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MACDONALD POLYNOMIALS FROM SKLYANIN ALGEBRAS:

arXiv:hep-th/9110066v1 24 Oct 1991October 1991EFI 91-43MACDONALD POLYNOMIALS FROM SKLYANIN ALGEBRAS:A CONCEPTUAL BASIS FOR THE p-ADICS-QUANTUM GROUPCONNECTION1Peter G. O. FreundEnrico Fermi Institute and Department of PhysicsUniversity of Chicago, Chicago, IL 60637andAnton V. ZabrodinInstitute of Chemical PhysicsKosygina Str. 4, SU-117334, Moscow USSRABSTRACTWe establish a previously conjectured connection between p-adicsand quantum groups.

We find in Sklyanin’s two parameter ellipticquantum algebra and its generalizations, the conceptual basis forthe Macdonald polynomials, which “interpolate” between the zonalspherical functions of related real and p-adic symmetric spaces. Theelliptic quantum algebras underlie the Zn-Baxter models.

We showthat in the n →∞limit, the Jost function for the scattering of firstlevel excitations in the Zn-Baxter model coincides with the Harish-Chandra-like c-function constructed from the Macdonald polynomi-als associated to the root system A1. The partition function of theZ2-Baxter model itself is also expressed in terms of this Macdonald--Harish-Chandra c-function, albeit in a less simple way.

We relatethe two parameters q and t of the Macdonald polynomials to theanisotropy and modular parameters of the Baxter model. In partic-ular the p-adic “regimes” in the Macdonald polynomials correspondto a discrete sequence of XXZ models.

We also discuss the possibilityof “q-deforming” Euler products.1IntroductionA connection between p-adics and quantum deformations has been noticed [1-5] in a variety of contexts over the past few years. The possibility of such a1Work supported in part by the NSF: PHY-90003861

connection emerges from work on p-adic strings [6,7] and q-strings [8,9]; fromwork on scattering on real [10,11], p-adic [4] and quantum [12] symmetric spaces;and from work on Macdonald polynomials associated to “admissible pairs” ofroot systems [1,2].All this evidence points in the direction of quantum group-like objects withtwo deformation parameters and the corresponding quantum symmetric spacesas underlying this “p-adics quantum deformation connection” [2,3,12]. Essen-tially, this is how such a connection expected to work.

Corresponding to a rootsystem R (or more generally to an “admissible pair” of root systems), one con-structs a two parameter family of quantum symmetric spaces, such that theirzonal spherical functions (zsf’s) “interpolate” between the zsf’s of ordinary realand p-adic symmetric spaces in the following sense [1,2].If we call the twoparameters q and t thena) for q = 0, t = 1/p, p = prime, these zsf’s essentially reduce to the zsf’sof a p-adic symmetric space whose restricted root system is R∨, the dual of thechosen root system R;b) for t = ql →1, with a certain value of l, these zsf’s reduce to the zsf’s ofthe real symmetric space with restricted root system R.There has been some progress [12-15] in exploring property b) in the contextof one-parameter quantum groups, obtained form the full two-parameter groupsby imposing t = ql, though not necessarily with t near one. However, none ofthe p-adic cases of a) above can be reached this way.

We therefore have toaddress the full two-parameter problem.Here we shall do just that and find that the two parameter quantum alge-bra of Sklyanin and its generalizations provide the conceptual understanding ofthe p-adics-quantum deformation connection. Specifically, we shall consider theZn-Baxter model of statistical mechanics [16-19] on a square lattice for whichthe underlying algebra is of the (generalized) Sklyanin type [20-24].

For thismodel, in a regime such that the equivalent magnetic chain is antiferromagneticwith finite gap, we will study the scattering of two first level excitations andwill find that, in the n →∞limit, the corresponding Jost function coincideswith the Harish-Chandra-like c-function [25] obtained from Macdonald’s poly-nomials for root system A1 (see Eq. (5.8)).

The anisotropy parameter and the2

modular parameter of the Baxter model, are then related with the parametersq and t according to the relations (5.9). This way of establishing the connectionis like “fishing” for SU(2) inside SU(∞).

One can also establish a connectiondirectly between the Z2-Baxter model and the A1-Macdonald-Harish-Chandrac-function, but this relation is less simple (see Eqs. (5.10)).

These connectionsare our main results. We suspect that both, the complexity of Eqs.

(5.10), andthe need for the n →∞limit before the transparent Eq. (5.8) is captured, areconnected with the involved coproduct situation in elliptic quantum algebras[24].

One may wonder what physics corresponds to (q, t) = (0, 1/p) in whichcases the Macdonald polynomials yield zsf’s of p-adic symmetric spaces (casea) above). It is readily checked that the choice of parameters (q, t) = (0, 1/p)corresponds to an XXZ model, with a particular value of the anisotropy param-eter.Mathematically, the most remarkable feature of our result is that Sklyanin’selliptic quantum group and its generalizations, unify the p-adic and real versionsof a Lie group (SL(2) in this case).

Of course a unification of SL(2, R) withthe SL(2, Qp)’s occurs also in the adelic [26] context. But the unification whichwe have in mind here is of a completely different nature.

It does not involveEuler products, but rather two real parameters which can be “dialed” for anyarchimedean or non-archimedean case. One can nevertheless ask the questionabout how this new unification relates to the adelic one.

We shall thereforediscuss the possibility of q-deforming Euler products!2Macdonald polynomials for the root systemA1Starting from any “admissible pair” (R, S) of root systems [27], Macdonald [1,2]has constructed a corresponding family of orthogonal polynomials enjoying sometruly remarkable features. This construction is very general, the root system[27] R need not even be reduced, it can be of the BCn type.

For our purposes, itwill suffice to describe the Macdonald construction in the simplest of all cases,where both R and S are reduced and of rank 1, so that R = S = A1.The root system A1 has one positive root α and one negative root −α. The3

root lattice is Λr = {nα : n ∈Z} and its positive “side” is Λ+r = {nα : n ∈Z, n > 0}. The weight lattice Λ of A1 is Λ = { n2 α : n ∈Z} and the set ofdominant weights is Λ+ = { n2 α : n ∈Z, n ≥0}.Obviously Λ+r ⊂Λ+.

On the weight lattice Λ a partial order is definedλn > λm ←→λn −λm ∈Λ+r(2.1a)or more explicitlyn2 α > m2 α ←→0 < n −m ∈2Z(2.1b)The Weyl group W of A1 is W = Z2 = {1, σ}, with 1 the identity element andσ the reflection which takes ±α into ∓α. The weight lattice Λ is an abeliangroup under addition.

Its group algebra A over R is suggestively presented interms of formal exponentialsλ = n2 α ∈Λ −→eλ = en2 α ∈A(2.2)so thatem2 αen2 α = em+n2αen2 α−1 = e−n2 αe0 = 1(2.3)These en2 α form an R basis of A. The Weyl group action on Λ defines aW-action also on the group algebra Aw(eλ) := ewλw ∈W, λ ∈Λ, eλ ∈A(2.4)The Weyl-invariant elements of A span a subalgebraAW = {a ∈A : wa = a ,∀w ∈W}(2.5)of A.

Obviously the elementsmn = en2 α + e−n2 αn ∈Z+(2.6)provide an R-basis of AW .Define the Weyl charactersχn = e(n+1) α2 −e−(n+1) α2eα2 −e−α2(2.7)Then theχn withn ≥0(2.8)4

also provide an R-basis of AW .Beside these R-bases of AW , there exists a much less obvious two-parameterfamily of R bases of AW . This family of Macdonald bases comes into being dueto the existence of a two parameter family of positive-definite scalar productson A.

They are constructed as follows. Call the two real parameters t and qand consider the element ∆(t, q) in A defined by∆(t, q) = (eα; q)∞(teα; q)∞(e−α; q)∞(te−α; q)∞(2.9)where e±α are the elements in A corresponding to the roots ±α.Here andthroughout this paper we adopted the notation [28, 29](a; q)n = (1 −a)(1 −aq) .

. .

(1 −aqn−1)(2.10a)and(a; q)∞=∞Yk=0(1 −aqk),(2.10b)so that(a; q)n =(a; q)∞(aqn; q)∞. (2.10c)To eachf =Xλ∈Λfλeλ ∈A(fλ ∈R)(2.11a)we associate its “conjugate”¯f =Xλ∈Λfλe−λ ∈A .

(2.11b)Now we consider the 1-torus (circle) T = R/Λ∨r , where Λ∨r is the root lattice ofthe dual root system. Obviously any x ∈R then has an image xT on the circleT .

Each eλ ∈A can therefore be viewed as a character of T viaeλ(xT ) = ei2π<λ,x> . (2.12)Macdonald’s two parameter family of positive-definite scalar products on A isthen given by< f, g >t,q= 12ZTf¯g∆(t, q),(2.13)5

the measure on T being the (normalized) Haar measure.Finally, for each scalar product < ,>t,q in the family (2.13), we define aMacdonald basis of AW by the following three requirementsa)Pm = mm +X0≤nt,q onA, i.e. < Pm, Pn >t,q= 0 for m ̸= nThese Pn are clearly polynomials in m1 = eα/2 + e−α/2, or equivalentlyLaurent polynomials in eα2 .

For the A1 case under discussion, they are explicitlygiven in terms of the famous Rogers-Askey-Ismail (RAI) polynomials [30,31]Cn(x; t|q). SpecificallyPneα2 ; t|q= (q; q)n(t; q)nΦn(eα2 ; t|q)(2.14a)whereΦn(x; t|q) =Xa+b=na,b∈Z+(t; q)a(t; q)b(q; q)a(q; q)bxa−b(2.14b)with the q-shifted factorial (t; q)a defined by Eq.

(2.10a)Finally, the connection between the Φn’s and the RAI polynomials isΦn(eiθ; t|q) = Cn(cos θ; t|q) . (2.14c)For future reference we give here the expression [31] for the RAI polynomials interms of the q-hypergeometric function 3φ2.Cn(cos θ; t|q) = t−ne−inθ (t2; q)n(q; q)n3φ2(q−n, t, te2iθ; t2, 0|q, q)(2.14d)The Macdonald scalar product (2.13) now reduces to the usual scalar producton RAI polynomials< Pm, Pn >t,q=Z +1−1Pm(eiθ; t|q)Pn(eiθ; t|q).· w(cos θ; t|q)(sin θ)−1/2d cos θ =δmn(tqn; q)∞(tqn+1; q)∞(t2qn; q)∞(qn+1; q)∞(2.15a)6

with the weight function w given byw(cos θ; t|q) = 12π(e2iθ; q)∞(e−2iθ; q)∞(te2iθ; q)∞(te−2iθ; q)∞(2.15b)With the notation (our l is Macdonald’s k)t = qlorl = log t/ log q(2.16)we can also write||Pn||2 =< Pn, Pn >t,q= (qnt; q)l(qn+1; q)l= (qn+l; q)∞(qn+l+1; q)∞(qn+2l; q)∞(qn+1; q)∞= Γq(n + 2l)Γq(n + l)Γq(n + 1)Γq(n + 1 + l)(2.17)where we used Eq. (2.10c) and the definition of the q-gamma function [28, 29]Γq(x) = (q; q)∞(qx; q)∞(1 −q)1−x(2.18)Introducing the Harish-Chandra-like c−function of Macdonaldc(x; l|q) =Γq(x)Γq(x + l)(2.19)we can finally recast Eq.

(2.12) in the form||Pn||2 = c(n + 1; l|q)c(n + l; l|q)(2.20)Through this formula the Macdonald polynomials Pn yield a Macdonald- Harish-Chandra c-function. We shall make extensive use of this fact in the followingsections.In Section 4 we shall show that this c-function as it emerges from the Mac-donald scalar product via Eq.

(2.20), can be obtained directly from the large nbehavior of the RAI polynomials along the usual lines of Harish-Chandra theory[25].3Macdonald’s miraclesIn spite of its relative ease, the A1 case considered here, already exhibits anumber of “miracles”, which as shown by Macdonald, generalize to all admissiblepairs of root systems.7

A) We havelimt=q1/2→1 Pn(eiθ; t|q) =Γ(1/2)Γ(n + 1)Γ(n + 1/2)Pn(cos θ)(3.1)where Pn(cos θ) are the ordinary Legendre polynomials, the zsf’s on the ordinarycompact archimedean symmetric space SU(2)/SO(2), the 2-sphere.B) By contrast, for q = 0 and t = 1/p with p a rational primePn(x; 1p|0) =(1 + 1pδn0)−1pn−22 (1 + p)ρx(n)(3.2a)withρx(n) = xn(px −x−1) + x−n(x −px−1)pn/2(p + 1)(x −x−1),(3.2b)the Mautner-Cartier polynomials [32,33], the zsf’s on the non-compact p-adicsymmetric space H(p) = SL(2, Qp)/SL(2, Zp),the p-adic hyperbolic plane.Remark: There is a big difference between the interpretations of the two“left-over” variables x and n in the two cases A) and B) above. In the archimedeanlimit A), the variable cos θ = (x + x−1)/2 is the “radial” coordinate on the realcompact symmetric space SU(2)/SO(2) and the quantized (angular) momen-tum variable n is related to the eigenvalue of the laplacian n(n + 1)(= l(l + 1)in more familiar notation).

Things are reversed in the p-adic case B). Therethe discrete variable n plays the role of “radial distance” on the non-compactp-adic symmetric space H(p), which is a discrete space, a Bruhat-Tits tree [34](or Bethe lattice).

Conversely, it is now the variable x which is related to theeigenvalue of the laplacian on the tree. This switch of variables between thecases A) and B) has a counterpart for all other root systems [1,2].

Specificallyfor q = 0, t = 1/p (case B) one obtains the zsf’s of the p-adic group G rela-tive to a maximal compact subgroup K such that the restricted root system ofthis (G/K)p−adic is R∨, the dual of the root system R which underlies the realsymmetric space (G/K)real whose zsf’s one reproduces in the archimedean caseA) (t = q1/2 →1). In the case at hand, R = R∨= A1, so that the differencebetween the real and p-adic cases is reflected only in the exchanged rˆoles of the xand n variables.

This is a very important point which will be further developedin the next Section.8

C) For t = 1, the Pn’s take the simple q-independent formPn(eα2 ; 1|q) = en α2 + e−n α2(3.3)in other words they reduce to the mn s of Eq. (2.6).D) For q = t the Pn’s are again q-independent and this time they reduce tothe Weyl charactersPn(eα2 ; q|q) = χn(3.4)with the χn given by Eq.

(2.7).Macdonald assumed both q and t real. If we relax this restriction, we canconsider the case of q an sth root of unity.E) For qs = 1, the Macdonald polynomials Pn(y; t|s) are quasi-periodic inn:Pns+k = PnsPk ,n ∈Z+ , k ∈{0, 1, 2, .

. ., s −1}(3.5)To see this, recall the recursion relation for Macdonald polynomials (see [31]Eq.

(2.15))Pn+1 = (y + y−1)Pn −Cn−1Pn−1(3.6a)withCn−1 = 1 −t2qn−11 −tqn−11 −qn1 −tqn =||Pn||2||Pn−1||2(3.6b)Notice that qs = 1 then impliesCns+k = CkCns−1 = 0n ∈Z+, k ∈{0, 1, 2, . .

., s −1}(3.7)Now from Eqs. (2.14) P0 = 1, P1 = y + y−1, so that from Eqs.

(3.6), (3.7) itfollows thatPns = PnsP0Pns+1 = PnsP1(3.8)Inserting (3.8) and (3.7) into the linear Eqs. (3.6a) then yields the quasi-perio-dicity (3.5).F) For t = q1/2 ̸= 1, the Pn’s become essentially continuous q-Legendrepolynomials, which can be interpreted [15] as “quasi-spherical” functions of theone-parameter quantum group SU(2)q.G) In the limit t = q(m−2)/2 →1 the RAI polynomials yield [35] the Gegen-bauer polynomials, zsf’s on the (m −1)-sphere Sm−1 = SO(m)/SO(m −1).9

4Further zonal spherical function-like proper-ties of the Rogers-Askey-Ismail polynomialsIn the remark following properties A) and B) in Section 3, we described theremarkable interchange of (radial) coordinate and momentum variables betweenthe archimedean case A) and the p-adic case B). This raises the question ofwhat the coordinate and momentum variables are for generic values of the twoparameters q and t.This question can be answered by noting a remarkable “self-duality” prop-erty of the RAI polynomials.To explain this, let us first observe that theMacdonald polynomials Pn(x; t|q) yielded familiar spherical functions in thetwo limiting cases A) and B) above, only up to the numerical factors in squarebrackets in formulae (3.1) and (3.2a).

These inconvenient factors can be elim-inated at the expense of relaxing condition b of Section 2 to rationality in thevariables t1/2 and q rather than in t and q. Then, instead of the Pn’s we candefineΨn(eiθ) = Φn(eiθ; t|q)Φn(t1/2; t|q)(4.1)These Ψn’s, rather than the Macdonald polynomials Pn themselves, are the can-didates for spherical functions of some, as yet hypothetical, quantum symmetricspace.

It is worth noting thatΦn(t1/2; t|q) = t−n/2 (t2; q)n(q; q)n.(4.2)Defineν = θ/ log q . (4.3)Recalling the definition of l from Eq.

(2.16), and combining it with Eqs. (4.1),(4.2), (4.3) and (2.14d), we then obtainΨn(qiν) = q−n(2iν+l)/23φ2(q−n, ql, q2iν+l; q2l, 0|q, q).

(4.4)Since the prefactor and the q-hypergeometric [28,29] function 3φ2 are both in-variant under the exchange−n ↔2iν + l(4.5)10

it then follows thatΨn(qiν) = Ψ−2iν−l(q−(n+l)/2)(4.6)where the right hand side is to be understood as obtained by analytic continua-tion. In Eq.

(4.6) the left-hand side is relevant for the compact case, the right--hand side (an analytic continuation) applies to the non-compact case. In par-ticular, this explains the rˆole exchange of the x and n variables between the twoextreme cases A) and B) above (SU(2)/SO(2) is compact, SL(2, Qp)/SL(2, Zp)is not).We can now use Eq.

(4.6) to give a conceptual definition of the Macdon-ald-Harish-Chandra c-function c(x; l|q) of Eq. (2.19).

Going to large “distance”in the non-compact case, means n →∞in Ψ−2iν−l(q−(n+l)/2).Accordingto Eq. (4.6) this means going to large n in Ψn(eiν log q).

Using Eqs. (4.1) and(2.14c), this means going to large n in the RAI polynomials Cn(cos θ; t|q), whereθ = ν log q.

But this large n-asymptotics of the RAI polynomials follows fromthe q-integral representation of these polynomials. Specifically, for large n [35,28]Cn(cos θ; t|q) = (1 −q)−l (t; q)∞(q; q)∞Γq(iu(θ))Γq(iu(θ) + l)e−inθ ++Γq(−iu(θ))Γq(−iu(θ) + l)einθ(4.7a)withl = log t/ log qandu(θ) =2θlog q.

(4.7b)According to Harish-Chandra we expect the coefficients of e∓inθ to be c(±iu; t|q).Comparing with Eq. (2.19), we see this is indeed the case.We are concerned in this paper with interpreting the RAI polynomials, ormore precisely the Ψn’s (Eq.

(4.1)) as zonal spherical functions (zsf’s) of aquantum symmetric space.In the classical case, a complex valued functionφ(g), g ∈G on a Lie group G is a zsf of G relative to its compact subgroup K ifi) φ is regular at the identity element e of G and suitably normalized thereφ(e) = 1;ii) φ is K biinvariant, i.e., φ(k1gk2) = φ(g) for all g ∈G and all k1, k2 ∈K;11

iii) φ obeys the functional equationφ(g1)φ(g2) =ZKφ(g1kg2)dHaark. (4.8)According to a classical theorem, condition iii) is tantamount to requiring thatφ(g) be a pull-back to G of a function on the symmetric space G/K which is aneigenfunction of each G-invariant differential operator on G/K.

As an examplefor Legendre polynomials Pn(cos θ), the zsf’s of SO(3)/SO(2), Eq. (4.8) becomesPn(cos α)Pn(cos β) = 12πZ 2π0Pn(cos α cos β −sin α sin β cos γ)dγ(4.9)Now if the RAI polynomials are zsf’s of a quantum symmetric space, then weexpect them to obey a relation of the type (4.8).

As a matter of fact they do[35].5Macdonald polynomials, Sklyanin algebras andZn-Baxter modelsOur aim is to find the two-parameter quantum group whose zonal sphericalfunctions are the Macdonald polynomials for the root system A1 i.e. the RAIpolynomials.

To explain our way of dealing with this question, let us consider,by analogy, a more familiar problem. Suppose we are given the Mautner-Cartierpolynomials Eq.

(3.26) and we want to find out whether they are the zsf’s ofSL(2, Qp) relative to SL(2, Zp). The simplest way to do this would be to con-sider “S-wave” scattering on the p-adic hyperbolic plane SL(2, Qp)/SL(2, Zp)and to find the corresponding scattering matrix element Sp(u), which is ex-pressed in terms of the Jost function Jp(iu)Sp(u) = Jp(iu)Jp(−iu)(5.1)If the Mautner-Cartier polynomials are the appropriate zsf’s, then the Jostfunction Jp(iu) must coincide with the Harish-Chandra c-function derived fromthe large u behavior of the Mautner-Cartier polynomials (for the chosen valueof p).Similar considerations apply to the continuation to complex n of theLegendre polynomials Pn(cos θ).12

In our problem we want to see whether the RAI polynomials are sphericalfunctions of a Sklyanin type quantum group. To this end we choose a phys-ical system for which the underlying algebra is of the Sklyanin type.Thenfor this system we set up an appropriate scattering problem (of certain excita-tions) such that the corresponding Jost function coincides with the Macdonald-Harish-Chandra c-function (Eq.

(2.19)) derived from the RAI polynomials (seeEqs. (4.7)).The appropriate physical system is the Zn-Baxter model (Bn for short) ofstatistical mechanics on a square lattice [16-19].

The n2 × n2 R-matrix of thismodel was parametrized by Belavin [18] in terms of Jacobi theta functions. Thealgebra which allows a solution of the (intertwining) Yang-Baxter equations,thus leading to the existence of infinitely many commuting transfer matricesand therefore to the integrability of the model, was studied by Sklyanin [20,21],Cherednik [22,23], and by Odeskii and Feigin [24], in whose notation the alge-bra is Qn2,n−1(E, γ), which we shall call Qn for short.

Its data are the integern, an elliptic curve E and a point γ on E. In particular Q2(= Q4,1(E, γ)) isthe original Sklyanin algebra [20,21] of the 8-vertex model [16,17]. We do notneed the detailed form of the Belavin R-matrix elements.

The essential thingis that the statistical weights depend on three independent parameters: thespectral parameter z, the anisotropy parameter γ and the modular parameterτ. As is customary, we treat z as a variable and γ, τ as parameters.

In fact itis convenient to treat n as a parameter on equal footing with γ and τ. Alongwith Bn we also find it useful to think in terms of the corresponding (1 + 1)--dimensional field theoretical models Mn. The hamiltonian of Mn is obtainedfrom the transfer matrix T (z) of Bn through logarithmic differentiation at aspecial point.

Mn is known as the generalized magnetic model [36]. Note thatB2 is Baxter’s famous eight-vertex model, and M2 the familiar XYZ chain.We shall be interested in the antiferromagnetic regime with finite gap, so thatthe ground state is constructed by filling the false (ferromagnetic) vacuum withquasiparticles.

The partition function t(z) of the Bn model in the thermody-namic limit (the Perron-Frobenius dominant eigenvalue of T (z)) was obtained13

by Richey and Tracy [19]. Up to some irrelevant factors, it is of the formt(z) = ϑ1/21/2(z −iγπ , τ) exp−i( n−1n )2πz−−iF(z; γ; n; −iπτγ)i.

(5.2a)whereF(z; γ; n, −iπτγ) = 2∞Xk=11ksinh kγ( −iπτγ−1)sinh kγ( −iπτγ)sinh kγ(n −1)sinh kγn· sin 2πkz (5.2b)and ϑ1/21/2is the standard odd theta function with modular parameter τ(τ ∈iR+). Notice the remarkable symmetry of F(z; γ; n, −iπτγ) in its last two argu-mentsF(z; γ; n, −iπτγ) = F(z; γ; −iπτγ, n) .

(5.3)This is our first signal to pay special attention to the variable −iπτγor its inverseiγ/πτ. In fact, it will turn out that precisely this combination iγ/πτ is to beidentified with the parameter l in Macdonald’s polynomials (Eq.

(2.16)). In thenext Section we shall make good use of the symmetry property (5.3).A remarkable fact [37] in quantum integrable models is that the partitionfunction, as a function of the spectral variable z, coincides up to some simplefactors and/or redefinitions of parameters with a two-particle dressed S-matrix,the spectral parameter acquiring the interpretation of relative rapidity of thescattering particles.

We have to be more specific, there being n −1 types ofdressed excitations in Mn. We therefore briefly recall the picture of these ex-citations in the context of the nested Bethe ansatz (BA).

The ground state, aswas already mentioned is found by filling the false vacuum with n −1 types ofquasiparticles, each type at its own “level”. The momenta are distributed con-tinuously in segments [−π, +π] at each level.

Excitations are viewed as “holes”in these distributions.The type of physical excitation is determined by thelevel at which the hole was created. In terms of the system of interacting parti-cles on the lattice associated to the Mn model in the usual way, the first levelcorresponds to charge excitations, while the others to “isotopic” excitations.The levels are naturally ordered according to the sequence of the nested BA.

Itturns out that t(z) of Eq. (5.2) is essentially the (scalar) S-matrix S(n)1for the14

scattering of two first level dressed excitations. More preciselyS(n)1= exp−i(n −1n)2πz −iF(z; γ; n, −iπτγ)(5.4)The S-matrix elements for the scattering of other types of excitations are morecomplicated.Therefore we restrict ourselves to the first level sector and itsS-matrix S(n)1.

At this point it pays to introduce new variablesl = iγπτq = ei2πτu = −izτ(5.5)(notice that l is the combination signaled already in the context of the symmetryproperty (5.3); our q is the usual one, i.e. the square of the one in [19]).

Thenwe can write e−iF in the formexp−iF(z; γ; n, −iπτγ)= σ(iu; l; n|q)σ(−iu; l; n|q)(5.6a)withσ(iu; l; n|q) =∞Yk=0Γq(iu + nl(k + 1))Γq(iu + nlk + l)Γq(iu + nlk + 1)Γq(iu + nlk + (n −1)l + 1)(5.6b)Now let n go to infinity. Using the definition (2.18) and keeping in mind thatas x →∞for q < 1, we have (qx, q)∞→1, it is then readily seen thatσ(iu; l, ∞|q) = [iu]qc(iu; l|q)(5.7a)with[iu]q = 1 −qiu1 −q(5.7b)the “q-analogue” of iu and c(iu; l|q) the Macdonald-Harish-Chandra c-functionfor root system A1 Eq.

(2.19)! Combining Eqs.

(5.7), (5.6a) and (5.4) we findS(n)1n=∞= −c(iu; l|q)c(−iu; l|q)(5.8)and we see the Macdonald c-function assuming the rˆole of Jost function in thisscattering process. This clearly establishes the connection between the n →∞limit of the Sklyanin-Cherednik-Odeskii-Feigin algebras Qn, which underlie the15

Bn models on the one hand, and the Macdonald polynomials for the root systemA1 on the other hand. As was mentioned above, the data for the Qn algebraare an elliptic curve E = C/Z + ZτE, characterized by the modular parameterτE, or equivalently qE = exp(i2πτE), and a point γ on E. The data for theset of A1-Macdonald-RAI polynomials are the two parameters t and q. Theconnection between the elliptic and Macdonald parameters is thenq = qEt = e−2γ(5.9)the second equation following directly from Eqs.

(2.16) and (5.5). This con-nection between Qn algebras (n →∞) and Macdonald polynomials is our mainresult.At this point the question arises as to why the n →∞limit had to be taken.On the face of it, all we should have had to deal with should have been theelliptic algebra Q2 and the models which it underlies B2 and M2.

Going toQn, Bn, Mn and then letting n →∞is like searching for SU(2) inside SU(∞).For ordinary Lie groups this would be a detour, for elliptic quantum algebrasthis may be needed on account of the complicated coproduct situation [24]. Butonce in Q∞, how is it we only found the Macdonald polynomials for root systemA1 and not those for higher An root systems?

The point is that we only lookedat the scattering of two first level excitations.After this discussion, we would like to see what would happen, were we tochoose n = 2, as naively indicated for root system A1, instead of letting n →∞.From Eqs. (5.4) - (5.6) we can see that for n = 2 we findS(n)1n=2 = qiu/2∞Yk=0c(iuk −l + 1; l|q)c(iuk; l|q)·c(−iuk; l|q)c(−iuk −l + 1; l|q)(5.10a)withiuk = iu + l(2k + 1) .

(5.10b)We see that the building block of S(n)1|n=2 is again the Macdonald-Harish-Chandra c-function of Eq. (2.19), but this time in a pattern not as conceptuallysimple as that of Eq.

(5.8). Yet we shall have more to say about this case in thenext Section.16

Whether or not the n →∞limit is taken, it would be nice to have a deriva-tion of the Macdonald-Harish-Chandra c-function of Eq. (2.19) directly fromsymmetric spaces constructed from Qn quantum groups, along the standardlines of Harish-Chandra theory (see e.g.

[25]) and without any reference to thephysics of the Zn-Baxter models. Conversely it would be of interest to find thegeometric object for which the infinite product σ(iu; l; n|q) of Eq.

(5.6b) is thec-function. To steer all this into more familiar territory, notice that in the limitq →1 the q-gamma functions in the infinite product reduce to ordinary gammafunctions and the full infinite product (5.6a) becomes essentially that which ap-pears [38] in the soliton-soliton scattering S-matrix in the sine-Gordon model,provided one relates our parameter nl to the sine-Gordon coupling constant βvianl = 28πβ2 −1(5.11)Thus the problem of understanding the “geometric” interpretation of the infiniteproducts has as an important special case soliton-soliton scattering in the sine-Gordon model.

Conversely, we can regard the S-matrix given by Eqs. (5.4)-(5.6)as a “q-deformation” Γ →Γq of the sine-Gordon soliton scattering matrix of[38].

The Sklyanin algebra (n = 2) looks like the further deformation of quantumSL(2) by a new parameter.To conclude, let us mention that the expression of the Perron-Frobeniusdominant eigenvalue of Baxter’s zero-field 8-vertex model (B2) transfer matrix,has been recast in terms of the c-function (2.19).6Interesting special casesWith the just-established connection between Bn or Mn systems and Macdonaldpolynomials it becomes interesting to see what happens in the regime in whichthe polynomials, “go” p-adic, i.e., in case B) of Section 3. For the n →∞situation of Eq.

(5.8) this corresponds toq = 0,t = e−2γ = 1/p . (6.1a)so thatγ = log √p(6.1b)17

Eqs. (5.8), (5.5), (2.19) and (2.18) then yieldS(n)1n=∞,q=0,t=1/p = pei2πz −1ei2πz −p ,(6.2)which coincides with the bare S-matrix in the XXZ model with the same valueof γ.

Could this result also be obtained from a model on a Bethe lattice withp + 1 edges incident at each vertex?A direct study of M∞models using the powerful quantum inverse scatteringmethod or the Bethe ansatz is highly desirable.The other interesting case ist = q1/2 ,(6.3a)so thatl = 1/2 . (6.3b)This corresponds, according to Section 3F, to the familiar one-parameter quan-tum group SU(2)q.

In the limit q →1 it then yields the ordinary Lie groupSU(2) (Section 3A). For the SU(2)q case we have the direct treatment by oneof us [12].

An immediate comparison with the results of [12] is not possible,since there n = 2, whereas for usn →∞,l−1 = −iπτγ= 2(6.4)It is clear from Eq. (5.4) that for large n, the S-matrix depends only on thefunction F(z; γ; n; iπτγ ).

But, as we saw in Eq. (5.3) this function is symmetricunder the interchange of its last two arguments.

Therefore instead of the case(6.4) we can deal with the equivalent case ofn = 2,−iπτγ→∞(6.5)which is then in line with [12], provided one replaces (5.5) byl = 1n ,q = e−2γnu = πzγn ,(6.6)so that, yet againt = ql = e−2γ . (6.7)18

As in [12], we find the XXZ model in this case.We can also view the p-adic and SU(2)q cases directly on the B2 or 8-vertexmodel or on the equivalent XYZ model, `a la (5.10). In terms of Baxter’s pa-rameters [16] the p-adic case corresponds to the 6-vertex model in the principalantiferroelectric regime withΓ = 1,∆= −p1/2 + p−1/22,(6.8)In terms of the XXZ chain this corresponds to the antiferromagnetic XXZ chain(Γ = 1) with asymmetry ∆given by (6.8) (remember Jx : Jy : Jz = 1 : Γ : ∆).Similarly the case t = q1/2 of the ordinary one-parameter quantum groupSU(2)q corresponds toΓ = 0∆= −1 + k2√k(6.9)where k is the modulus of the elliptic Jacobi functions of nome q.

This is theXZ model. If we now let t = q1/2 →1, corresponding to the ordinary SU(2) Liegroup (Case A of Section 2) then the comodulus k′ →0, so that the modulusk →1 and ∆→−1.7Applications and generalizationsa) A large class of elliptic quantum algebras has already been brought into playin the context of the Zn-Baxter models and the simplest Macdonald polyno-mials.

The question then naturally arises as to a full classification of ellipticquantum symmetric spaces, in correspondence with admissible pairs of rootsystems.There is one more aspect to this. The parameters q, t of Macdonald translateon the “Sklyanin side” into an elliptic curve and a point on it.

Could one makethe connection with elliptic curves explicit directly on the “Macdonald side”?b) For generic t and q, the orthogonal RAI polynomials obey of course, athree term recursion relation (Eqs. (3.6)).

In the p-adic regime (q = 0, t =1/p) this recursion relation becomes precisely the condition that the zsf’s beeigenfunctions of the laplacian. On the Bruhat-Tits tree, corresponding to thiscase, the laplacian has a simple interpretation as a difference operator obeying19

the mean value theorem. It is then natural to expect that for generic t and q,the recursion relation (3.6) also corresponds to the requirement that the RAIpolynomials be eigenfunctions of the laplacian on some “non-arboreal” discretespace, which reduces to a tree in the p-adic regime.

It would be nice to finda simple geometric description of this generic discrete space which in the p--adic case becomes a tree whereas for t = q1/2 →1 becomes a (continuous)sphere of (real) dimension 2. In short then it would be interesting to have adirect geometric picture of the quantum symmetric space, not only of its zonalspherical functions.c) The interpolation between real and p-adic symmetric spaces by varyingthe parameters q and t, makes one recall another real-p-adic connection, at theadelic level [26], via Euler products [38].

In fact, for q = 0, t = 1/p the c-functionis a ratio of local (p-adic) zeta functions.ζp(s) =11 −p−s(7.1)whereas for t = q1/2 →1, the c-function is a ratio of (real) local zeta-functionsζ∞(s) = π−s/2Γs2(7.2)Taking the Euler product yields the adelic zeta-functionΛ(s) = ζ∞(s)Ypζp(s) = π−s/2Γ(s2)ζ(s)(7.3)which involves the Riemann zeta function ζ(s) and obeys the simple functionalequation Λ(s) = Λ(1−s). Can this construction be q-deformed?

Is there such athing as a “q-Euler product”? To answer these questions, notice that the Eulerproduct runs one of the Macdonald variables (namely t) over all reciprocal primevalues, while the other stays fixed.

As the other Macdonald variable one canchoose either q or l = −log p/ log q, as both of them stay fixed at zero. It willturn out that for us the sensible choice is l. So we view an Euler product asa product over t = 1/2, 1/3, 1/5, .

. .

while l is fixed at zero. A deformed Eulerproduct then should do the same but with l fixed at some value other than zero.At this point we have to find the deformations of the local zeta functionsζp(s) and ζ∞(s).

If we call ζ(s; t, l) the(two-parameter) “deformed local zeta20

function”, then we must imposeζ(s; 1p, 0) = ζp(s)(7.4)andζ(s; 1, 1/2) = ζ∞(s)It is easy to see that (q = t1/l)ζ(s; t, l) = πqlsΓt1/l(ls)(7.5)fits the bill. We haveζ(s; t, l) = π−qls (t1/l; t1/l)∞(1 −t1/l)1−ls(ts, t1/l)∞== π−qls(1 −t1/l)1−lsQ∞n=1(1 −tn/l)Q∞n=0(1 −ts+n/l)(7.6)which for t = 1/p < 1, l →0+, q →0 does indeed become11−p−s = ζp(s).

Onthe other hand, for l = 1/2, t →1 (so that also q →1),ζ(s; 1, 1/2) = π−s/2Γs2= ζ∞(s)(7.7)To get a meaningful q-Euler product starting from (7.5), we have to unload theπ−qls factor. So the q-deformed Euler factor will be πqlsζ(s; t, l)|l=fixed,t=1/p,and our “q-Euler product” or more appropriately “l-Euler product” will be(q = t1/l)E(s; l) =Ypπqlsζ(s; 1p; l) ==Yp(1 −p−1/l)1−ls Q∞n=1(1 −p−n/l)Q∞n=0(1 −p−s−n/l)= ζ(l−1)1−ls Q∞n=1 ζ(nl−1)Q∞n=0 ζ(s + nl−1)(7.8)an interesting combination.The special rˆole in all this of the q-gamma function Γq(ls) as “interpolator”between the local zeta functions at the finite and infinite places, can be betterunderstood by tracing it back to a remarkable property of the q-exponential andto a remarkable property of Jackson’s q-integral.

The point is that the q-gammafunction admits a q-integral supresentation [29], which upon a standard change21

of variables turns into a ql(= t)-integral representation. This is essentially at-Mellin transform of eq(−x1/l).

Here the q-exponential eq(y) is defined as [28],[29]eq(y) =∞Xa=0ya[a]q! ,[a]q = 1 −qa1 −q ,[a]q!

= [a]q[a −1]q · · · [1]q(7.9)Not surprisingly for t →1, l →1/2 (the archimedean regime) this t-Mellintransform reduces to the ordinary real Mellin transform and eq(−x1/l) becomesthe real gaussian (a well known representation of the gamma function of half--argument). The p-adic regime, surprisingly allows a similar interpretation.

At-integral is a sum [29]Zf(x)dtx =+∞Xn=−∞f(tn)[(1 −t)tn] . (7.10)In the p-adic regime q →0 t →1/p, the factor in square brackets coincides withthe volume of the “shell” In = {ξ ∈Qp, |ξ|p = p−n} of p-adic integration, sothat the sum over n, itself amounts to an integration over Qp of the complexfunction f(|ξ|p) of the p-adic variable ξ (f depends on ξ only through its norm|ξ|p).Under the sum (7.10), eq(−x1/l) becomes eq(−tn/l) = eq(−qn) (sincet = ql).

As q →0, the q-analogues of all nonnegative integers go to one, so thateq(−qn) behaves like the geometric sum P∞a=0(−qn)a = (1 + qn)−1 and thusequals one for n > 0 and zero for n negative. When the case n = 0 is included,this shows that the function eq(−x1/l), under t-integration dtx, translates intothe characteristic function χp(ξ) of the p-adic integersχp(ξ) = 1for|ξ|p ≥10for|ξ|p > 1(7.11)under p-adic integration.

But this χp(ξ) is the “p-adic gaussian”, that is theFourier self similar complex-valued function of the p-adic variable ξ.We thus come to realize that t-integration “interpolates” between real Rie-mann integration and p-adic integration, while at the same time the functioneq(−x1/l) plays the rˆole of a “quantum gaussian” which interpolates betweenthe ordinary real gaussian and the step-functions χp(ξ), the p-adic gaussians.All this clearly begs for a q- and/or l-deformation of Tate’s Fourier analysis onlocal fields.22

d) Does any of this work bear on string theory? Yes, we can construct two-parameter deformations of string theory which for t = 1/p, q = 0 reproduce theknown p-adic strings, for t = q1/2 →1, the ordinary Veneziano string, and fort = q1/2 ̸= 1 involve q-strings.

We shall return to this elsewhere.We wish to thank S. Bloch, L. Chekhov, L. Mezincescu and M. Olshanetskyfor valuable discussions.REFERENCES1. Macdonald, I. G.: in Orthogonal Polynomials: Theory and Practice, P.

Nevaied., Kluwer Academic Publ., Dordrecht, 19902. Macdonald, I. G.: Queen Mary College preprint 19893.

Freund, P. G.O.

: in Superstrings and Particle Theory, L. Clavelli andB. Harms eds., World Scientific, Singapore, 19904.

Freund, P. G. O.: Phys. Lett.

257B, 119 (1991)5. Freund, P. G. O.: in Quarks, Symmetries and Strings, A Symposium inHonor of Bunji Sakita’s 60th Birthday, M. Kaku, A. Jevicki and K.

Kikkawaeds., World Scientific, Singapore, 19916. Brekke, L., Freund, P. G. O., Olson, M., and Witten, E.: Nucl.

Phys.B302, 365 (1988)7. Zabrodin, A. V.: Comm.

Math. Phys.

123, 463 (1989)8. Coon, D. D.: Phys.

Lett. 29B, 1422 (1969)9.

Romans, L. J.: preprint USC-88/HEP014 (1988)10. Olshanetsky, M. A., and Perelomov, A. M.: Phys.

Reports 94, 313 (1983)11. Wehrhahn, R. F.: Phys.

Rev. Lett.

65, 1294 (1990)12. Zabrodin, A. V.: Moscow preprint 199113.

Ueno, K.: Proc. Jap.

Acad. 66A, 42 (1990)23

14. Vaksman, L. L., and Korogodsky, L. I.: Moscow preprint (1990)15.

Koornwider, T.: in Orthogonal Polynomials Theory and Practice, P. Nevai,ed., Kluwer Academic Publ., Dordrecht, 199016.

Baxter, R. J.: Exactly Solved models in Statistical Mechanics, Acad. Press,N.Y., 198217.

Gaudin, M.: La Fonction D’Onde de Bethe, Masson, Paris 198318. Belavin, A.

A.: Nucl. Phys.

B180, 189 (1981)19. Richey, M. P., and Tracy, C. A.: J. Stat.

Phys. 42, 311 (1986)20.

Sklyanin, E. K.: Funk. Anal.

Appl. 16,263 (1982)21.

Sklyanin, E. K.: Funk. Anal.

Appl. 17, 273 (1983)22.

Cherednik, I. V.: Yad. Fiz.

36, 549 (1982)23. Cherednik, I. V.: Funk.

Anal. Appl.

19, 77 (1985)24. Odeskii, A. V., and Feigin, B. L.: Funk.

Anal. Appl.

23, 207 (1989)25. Helgason, S.: Topics in Harmonic Analysis on Homogenous Spaces, Birkh-¨auser, Basel, 198126.

Weil, A.: Adeles and Algebraic Groups, Birkh¨auser, Basel, 198227. Humphreys, J. E.: Introduction to Lie Algebras and Representation The-ory, Springer, Berlin, 197028.

Gasper, G., and Rahman, M.: Basic Hypergeometric Series, CambridgeUniv. Press, Cambridge, 199029.

Exton, H.: q-Hypergeometric Functions and Applications, John Wiley,N.Y., 198330. Rogers, L. J.: Proc.

London Math. Soc.

26, 318 (1895)31. Askey, R., and Ismail, R. E. H.: in Studies in Pure Math.

P. Erd¨os, ed.Birkh¨auser, Basel, 198324

32. Mautner, F.: Am.

J. Math.

80, 441 (1958)33. Cartier, P.: Proc.

Symp. Pure Math.

Vol. 26, A. M. S. Providence, 197334.

Bruhat, F., and Tits, J.: Publ. Math.

I.H.E.S. 41, 5 (1972)35.

Rahman, M., and Verma, A.: SIAM J. Math.

Anal. 17, 1461 (1986)36.

Kulish, P. P., and Reshetikhin, N. Yu.

: Soviet Phys. JETP 53, 108 (1981)37.

Zamolodchikov, A. B.: Comm.

Math. Phys.

69, 165 (1979)38. Zamolodchikov, A.

B., and Zamolodchikov, Al. B.: Ann.

Phys. (N.Y.)120, 253 (1979)39.

Langlands, R. P.: Euler Products, Yale University Press, New Haven, 197140. Tate, J.: Thesis, Princeton 1950, reprinted in Algebraic Number Theory,J.

W. S. Cassels and A. Fr¨ohlich, eds., Academic Press, N.Y., 196741. Gel’fand, I. M., Graev, M. I., and Pyatetskii-Shapiro, I. I.: RepresentationTheory and Automorphic Functions, Saunders, London, 196625

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