1,j} lies above the domain of integration over the variables γ_{nj}. Thus, we arrive at the analytic continuation of the wave function described in the theorem.

The explicit form of the eigenfunction and absolute convergence of the integral leads to recursive relation between the wave functions corresponding to 𝔤𝔩(n) and 𝔤𝔩(n − 1) ; this relation is a direct consequence of the Gelfand-Tsetlin inductive procedure:

Corollary 5.1 Let us fix the solution ψ_{γ_{11}}(x_{1}) = e^{\frac{i}{ℏ}x_{1}γ_{11}} of the GL(1, ℝ) Toda chain. For any n = 2, …, N there is the recursive relation between the wave functions ψ_{γ_{n}} and ψ_{γ_{n−1}}

\begin{matrix}ψ_{γ_{n}}(x_{1}, …, x_{n}) \\ = ∫_{S_{n}} μ^{(n−1)}(γ_{n−1})f_{n}(γ_{n}, γ_{n−1}) e^{− \frac{i}{ℏ}x_{n}( ∑_{j=1}^{n} γ_{nj}− ∑_{k=1}^{n−1} γ_{n−1,k})}ψ_{γ_{n−1}}(x_{1}, …, x_{n−1}) ∏_{j=1}^{n−1} dγ_{n−1,j}, \end{matrix} (5.7)

where

μ^{(n−1)}(γ_{n−1}) = ∏_{s≠p} [Γ(\frac{γ_{n−1,s} − γ_{n−1,p}}{iℏ} )]^{−1}

and

f_{n}(γ_{n}, γ_{n−1}) = ∏_{k=1}^{n−1} ∏_{m=1}^{n} ℏ^{\frac{γ_{n−1,k}−γnm}{iℏ}}Γ(\frac{γ_{n−1,k} − γ_{nm}}{iℏ} ).  (5.8)

The domain of integration S_{n} is defined by conditions min_{j}{Im γ_{n−1,j}} > max_{m}{Imγ_{nm}}.

The above results are in complete agreement with the recursive relations between the wave functions and the explicit solution in terms of Mellin-Barnes representation obtained previously by the QISM-based approach in [20].

5.2 The hyperbolic Sutherland model

In this section we give a short derivation of the representation through integrals for a wave function of the hyperbolic Sutherland model.

The quantum hyperbolic Sutherland model corresponding to 𝔤𝔩(N) is the integrable quantum mechanical system with N degrees of freedom ( see [2] and references therein). It has N mutually commuting Hamiltonians, the first two of which are given by

\begin{matrix}h_{1} = ∑_{n=1}^{N} p_{n} , \\ h_{2} = ∑_{m<n} \{p_{n}p_{m} + \frac{ℏ^{2}/4}{sinh^{2}(x_{m} − x_{n})}\}. \end{matrix}  (5.9)

The eigenfunction of the Hamiltonians of the quantum hyperbolic Sutherland model is defined by the equations:

∑_{n=1}^{N} p_{n} Ψ = σ_{1}(γ_{N})Ψ ,

∑_{m<n} \{p_{n}p_{m} + \frac{ℏ^{2}/4}{sinh^{2}(x_{m} − x_{n})} \}Ψ = σ_{2}(γ_{N})Ψ .

(5.10)

Similarly to the open Toda chain, the hyperbolic Sutherland model could be explicitly solved using the Representation Theory approach. Let V be a representation of U(𝔤𝔩(N, ℝ)) and v_{N} is a vector in V such that

(E_{n,n+1} − E_{n+1,n})v_{N} = 0 ,  (n = 1, …, N − 1).  (5.11)

Consider the matrix element Φ(x_{1}, …, x_{N}) of the form

Φ = ⟨v_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}v_{N}⟩ .  (5.12)

One could show that (5.12) gives (up to a simple factor) the eigenfunction of the Hamiltonian of the hyperbolic Sutherland model. To derive this let us consider the more general matrix elements:

Φ_{r} = ⟨v_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}a_{Nr}v_{N}⟩ ,  (5.13)

where a_{Nr} are the coefficients of the expansion of A_{N}(λ) (see (2.17)): A_{N}(λ) = λ^{N} − iℏλ^{N−1}a_{N1} + λ^{N−2}a_{N2} + …. The first two have the form:

\begin{matrix}a_{N1} = −iℏ ∑_{n=1}^{N} E_{nn} , \\ a_{N2} = −ℏ^{2}\{ ∑_{m<n} E_{mm}E_{nn} − ∑_{n=1}^{N} ρ_{n}^{(N)}E_{nn} − ∑_{m<n} E_{nm}E_{mn} + σ_{2}(ρ^{(N)})\}. \end{matrix}  (5.14)

This should be compared with the discussion of the open Toda chain (5.3). The same line of reasoning shows that the matrix element (5.12) satisfies the differential equations

iℏ ∑_{n=1}^{N} ∂_{x_{n}}Φ = σ_{1}(γ_{N})Φ ,

−ℏ^{2} ∑_{m<n} \{∂_{x_{m}}∂_{x_{n}} − \frac{1}{2} coth(x_{m} − x_{n})(∂_{x_{m}} − ∂_{x_{n}})\}Φ = (σ_{2}(γ_{N}) + ℏ^{2}σ_{2}(ρ^{(N)}))Φ .

(5.15)

Finally, the wave function Ψ defined by

Ψ := Φ ∏_{j<k} sinh^{1/2}(x^{j} − x^{k})  (5.16)

satisfies the equations (5.10).

Now we give explicit integral formula for the eigenfunction of the hyperbolic Sutherland model using the GT representation discussed in section 2. First, let us find the vector v_{N} satisfying (5.11).

Lemma 5.1 Let

ϕ_{n}(γ_{n}, γ_{n+1}) = ∏_{k=1}^{n} ∏_{m=1}^{n+1} ℏ^{\frac{γ_{nk}−γ_{n+1,m}}{2iℏ} + \frac{1}{4}} Γ(\frac{γ_{nk} − γ_{n+1,m}}{2iℏ} + \frac{1}{4}).  (5.17)

Then the vector

v_{N} = e^{− \frac{π}{2ℏ} ∑_{n=1}^{N−1}(n−1) ∑_{j=1}^{n} γ_{nj}} ∏_{n=1}^{N−1} ϕ_{n}(γ_{n}, γ_{n+1})  (5.18)

satisfies the equations (5.11).

Proof. Straightforward check.

Now we have the following

Proposition 5.2 The solution to (5.10) admits the integral representation:

\begin{matrix}Ψ_{γ_{N1},... ,γ_{NN}} (x_{1}, …, x_{N}) = ∏_{j<k} sinh^{1/2}(x_{j} − x_{k}) \\ × ∫_{ℝ^{\frac{N(N−1)}{2}}} ∏_{n=1}^{N−1} \frac{∏_{k=1}^{n} ∏_{m=1}^{n+1} |Γ(\frac{γ_{nk}−γ_{n+1,m}}{2iℏ} + \frac{1}{4})|^{2}}{∏_{s<p} |Γ(\frac{γ_{ns}−γ_{np}}{iℏ})|^{2}} e^{\frac{i}{ℏ} ∑_{n,j=1}^{N}(γ_{nj}−γ_{n−1,j})x_{n}} ∏_{{n=1 \atop j≤n}}^{N−1} dγ_{nj}. \end{matrix} (5.19)

Proof. The only non obvious thing is a convergence of the integral. The integral converges absolutely due to the inequality (2.23).

6 Connection with QISM-based approach

In section 5 we have outlined the derivation of the integral representation for the wave function of open Toda chain based on the GT representation of U(𝔤𝔩(N, ℝ)) . As was stressed in section 4, this representation has a direct relation to Y (𝔤𝔩(N, ℝ)) . This provides us with another point of view on the way of obtaining the wave functions for the the open Toda chain. In particular, the algebra generated by the images of the polynomials A_{n}(λ) , B_{n}(λ) , C_{n}(λ) under homomorphism π_{N} : Y (𝔤𝔩(N)) → U(𝔤𝔩(N)) could be successfully used to derive directly the recursive relations for the wave functions.

We begin with a short summary of QISM-based approach to GL(N, ℝ) Toda chain due to Sklyanin [18]. The model is described with the help of auxiliary 2×2 matrix Lax operator with spectral parameter:

L_{n}(λ) = ( \begin{matrix}λ − p_{n} & e^{−x_{n}} \\ −e^{x_{n}} & 0 \end{matrix} ) .  (6.1)

The operator (6.1) satisfies the usual quadratic relation

R_{2}(λ − μ)(L_{n}(λ)) ⊗ I)(I ⊗ L_{n}(μ)) = (I ⊗ L_{n}(μ))(L_{n}(λ) ⊗ I)R_{2}(λ − μ)  (6.2)

with the rational 4 × 4 R -matrix (see (4.2)). The monodromy matrix

T_{n}(λ) = L_{n}(λ) … L_{1}(λ) ≡ ( \begin{matrix}A_{n}(λ) & B_{n}(λ) \\ C_{n}(λ) & D_{n}(λ) \end{matrix} )  (6.3)

also satisfies the relation:

R_{2}(λ − μ)(T_{n}(λ) ⊗ I)(I ⊗ T_{n}(μ)) = (I ⊗ T_{n}(μ))(T_{n}(λ) ⊗ I)R_{2}(λ − μ).  (6.4)

In particular, the following commutation relations hold:

\begin{matrix}[A_{n}(λ), A_{n}(μ)] = [B_{n}(λ), B_{n}(μ)] = [C_{n}(λ), C_{n}(μ)] = 0, \\ (λ − μ + iℏ)A_{n}(μ)C_{n}(λ) = (λ − μ)C_{n}(λ)A_{n}(μ) + iℏA_{n}(λ)C_{n}(μ) , \\ (λ − μ + iℏ)A_{n}(λ)B_{n}(μ) = (λ − μ)B_{n}(μ)A_{n}(λ) + iℏA_{n}(μ)B_{n}(λ). \end{matrix}  (6.5)

The polynomial A_{n}(λ) is a generating function for the quantum Hamiltonians of GL(n, ℝ) Toda chain and the wave functions ψ_{γ_{n}}(x_{1}, …, x_{n}), n = 1, …, N are defined as solutions of the differential equations

A_{n}(λ)ψ_{γ_{n1},... ,γ_{nn}} = ∏_{j=1}^{n}(λ − γ_{nj}) ψ_{γ_{n1},... ,γ_{nn}} .  (6.6)

The idea of the QISM approach is to solve iteratively these equations for ψ_{γ_{n}}, n = 1, …, N using the additional operators B_{n}(λ) and C_{n}(λ) . ¿From the results of [20] the following expressions for the operators A_{n}(λ), B_{n}(λ), and C_{n}(λ) in γ -representation can be obtained:

\begin{matrix}A_{n}(λ) = ∏_{j=1}^{n}(λ − γ_{nj}) , \\ B_{n}(λ) = i^{n−1} ∑_{j=1}^{n} ∏_{s≠j} \frac{λ − γ_{ns}}{γ_{nj} − γ_{ns}} e^{−iℏ∂_{γ_{nj}}} , \\ C_{n}(λ) = i^{−n−1} ∑_{j=1}^{n} ∏_{s≠j} \frac{λ − γ_{ns}}{γ_{nj} − γ_{ns}} e^{iℏ∂_{γ_{nj}}} . \end{matrix}  (6.7)

The simple check shows that thus defined operators (6.7) satisfy the relations (6.5). Hence, one establishes the connection between the generators A_{n}(λ), B_{n}(λ) , and C_{n}(λ) entering the description of the GT representation (4.12) and the corresponding QISM generators (6.7):

\begin{matrix}A_{n}(λ) = A_{n}(λ) , \\ B_{n}(λ) = g_{n}^{−1} ∘ B_{n}(λ) ∘ g_{n} , \\ C_{n}(λ) = g_{n−1}^{−1} ∘ C_{n}(λ) ∘ g_{n−1} , \end{matrix}  (6.8)

where g_{n} is defined by (2.10).

At the end of the section we illustrate the connection between the differential operator A_{n}(λ) entering in (6.3) and generator A_{n}(λ) defined by (2.17). First, let us note that (6.1), (6.3) define the relations between the operators A_{n}(λ) for different levels:

A_{n}(λ) = (λ−p_{n})A_{n−1}(λ) − e^{x_{n−1}−x_{n}}A_{n−2}(λ) ,  (6.9)

where p_{n} = −iℏ∂_{x_{n}} .

Lemma 6.1 The operators A_{n}(λ) and A_{n}(λ) are related as

A_{n}(λ)ψ_{γ_{N1},... ,γ_{NN}} = e^{−x·ρ^{(N)}}⟨w′_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}A_{n}(λ − \frac{i(N−n)ℏ}{2})w_{N}⟩.  (6.10)

Proof. We prove the statement by showing that the operators A_{n}(λ) defined by (6.10) satisfy the recursive relations (6.9). Consider the relation (2.19). The generators E_{nn} and E_{n,n−k+1} commute with A_{n−1}(λ) and A_{n−k}(λ) respectively. Therefore, due to (2.27) and (2.7)

\begin{matrix}⟨w′_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}A_{n}(λ − \frac{i(N−n)ℏ}{2})w_{N}⟩ \\ = (λ − iℏρ_{n}^{(N)} + iℏ∂_{x_{n}})⟨w′_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}A_{n−1}(λ − \frac{i(N−n+1)ℏ}{2})w_{N}⟩ \\ − e^{x_{n−1}−x_{n}}⟨w′_{N}, e^{− ∑_{k=1}^{N} x_{k}E_{kk}}A_{n−2}(λ − \frac{i(N−n+2)ℏ}{2})w_{N}⟩. \end{matrix}  (6.11)

We leave a more systematic discussion of the interrelation between two sets of operators A_{n}(λ) , B_{n}(λ) , C_{n}(λ) and A_{n}(λ) , B_{n}(λ) , C_{n}(λ) for the future publication.

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