From Quantum $A_N$ to $E_8$ Trigonometric Model: Space-of-Orbits View

A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for $(A-B-C{-D)$-models, both rational and trigonometric, is related to the universal enveloping algebra $U_{gl_n}$. For the exceptional $(G-F-E)$-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with $BC_1\equiv(\mathbb{Z}_2)\oplus T$ symmetry. In particular, the $BC_1$ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra $sl(2)\oplus sl(2)$.


Introduction
In this article we attempt to overview our constructive knowledge of (quasi)-exactly-solvable potentials having the form of a meromorphic function in trigonometric variables. Any model with such a potential is characterized by a discrete symmetry group, and possesses an (in)finite set of polynomial eigenfunctions in a certain trigonometric variables. In the case of exactlysolvable potentials an infinite discrete spectra is quadratic in the quantum numbers. All of these models are characterized by the appearance of a hidden (Lie) algebraic structure. They do not admit a separation of variables, they are completely-integrable and possess a commutative algebra of integrals. So far, no super-integrable models with trigonometric potentials are known, although all of them admit at least one particular integral [22].
A similar overview of the rational models (with a potential in the form of a meromorphic function in Cartesian coordinates) was given in [21]. Unlike the trigonometric models the rational models do admit a separation out of radial coordinate and hence, the emergence of integral of the second order leading to super-integrability. For exactly-solvable rational models their , (1.2) in the exactly-solvable case, where R + is a set of positive roots in the root space ∆ of dimension N , β is a parameter and µ |α| are coupling constants which depend on the root length. For roots of the same length the constants ν |α| are equal. Thus, the potential in (1.2) is a superposition of the Weyl-invariant functions, each defined as a sum over roots of the same length. The configuration space is the Weyl alcove. The ground state wave function has a form (1. 3) The ground state energy has a form E 0 = β 2 0 (ν) and is known explicitly. Let us take the Hamiltonian in R d which is symmetric with respect to the (maximal) discrete group G. One can construct invariants of G using a procedure of averaging some function over orbit(s). The d linearly independent invariants span a linear space called the space of orbits. These invariants are generating elements of the algebra of invariants. The main idea of this paper is to study the Hamiltonian in a space of orbits (space of invariants). Technically, it implies a change of variables from the original coordinates to the invariants. Conceptually, it means factoring out the discrete symmetry of the problem. It reveals a "primary" operator of the system which being dressed by the discrete symmetry becomes the Hamiltonian.
We consider some models from the list of ones known so far.

Generalities
Many years ago, as the state-of-the-art, Sutherland found a many-body exactly-solvable and integrable Hamiltonian with trigonometric potential [18]. A few years later the Hamiltonian reduction method was introduced (for review and references see e.g. Olshanetsky-Perelomov [13]). In this method an extended family of integrable and exactly-solvable Hamiltonians with trigonometric potentials, associated with affine Weyl (Coxeter) symmetry, was found. The Sutherland model appeared as one of its representatives, the A N trigonometric model. The idea of the Hamiltonian reduction method is beautiful: • Take a simple group G.
• Define the Laplace-Beltrami (invariant) operator on its symmetric space (free motion).
• Radial part of the Laplace-Beltrami operator is the Olshanetsky-Perelomov Hamiltonian relevant from physical point of view. The emerging Hamiltonian is affine Weyl-symmetric, it can be associated with root system, it is integrable with integrals given by the invariant operators of order higher than two with a property of solvability.
Trigonometric case. This case appears when the coordinates of the symmetric space are introduced in such a way that a negative-curvature surface occurs. Emerging the Calogero-Moser-Sutherland-Olshanetsky-Perelomov Hamiltonian in the Cartesian coordinates has the form (1.2) with the ground state given by (1.3). In the Hamiltonian reduction, the parameters ν |α| of the Hamiltonian take a set of discrete values, however, they can also be generalized to any real value without loosing the property of integrability as well as solvability with the only constraint being the existence of L 2 -solutions of the corresponding Schrödinger equation. The configuration space for (1.2) is the Weyl alcove.
The Hamiltonian (1.2) is completely-integrable: there exists a commutative algebra of integrals (including the Hamiltonian) of dimension which is equal to the dimension of the configuration space (for integrals, see Oshima [14] with explicit forms of those). The Hamiltonian (1.2) is invariant with respect to the affine Weyl (Coxeter) group transformation, which is the discrete symmetry group of the corresponding root space, see e.g. [13].
The Hamiltonian (1.2) has a hidden (Lie)-algebraic structure. In order to reveal it (see [1,2,3,4,12,15,16]) we need to • Gauge away the ground state eigenfunction making a similarity transformation • If the state-of-art variables are introduced for trigonometric models A N , BC N , G 2 and F 4 (see [1,4,15,16], respectively), the Hamiltonian h becomes algebraic, however, • It can be checked, which, in fact, looks evident, that parameterizing the space of orbits of the Weyl (Coxeter) group by taking the Weyl (Coxeter) fundamental trigonometric invariants, where Ω a is an orbit generated by fundamental weight w a , a = 1, 2, . . . , N (N is the rank of the root system); y is N -dimensional auxiliary vector which defines the Cartesian coordinates, as coordinates we arrive at the conclusion that the state-of-the-art variables [1,4,15,16]  From physical point of view, the expression (2.1) is a Weyl-invariant non-linear superposition of plane waves with momenta proportional to β.
The fundamental trigonometric invariants τ (β) taken as coordinates always lead to the gaugerotated trigonometric Hamiltonian h in a form of algebraic differential operator with polynomial coefficients. It is proved by demonstration. It is worth emphasizing a surprising fact that the period(s) of the invariants τ (β) is half of the period(s) of the Hamiltonian (1.2) and the ground state function (1.3). It seems correct (this can be proved by demonstration) that the original Hamiltonian H (1.2) written in terms of the fundamental trigonometric invariants τ (β) takes the form is the Laplace-Beltrami operator with a metric g ij (τ ) with polynomial in τ matrix elements, hence with polynomial in τ coefficient functions in front of the second derivatives, and with the property that the coefficient functions in front of the first derivatives are also polynomials in τ ; V (τ ) is a rational function, see below e.g. (2.7), (2.21). The form (2.2) can be called the rational form of the trigonometric model. It is evident that the similar rational form (2.2) appears for rational models when polynomial Weyl invariants are used as new coordinates to parameterize the space of orbits. In turn, the gauge-rotated Hamiltonian h in τ -variables takes the form where C a (τ ) are polynomials in τ , see below e.g. (2.9), (2.27). The same representation is valid for the rational models.

A 1 /BC 1 case or trigonometric Pöschl-Teller potential
The BC 1 trigonometric Hamiltonian reads 1 where β, g 2 , g 3 are parameters. Symmetry: As for the configuration space, it can be taken the interval [0, π β ]. If g 2 = 0 the interval can be extended to [0, 2π β ]. At g 3 = 0 (or g 2 = 0) the Hamiltonian (2.3) degenerates to the A 1 trigonometric Hamiltonian, which describes the relative motion of two particle system on a line.
The BC 1 trigonometric Hamiltonian (2.3) is easily related to the trigonometric Pöschl-Teller (PT) Hamiltonian Replacing in (2.5) β → iβ, we arrive at the general hyperbolic Pöschl-Teller Hamiltonian while the one-soliton Hamiltonian appears at α 2 = 1 4 . In the case of the BC 1 trigonometric Hamiltonian under the replacement β → iβ the BC 1 Hyperbolic Hamiltonian occurs.
Let us introduce a new variable ). It appears that with amazingly simple meromorphic potential, where is the flat Laplace-Beltrami operator with metric g 11 = τ 2 −1 . Overall multiplicative factor β 2 in ( (2.8) At ν 2 = 1 and ν 3 = 0 it coincides to the Jacobian. Now let us make a gauge rotation with Ψ 0 given by (2.4) and write the result in the variable τ . After a simple calculations it reads which is the algebraic form of the BC 1 Hamiltonian (2.3). Its eigenvalues are p = p 2 + (2ν 2 + ν 3 )p, p = 0, 1, 2, . . . , (2.10) being quadratic in quantum number p, while the eigenfunctions are the Jacobi polynomials, ϕ p = P (ν 2 +ν 3 − 1 2 ,ν 2 − 1 2 ) p (τ ). Eventually, the explicit form of an eigenfunction of the Hamiltonian (2.3) is It can be easily checked that the gauge-rotated Hamiltonian h BC 1 (τ ) has infinitely many finite-dimensional invariant subspaces P n = τ p |0 ≤ p 1 ≤ n , n = 0, 1, 2, . . . , (2.11) hence, the infinite flag P, with the characteristic vector f = (1) (see below), is preserved by h BC 1 . Thus, the eigenfunctions of h BC 1 are elements of the flag P. Any subspace P n contains (n + 1) eigenfunctions which is equal to dim P n . Take the algebra gl 2 in (n + 1)-dimensional representation realized by the first order differential operators where n = 0, 1, . . . and T 0 is the central element. Its finite-dimensional representation space is the space of polynomials P n (2.11). Hence, the finite-dimensional invariant subspaces of the Hamiltonian h BC 1 coincide with the finite-dimensional representation spaces of gl 2 (2.12) for n = 0, 1, 2, . . .. It immediately implies that the algebra gl 2 is the hidden algebra of the BC 1 trigonometric Hamiltonian -it can be written in terms of gl 2 generators (2.12) Thus, the Hamiltonian h BC 1 is an element of the universal enveloping algebra U gl 2 .
Among the generators of the algebra gl 2 (2.12) there is the Euler-Cartan operator, which has zero grading; it maps a monomial in τ to itself. It defines the highest weight vector. This generator allows us to construct a particular integral -π-integral of zero grading of the (n + 1)th order (see [22]) i J 0 n + j , (2.14) Making the gauge rotation of the π-integral (2.14) with Ψ −1 0 (τ ) given by (2.4) and changing variables τ back to the Cartesian coordinate we arrive at the quantum π-integral acting in the Hilbert space, Under such a gauge transformation the triangular space of polynomials P n becomes the space par,BC 1 (x) over this space Any eigenfunction Ψ ∈ V n is zero mode of the π-integral I (n) par,BC 1 (x). It is worth noting a connection of the BC 1 trigonometric model with the so-called Tremblay-Turbiner-Winternitz (TTW) model [19] and, in particular, with the I 2 (k) rational model (see e.g. [21]). In order to see it let us combine the BC 1 trigonometric Hamiltonian H BC 1 (φ) (2.3) as the angular part and the radial part of two-dimensional spherical-symmetrical harmonic oscillator Hamiltonian as the radial part forming the 2D Hamiltonian which is nothing but the Hamiltonian of the TTW model [19]. If β = k is integer, this Hamiltonian corresponds to the I 2 (k) rational model [13]. Since the both Hamiltonians are exactly-solvable, the TTW model is also exactly-solvable but with spectra of two-dimensional anisotropic (!) harmonic oscillator with frequency ratio 1 : β. Any eigenfunction of (2.15) has the form of a polynomial p(r 2 , cos(βφ)) in variables r 2 and cos(βφ) multiplied by a ground state function namely, If in the construction (2.15) instead of two-dimensional radial harmonic oscillator, the radial Hamiltonian of the sextic QES 2D central potential (see e.g. [23]) is taken, the quasi-exactlysolvable extension of the TTW model occurs [19] H (qes) cf. (2.15), here n is non-negative integer and a > 0 is a parameter. In this Hamiltonian a finite number of eigenstates can be found explicitly (algebraically). Their eigenfunctions have the form of a polynomial p(r 2 , cos(βφ)) of degree n in r 2 multiplied by a factor Ψ (qes,TTW) 0 The factor (2.18) is the ground state eigenfunction of the Hamiltonian (2.17) at n = 0. If β is equal to non-negative integer k, a polynomial p(r 2 , cos(βφ)) belongs to the space P (1,k) with the characteristic vector f = (1, k), see below.
2.3 Quasi-exactly-solvable BC 1 case (or QES trigonometric Pöschl-Teller potential) The Hamiltonian h BC 1 (τ ) (2.9) is gl(2)-Lie-algebraic operator (2.13) which has infinitely-many finite-dimensional invariant subspaces in polynomials (2.11). By adding to h BC 1 (τ ) (2.9) the operator where b is a parameter and n is non-negative integer, as a result we get the operator h (qes) which has a single finite-dimensional invariant subspace of the dimension (n + 1). Hence, this operator is quasi-exactly-solvable -it can be written in terms of gl 2 generators in (n + 1)-dimensional representation (2.12), h (qes) Making the gauge rotation of (2.19) with and the change of variable τ = cos(βx) we arrive at the BC 1 -trigonometric QES Hamiltonian [23] H (qes) 3), where b, ν 2 , ν 3 , β are parameters, n is non-negative integer. In τ -variable (2.6) the BC 1 -trigonometric QES Hamiltonian appears in rational form In the Hamiltonian (2.20) the (n + 1) eigenfunctions are of a form where P n (τ ) is a polynomial of degree n, they can be found by algebraic means. It is evident that i (n) par (τ ) (2.14) remains the particular integral -π-integral of the BC 1 -trigonometric QES Hamiltonian (2.19) (see [22]) Interestingly, the BC 1 -trigonometric QES Hamiltonian (2.20) degenerates to the so-called Magnus-Winkler (MW) Hamiltonian or, in other words, to the QES Lame Hamiltonian (see e.g. [23]) where ν = 0, 1. For ν = 0 and given n there exist two families of eigenfunctions which correspond to periodic (anti-periodic) boundary conditions, correspondingly. These eigenfunctions describe lower (upper) edges of Brillouin zones, respectively. Polynomial factors in ϕ respectively (see (2.12)). For ν = 1 and given n there also exist two families of eigenfunctions respectively (see (2.12)). If in a construction (2.15) to obtain the TTW model we replace the BC 1 -trigonometric a new quasi-exactly-solvable extension of the TTW model is obtained If in the construction (2.15) instead of two-dimensional radial harmonic oscillator, the radial Hamiltonian of the sextic QES 2D radial potential [23] is taken and the BC 1 -trigonometric where n, m is non-negative integer and a > 0, b are parameters. In this Hamiltonian a finite number of eigenstates can be found explicitly (algebraically). Their eigenfunctions have the form of a polynomial p r 2 , cos(βφ) of degree n in r 2 and of degree m in cos(βx) multiplied by a factor Ψ (qes,TTW) 0 The factor (2.23) becomes the ground state eigenfunction of the Hamiltonian (2.22) at n = m = 0.

Case
This is the celebrated Sutherland model (A N −1 trigonometric model) which was found in [18]. It describes N identical particles on a circle (see Fig. 1 where h is the horde. The Hamiltonian is where g is the coupling constant and β is a parameter. The symmetry of the system is S N ⊕T ⊕Z 2 (permutations x i → x j , translation x i → x i + 2π/β and all x i → −x i ). The ground state of the Hamiltonian (2.24) reads 1.3)). Let us make the gauge rotation where E 0 is the ground state energy. Then introduce center-of-mass variables y i = 0, and then new permutationally-symmetric, translationally-invariant, periodic relative variables [16] ( are elementary symmetric polynomials, and The ground state function (2.25) in τ -variables takes a form of a polynomial in some power, e.g.

2.4.1
The gl d+1 -algebra acting by 1st order dif ferential operators in R d It can be checked by the direct calculation that the gl d+1 algebra realized by the first-order differential operators acting in R d in the representation given by the Young tableaux as a row (n, 0, 0, . . . , 0 where n is an arbitrary number. The total number of generators is (d + 1) 2 . If n takes the integer values, n = 0, 1, 2, . . ., the finite-dimensional irreps occur (2.28)). It is a common invariant subspace for (2.29). The spaces P (d) n at n = 0, 1, 2, . . . can be ordered Such a nested construction is called infinite flag (filtration) P (d) . It is worth noting that the flag P (d) is made out of finite-dimensional irreducible representation spaces P (d) n of the algebra gl d+1 taken in realization (2.29). It is evident that any operator made out of generators (2.29) has finite-dimensional invariant subspace which is finite-dimensional irreducible representation space.

Algebraic properties of the Sutherland model
It seems evident that the Hamiltonian (2.27) has to have a representation as a second order polynomial in generators (2.29 where the raising generators J + i are absent. Thus, gl(N ) (or, strictly speaking, its maximal affine subalgebra) is the hidden algebra of the N -body Sutherland model. Hence, h Suth is an element of the universal enveloping algebra U gl(N ) . The eigenfunctions of the N -body Sutherland model are elements of the flag of polynomials P (N −1) . Each subspace P (N −1) n is represented by the Newton polytope (pyramid). It contains C N −1 n+N −1 eigenfunctions, which is equal to the volume of the Newton polytope. They are orthogonal with respect to Ψ 2 0 , see (2.25). The Hamiltonian (2.24) is completely-integrable: there exists a commutative algebra of integrals (including the Hamiltonian and the momentum of the center-of-mass motion) of dimension N which is equal to the dimension of the configuration space (for integrals, see Oshima [14] with explicit forms of those). Each integral I k has a form polynomial in momentum of degree k ≤ N . Making gauge rotation with Ψ 2 0 , separating center-of-mass motion and changing variable to (2.26) any integral appears in a form differential operator with polynomial coefficients. Evidently, it preserves the flag of polynomials (2.30) and can be written as a non-linear combination of the generators (2.29) at d = N − 1 from its affine subalgebra. The explicit formulae of integrals in (2.29) are unknown. The spectra of the integral which is a polynomial in momentum of degree k is given by a polynomial in quantum numbers of the degree k. All eigenfunctions of the integrals are common.
Among the generators of the hidden algebra there is the Euler-Cartan operator, see (2.29), which has zero grading and plays a role of constant acting as identity operator on a monomial in τ . It defines the highest weight vector. This generator allows us to construct the particular integral -π-integral of zero grading (see [22]) Making the gauge rotation of the π-integral (2.31) with Ψ −1 0 (τ ) given by (2.25) and changing variables τ (see (2.26)) back to the Cartesian coordinates we arrive at the quantum π-integral, It is a differential operator of the (n + 1)th order. Under such a gauge transformation the triangular space of polynomials P

Case BC N
The BC N -Trigonometric model is defined by the Hamiltonian, where β, g, g 2 , g 3 are parameters. Symmetry: cf. (1.3), where ν, ν 2 , ν 3 are found from the relations Any eigenfunction has a form Ψ 0 ϕ, where ϕ is a polynomial in the C N fundamental trigonometric invariants τ (β) (2.1). Hence, Ψ 0 plays a role of multiplicative factor. The BC N Hamiltonian (2.32) degenerates to the B N Hamiltonian at g 2 = 0, to the C N Hamiltonian at g 3 = 0 and to the D N Hamiltonian at g 2 = g 3 = 0. For the B N Hamiltonian there exist two families of eigenfunctions with multiplicative factors respectively. For the D N Hamiltonian there exist three families of eigenfunctions with multiplicative factors

respectively. Let us make a gauge rotation
and then change variables [4] (x 1 , x 2 , . . . , x N ) → τ k = σ k (cos βx) k = 1, 2, . . . , N , where σ k is the elementary symmetric polynomial, τ 0 = 1 and τ k = 0 for k < 0 and k > N . It can be checked that τ k are C N trigonometric invariants with period 2π β . We arrive at [4] with coefficients cf. (2.9). This is an algebraic form of the BC N trigonometric Hamiltonian. For polynomial eigenfunctions we find the eigenvalues are where the raising generators J + i are absent. Hence, gl(N + 1) is the hidden algebra of the BC N trigonometric model, the same algebra as for the A N -rational model. The eigenfunctions of the BC N trigonometric model are elements of the flag of polynomials P (N ) . Each subspace P where ∆ g is the Laplace-Beltrami operator with a metric g ij (τ ) = A ij (see (2.36)) and V BC N (τ ) is a potential. The explicit expression for V BC 1 (τ ) is presented in (2.7) while the ground state eigenfunction Ψ (BC 1 ) 0 (τ ) is given by (2.8). The configuration space in τ coordinate is the interval, τ ∈ [−1, 1] (trigonometric case) or half-line, τ ∈ [1, ∞) (hyperbolic case). As for BC 2 case, 3 2 , and the configuration space is illustrated by Fig. 2. The Hamiltonian (2.32) is completely-integrable: there exists a commutative algebra of integrals (including the Hamiltonian) of dimension N which is equal to the dimension of the configuration space (for integrals, see Oshima [14] with explicit forms of those). Each integral I k has a form polynomial in momentum of degree 2k ≤ 2N . Making gauge rotation with Ψ 2 0 and changing variable to (2.26) any integral appears in a form differential operator with polynomial coefficients. Evidently, it preserves the flag of polynomials (2.30) and can be written as a non-linear combination of the generators (2.29) at d = N from its affine subalgebra. The explicit formulae of integrals in generators (2.29) are unknown. The spectra of the integral which is a polynomial in momentum of degree 2k is given by a polynomial in quantum numbers of the degree 2k. All eigenfunctions of the integrals are common.
It is evident that for the BC N trigonometric model there exists a particular integral -πintegral of zero grading (see [22]) Making the gauge rotation of the π-integral (2.31) with Ψ −1 0 (τ ) given by (2.33) and changing variables τ (see (2.34)) back to the Cartesian coordinates we arrive at the quantum π-integral, It is a differential operator of the (n + 1)th order.
Under such a gauge transformation the triangular space of polynomials P • Both A N -and BC N -trigonometric (and rational) models possess algebraic forms associated with preservation of the same flag of polynomials P (N ) . The flag is invariant with respect to linear transformations in space of orbits τ → τ + A. It preserves the algebraic form of Hamiltonian.
• Their Hamiltonians (as well as higher integrals) can be written in the Lie-algebraic form where Pol 2 is a polynomial of 2nd degree in generators J of the maximal affine subalgebra of the algebra b of the algebra gl N +1 in realization (2.29). Hence, gl N +1 is their hidden algebra. From this viewpoint all four models are different faces of a single model.
• Supersymmetric A N -and BC N -rational (and trigonometric) models possess algebraic forms, preserve the same flag of (super)polynomials and their hidden algebra is the superalgebra gl(N + 1|N ) (see [4]).
In a connection to flags of polynomials we introduce a notion 'characteristic vector'. Let us consider a flag made out of "triangular" linear space of polynomials where the "grades" f 's are positive integer numbers and n = 0, 1, 2, . . .. In lattice space P

Case G 2
Take the Hamiltonian where g, g 1 and β are parameters. It describes a trigonometric generalization of the rational Wolfes model of three-body interacting system or, in the Hamiltonian reduction nomenclature, the G 2 -trigonometric model [13]. The symmetry of the model is dihedral group D 6 ⊕ T . The ground state function is

Algebra g (2)
Let us consider the Lie algebra spanned by seven generators It is non-semi-simple algebra gl(2, R) R (2) (S. Lie [11, p. 767-773] at n = 0 and A. González-Lopéz et al. [9] at n = 0 (case 24)). If the parameter n in (2.41) is a non-negative integer, it has (2.40) Figure 3. Triangular diagram relating the subalgebras L, U and g 2 . P 2 (g 2 ) is a polynomial of the 2nd degree in g 2 generators. It is a generalization of the Gauss decomposition for semi-simple algebras.
as common (reducible) invariant subspace. By adding three operators to gl(2, R) R (2) (see (2.41)), the action on P n, (1,2) , all of them are differential operators of degree 2. These new generators have a property of nilpotency, and commutativity: The generators (2.41) plus (2.42) span a linear space with a property of decomposition: g (2) . Fig. 3).
The G 2 trigonometric Hamiltonian admits the integral in a form of the 6th order differential operator [14]. After gauge rotation with Ψ 0 in variables τ 1,2 the integral has to take the algebraic form which is not known explicitly. This integral preserves the same flag P (2) (1,2) as the Hamiltonian (2.39). It can be rewritten in term of generators of the algebra g (2) . In addition to it, there exists π-integral of zero grading (see [22]) n,(1,2) → 0.
Making the gauge rotation of the π-integral (2.31) with Ψ −1 0 (τ ) given by (2.33) and changing variables τ (see (2.34)) back to the Cartesian coordinates we arrive at the quantum π-integral, It is a differential operator of the (n + 1)th order. Under such a gauge transformation the triangular space of polynomials P (2) n, (1,2) becomes the space n, (1,2) .
The Hamiltonian H G 2 (x) commutes with I (n) par,G 2 (x) over this space Any eigenfunction Ψ ∈ V (N ) n is zero mode of the π-integral I (n) par,G 2 (x). Summarizing let us mention that in addition to the flag P (2) (1,2) the G 2 trigonometric Hamiltonian preserves two more flags: P (3,5) and P (5,9) , where their characteristic vectors (3,5) and (5,9) coincide to the Weyl vector and co-vector, respectively.

Cases F 4 and E 6,7
These three cases are described in some details in [2,12] and in [3, p. 1416], respectively.

Case E 8 (in brief )
In this Section a brief description of E 8 trigonometric case is given, all details can be found in [3].
The E 8 trigonometric model is completely-integrable -there exist seven algebraically independent mutually commuting differential operators of finite order that commute with the Hamiltonian (2.43) [10,13]. We are not aware on the existence of their explicit forms. It seems evident that any of these integrals after the gauge rotation with the ground state function Ψ 0 the space of orbits should take an algebraic form of a differential operator with polynomial coefficient functions. Any integral as well as the Hamiltonian is an element of the algebra e (8) . In addition to "global" integrals, there exists π-integral of zero grading (see [22])  given by

Conclusions
• For trigonometric Hamiltonians for all classical A N , BC N , B N , C N , D N and for exceptional root spaces G 2 , F 4 , E 6,7,8 , similar to the rational Hamiltonians including noncrystallographic H 3,4 , I 2 (k) (see [21]), there exists an algebraic form after gauging away the ground state eigenfunction, and changing variables from Cartesian to fundamental trigonometric Weyl invariants (see [1,2,3,4,12,15,16]). Their eigenfunctions are polynomials in these variables. They are orthogonal with respect to the squared ground state eigenfunction.
Coefficient functions in front of the second derivatives of these gauge-rotated Hamiltonians which are polynomials in fundamental trigonometric Weyl invariants define a metric A of flat space in the space of orbits. We will call this metric the V.I. Arnold metric, he was the first to calculate a similar metric in the case of polynomial Weyl invariants. This metric has a property that in the Laplace-Beltrami operator the coefficient functions in front of the first derivatives are polynomials in fundamental trigonometric invariants. This property is similar to one which occurs in the case of rational models. The (rational) Arnold metric for the space of orbits parameterized by polynomial Weyl invariants can be considered as an appropriate degeneration of the (trigonometric) Arnold metric for the space of orbits parameterized by fundamental trigonometric Weyl invariants.  • Any trigonometric Hamiltonian is characterized by a hidden algebra. These hidden algebras are U gl(N +1) for the case of classical A N , BC N , B N , C N , D N and new infinitedimensional but finite-generated algebras of differential operators for all other cases. All these algebras have finite-dimensional invariant subspace(s) in polynomials. Rational Hamiltonians are characterized by the same hidden algebra with a single exception of the E 8 case.
• The generating elements of any such hidden algebra can be grouped into an even number of (conjugated) Abelian algebras L i , U i and one Lie algebra B. They obey a (generalized) Gauss decomposition rule (see Fig. 5). A study and a description of all these algebras is in progress and will be given elsewhere.
• Any algebraic Hamiltonian h of a trigonometric model preserves one or several flags of invariant subspaces with characteristic vectors given by the highest root vector, the Weyl vector and the Weyl co-vector (see Table 1). With the single exception of the E 8 case the flags for rational and trigonometric models coincide.
• The original Weyl-invariant periodic Hamiltonian (1.1) written in the fundamental trigonometric invariants (2.1) corresponds to a particle moving in flat space with (trigonometric) Arnold metric A in a rational potential, where ∆ A is the Laplace-Beltrami operator, g k , k = 1, . . . , are coupling constants, is the number of different root lengths in the root space. V k (τ ) are rational functions. So far, we are unaware about the explicit form of the functions V k (τ ) for all root systems except for some particular cases (see (2.7), (2.21), (2.37), (2.38)).
• The existence of an algebraic form of the Hamiltonian h of a trigonometric model allows us to construct integrable discrete systems in the space of orbits with the same hidden algebra structure, having a property of isospectrality, on uniform, exponential and mixed uniform-exponential lattices following the strategy presented in [17] (uniform lattice) and [5] (exponential lattice).
• The space of orbits formalism allowed us to show that both rational and trigonometric models for any root system are essentially algebraic: the (appropriately) gauge-rotated Hamiltonians are algebraic operators, their invariant subspaces are spaces of polynomials.
A natural question to ask is: How the elliptic Calogero-Moser systems look like in a space of orbits formalism; are they algebraic just like rational and trigonometric systems? A main obstruction to get an answer is that, in general, it is not known how to construct elliptic invariants -the invariants with respect to a "double"-affine Weyl group (the Weyl group plus two translations) -on a regular basis. However, such invariants can be constructed explicitly for two particular root systems: A 1 /BC 1 [24] and BC 2 [20]. It can be shown that the corresponding elliptic systems are algebraic.