Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 30 April 2014

Notes and References

This is an excerpt of the paper Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory by Ivan Cherednik, in collaboration with Etsuro Date, Kenji Iohara, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Masatoshi Noumi, and Yoshihisa Saito.

Double affine Hecke algebras and Macdonald polynomials

Macdonald polynomials : the A1 case

The subject of this section is to show how the Hecke algebra technique is applied to the Macdonald polynomials. We will concentrate on the duality and the recurrence relations. The key notion will be the double affine Hecke. Let us start with A1.

The corresponding L-operator in the differential case reads as follows L(k)= 2u2+ 2k eu+e-u eu-e-u u +k2, (5.1) where k is a complex parameter. There are two special values of k when the operator L(k) is very simple. For k=0 we have L(0)=2/u2. When k=1, L(1)=d-1 2u2d, withd=eu- e-u. (5.2) Similarly, we can conjugate by dk for any k: dkL(k)d-k =2u2- 4k(k-1) (eu-e-u)2 . (5.3) Sometimes L is more convenient to deal with in this form.

Let us now consider the eigenvalue problem for the operator L(k): L(k)φ=λ2φ. (5.4) If k=1, the solution of this equation is immediate: φ(u;λ)= sinh(uλ) sinh(u)sinh(λ) . (5.5) In this normalization it is symmetric with respect to u and λ. Without sinh(λ) it generalizes the characters of finite-dimensional representations of SL2() (k=1).

If k=1/2, this operator is the radial part of the Casimir operator for the symmetric space SL2()/SO2(). It is the restriction of the Casimir operator C on the double coset space SO2\SL2/SO2 which is identified with a domain in */S2. If k=1,2, then L(k) corresponds in the same way to SL2()/SU(2) and SL2(𝒦)/SU2(𝒦) for the quaternions 𝒦.

For any k, one can find a family of even (u-u) solutions of the form pn=enu+ e-nu+lower integral exponents, (5.6) such that L(k)pn= (n+k)2pn (5.7) for n=0,1,2,. This family of hyperbolic polynomials satisfy the orthogonality relations Constant Term(pnpmd2k) =cnδnm. (5.8) They are called the ultraspherical polynomials.

We can also consider the rational limit (k)= 2u2+ 2kuu (5.9) of the operator L(k), switching from the Sutherland model to the Calogero model. The solutions of the rational eigenvalue problem are expressed in terms of the Bessel function. In this case, the solutions can be normalized to ensure the symmetry between the variable and the eigenvalue. In the trigonometric case it is possible only for two special values k=0,1. It is one of the main demerits of the harmonic analysis on the symmetric spaces.

In the difference theory, this symmetry holds for any root systems. This discovery is expected to renew the Harish-Chandra theory. The so-called group case (k=1) is an intersection point of the differential (classical) and difference (new) theories.

We now turn to the difference version. We set x=eu and introduce the 'multiplicative difference' Γq acting as Γq(f(x))=f(qx) and satisfying the commutation relation Γqx=qxΓq. The Macdonald operator L is expressed as follows: L= tx-t-1x-1 x-x-1 Γq+ tx-1-t-1x x-1-x Γq-1. (5.10)

The parameter k in the difference setup is determined from the relation t=qk. When q=t (or k=1), the operator L is simple: L=1x-x-1 (Γq+Γq-1) (x-x-1). (5.11) Compare this formula with (5.2) in the differential case and notice that (5.11) is easier to check than (5.2).

The eigenvalue problem Lφ=(Λ+Λ-1)φ (5.12) always has a self-dual family of solutions. When n=0,1,2,, there exists a unique family of the so-called q, t-ultraspherical (or Roger-Askey-Ismail) Laurent polynomials pn=xn+x-n +lower terms, (5.13) which are symmetric with respect to the transformation xx-1, and satisfy the equation Lpn= (tqn+t-1q-n) pn. (5.14) The following duality theorem is proved in the next section by using the double affine Hecke algebra.

Theorem 5.1 (Duality). pn(tqm)pm(t)=pm(tqn)pn(t) for any m,n=0,1,2,.

If we set πn(x)= pn(x)pn(t), (5.15) the duality can be rewritten as follows: πn(tqm)= πm(tqn) (m,n=0,1,2,). (5.16)

The Askey-Ismail polynomials are nothing but the Macdonald polynomials of type A1. There are three main Macdonald's conjectures for the Macdonald polynomials associated with root systems (see [Mac0011046, Mac1423624]):

(1) the scalar product conjecture,
(2) the evaluation conjecture,
(3) the duality conjecture.

One may also add the Pieri rules to the list. These conjectures were justified recently using the double affine Hecke algebras in [Che1993, Che1995-2].

A modern approach to q, t-ultraspherical polynomials

The duality from Theorem 5.1 can be rephrased as the symmetry of a certain scalar product. Actually this product is a difference counterpart of the spherical Fourier transform. For any symmetric Laurent polynomials f,g[x+x-1], we set {f,g}= (a(L)g)(t), (5.17) where a is a polynomial such that f(x)=a(x+x-1). So we apply the operator a(L) to g and then evaluate the result at x=t.

Theorem 5.2. {f,g}={g,f} for any f,g[x+x-1].

Theorem 5.1 follows from Theorem 5.2. Indeed, if f=πm and g=πn, we can compute the scalar product as follows: {πm,πn}= (a(L)πn) (t)=a (tqn+t-1q-n) πn(t)=πm (qtn). (5.18) Use Lπn=(tqn+t-1q-n)πn and the normalization πn(t)=1. Hence Theorem 5.2 implies πm(qtn)=πn(qtm). Actually the theorems are equivalent, since pn form a basis in the space of all symmetric Laurent polynomials.

Definition 5.1. The double affine Hecke algebra q,t of type A1 is the quotient q,t= X,Y,T/, (5.19) by the relations for the generators X,Y,T TXT=X-1, T-1YT-1= Y-1, (5.20) Y-1X-1 YXT2=q-1, (T-t) (T+t-1)=0. (5.21)

Here we consider q,t as numbers or parameters. The first point of the theory is the following statement of PBW type.

Any element of H can be uniquely expressed in the form H=i,je=0,1 ciejXiTeYj (ciej). (5.22) The second important fact is the symmetry of q,t with respect to X and Y.

Theorem 5.3. There exists an anti-involution ϕ: such that ϕ(X)=Y-1, ϕ(Y)=X-1 and ϕ(T)=T.

Indeed, ϕ transposes the first two relations and leaves the remaining invariant.

Next, we introduce the expectation value {H}0 of an element Hq,t by {H}0= i,je=0,1 ciejt-ite tj, (5.23) using the expression (5.22). The definitions of ϕ and {}0 give that {ϕ(H)}0= {H}0for any Hq,t. (5.24) Now we can introduce the operator counterpart of the pairing {f,g}={g,f} on q,t×q,t: setting {A,B}0= {ϕ(A)B}0 (5.25) for any A,Bq,t.

The ϕ-invariance of the expectation value (5.24) ensures that it is symmetric {A,B}0= {B,A}0. We also remark that this pairing is non-degenerate for generic q,t.

Theorem 5.2 readily follows from

Lemma 5.4. For any symmetric Laurent polynomials f(x),g(x)[x+x-1], {f(X),g(X)}0 ={f,g}.

To prove the lemma we need to introduce the basic representation of the double affine Hecke algebra q,t. Consider the one-dimensional representation of the Hecke algebra Y=T,Y sending Tt and Yt. We denote this representation simply by +. Then take the induced representation V=IndY(+) =/ { (T-t)+ (Y-t) } [x,x-1], (5.26) where the last isomorphism is xnXn mod (T-t)+(Y-t). Under this identification of V with the ring [x,x-1] of Laurent polynomials, the element X acts on [x,x-1] as the multiplication by x, while T and Y act by the operators Tˆ=ts+ t-t-1x2-1 (s-1)andYˆ =sΓqTˆ, (5.27) respectively. Here s(f)(x)=f(x-1), the equality Hf(x)=g(x) in V means that Hf(X)-g(X)(T-t)+(Y-t). The latter readily gives the desired formulas for Tˆ,Yˆ.

The expectation value is the composition αC [x,x-1] β, (5.28) where α is a residue mod (T-t)+(Y-t) and β(f)=f(t-1) is the evaluation map at t-1. Take any f,g[X,X-1]. Then {f(X),g(X)}0= {ϕ(f(X))g(X)}0= {f(Y-1)g(X)}0= f(Yˆ-1)(g)(t-1). (5.29) The last equality follows from (5.28). If f and g are symmetric and f(X)=a(X+X-1), then {f(X),g(X)}0= a(L)(g)(t)={f,g}, (5.30) since the operator Yˆ+Yˆ-1 acts on symmetric Laurent polynomials as L. It is straightforward. The duality is established.

This method of proving of the duality theorem can be generalized to any root system.

We now discuss the application of the duality to the Pieri rules, the recurrence formulas for πn's with respect to the index n. First we will discretize functions and operators.

Recall that the renormalized q,t-ultraspherical polynomials πn(x) are characterized by the conditions Lπn= (tqn+t-1q-n) πn,πn(t)=1, (5.31) where L= tx-t-1x-1 x-x-1 Γ+ tx-1-t-1x x-1-x Γ-1. (5.32) As always, Γx=qxΓ. Denote the set of -valued functions on by Funct(,). For any Laurent polynomial f[x,x-1] or more general rational function, we define fˆFunct(,) by setting fˆ(m)=f (tqm)for all m. (5.33) Considering 𝒜=(x),Γ as an abstract algebra with the fundamental relation Γx=qxΓ, the action of 𝒜 on φFunct(,) is as follows: xˆφ(m)=tqm φ(m),Γˆ φ(m)=φ(m+1). (5.34) The correspondence ffˆ, whenever it is well-defined (the functions f may have denominators), is an 𝒜-homomorphism (x)Funct(,).

Due to (5.31): Lˆπˆn(m)= (tq-n+t-1qn) πˆn(m). (5.35) The Pieri rules result directly from this equality. Indeed, the duality πˆn(m)=πˆm(n) implies: πˆm(n)= (tqn+t-1q-n) πˆm(n)= (xˆ+xˆ-1) πˆm(n). (5.36) Here is Lˆ acting on the indices m instead of n. Explicitly, t2qm-t-2q-m tqm-t-1q-m πˆm-1(n)+ q-m-qm t-1q-m-tqm πˆm-1(n)= (xˆ+xˆ-1) πˆm(n). (5.37) For generic q,t, the mapping ffˆ is injective. Hence one can pull (5.37) back, removing the hats: (x+x-1)πm= t2qm-t-2q-m tqm-t-1q-m πm+1+ qm-q-m tqm-t-1q-m πm-1. (5.38) This is the Pieri formula in the case of A1. See [AIs1983]. We remark that this formula makes sense when m=0, since the coefficient of πm-1 vanishes at m=0. Generally speaking the 'vanishing conditions' are much less obvious.

The Pieri rules obtained above can be used to prove the so-called evaluation conjecture describing the values of pn at x=t. Applying (5.38) repeatedly, we get the formula (x+x-1) πm=c,m πm++lower terms (5.39) for each =0,1,2,. The leading coefficient c,m can be readily calculated: c,m=i=0-1 t2qm+i-t-2q-m-i tqm+i-t-1q-m-i . (5.40) Let us look at (5.39) for m=0: (x+x-1)= c,0π+lower terms. (5.41) Comparing the coefficients of xl+x-l, we have 1=c,0/p(t), since p=x+x-+. Hence p(t)=c,0= i=0-1 t2qi-t-2q-i tqi-t-1q-i . (5.42)

This value is easy to calculate directly (the formulas for pn are known). However the method described in this section is applicable to arbitrary root systems. We need only the duality, which is the main advantage of the difference harmonic analysis in contrast to the classical Harish-Chandra theory.

The GLn case

In this last subsection, we will discuss the double affine Hecke algebra and applications for GLn. Since we have already clarified the A1 case in full detail, we will try to get concentrated on the main points only.

In the GLn case, the Macdonald operators M0=1, M1,,Mn are as follows: Mm=I(i1<<im) iIjI txi-t-1xj xi-xj Γi1Γim. (5.43) In this normalization, t=qk/2 (cf. the differential case). For instance, the so-called group case is for k=1/2 (in contrast to SL2 considered above).

The Macdonald polynomials pλ for GLn satisfy the Macdonald eigenvalue problem: Mmpλ=em ( tn-1qλ1,, t-n+1qλn ) pλ(m=0,1,,n), (5.44) where λ=(λ1,,λn) are partitions, i.e., sequences of integers λi such that λ1λ2λn0. Here em elementary symmetric function of degree m. Given λ, pλ=pλ(x) is a symmetric polynomial in x=(x1,,xn) of degree |λ|=i=1nλi in the form pλ(x)= x1λ1 xnλn+ lower order terms. (5.45) The lower order terms are understood in the sense of the dominance ordering. Namely, a partition obtained from λ by subtracting simple roots (0,,0,1,-1,0,,0) is lower than λ. For instance. (λ1,λ2,) >(λ1-1,λ2+1,) >(λ1-1,λ2,λ3+1) >. (5.46) We will use the abbreviation t2ρqλ= ( tn-1qλ1,, t-n+1qλn ) , (5.47) where 2ρ=(n-1,n-3,,-n+1). So Mmpλ=em(t2ρqλ)pλ. Using k: t=qk/2, and t2ρqλ=qkρ+λ.

Given a partition λ, we set πλ(x)= pλ(x) pλ(t2ρ) = pλ(x1,,xn) pλ(tn-1,tn-3,,t-n+1) . (5.48)

Theorem 5.5 (Duality). For any partitions λ and μ, we have πλ(t2ρqμ) =πμ(t2ρqλ). (5.49)

This duality theorem implies the following Pieri formula.

Theorem 5.6. em(x)πλ(x)= |I|=m iIjI t2(j-i)+1 qλi-λj- t-1 t2(j-i) qλi-λj -1 πλ+eI(x), (5.50) where eI=iIei (sum of unit vectors).

Here the summation is taken only over subsets I{1,2,,n} (|I|=m) such that λ+eI remain partitions (generally speaking, dominant). It happens automatically, since the coefficient of πλ+eI(x) on the right vanishes unless λ+eI is dominant.

We can also determine the value of the Macdonald polynomial pλ(x) at x=t2ρ exactly by the method used in the A1 case. The formula was conjectured by Macdonald and proved by Koornwinder. The above theorems (for GLn) are also due to Macdonald and Koornvvinder. See also [EKi1995]. For arbitrary roots they were established in my recent papers.

Our approach is based on the double Hecke algebras. The operators Mm appear naturally using the operators Δi from (4.32). The latter describe the action of the generators Yi in the induced representation IndHY(-) isomorphic to the algebra of Laurent polynomials [X1±1,,Xn±1]. So the analogy with (5.26) is complete.

The double affine Hecke algebra (DAHA) =q,t for GLn is the algebra generated by the following two commutative algebras of Laurent polynomials in n variables: [X1±1,,Xn±1] and[Y1±1,,Yn±1], (5.51) and the Hecke algebra of type An-1: =T1,,Tn-1 (5.52) with the standard braid and quadratic relations. The remaining relations are as follows: TiXiTi= Xi+1(i =1,,n-1), TiXj=Xj Ti(ji,i+1 ), (5.53) Ti-1Yi Ti-1=Yi+1 (i=1,,n-1 ),TiYj= YjTi(j i,i+1), (5.54) Y2-1X1Y2 X1-1=T12, (5.55) YXj=qXj YandX Yj=q-1Yj X. (5.56) Here X=i=1nXi and Y=i=1nYi. They commute with {T1,,Tn-1} thanks to 5.53 and 5.55.

When q=1, t=1 we come to the elliptic or 2-extended Weyl group of type GLn due to Saito [Sai1985]. If q=1 and there are no quadratic relations, the corresponding group is the elliptic braid group (π1 of the product of n elliptic curves without the diagonals divided by 𝕊n). It was calculated by Birman [Bir1969] and Scott.

Establishing the connection with (4.32), Xi=evi, Ti=Tˆi and Yi=Δi give the so-called polynomial (or basic) representation of .

There is another version of this definition, using the element π. It is introduced from the formula Y1=T1Tn-1 π-1 (5.57) and has the following commutation relations with Xi and Ti: πXi=Xi+1π (i=1,,n-1), πXn=q-1X1π (5.58) and πTi=Ti+1π (i=1,,n-2). (5.59) In the polynomial representation this element coincides with π from Lemma 4.3. Note that it acts on the functions Xi=evi through the action of π-1 on vectors v. Considered formally, π is the image of the element P from Lemma 4.1 with respect to the Kazhdan-Lusztig automorphism, sending TT-1, YY-1, tt-1.

Since Ti-1T1X1 T1Ti-1= Xi,T1 Ti-1YiTi-1 T1=Y1, (5.60) we can reduce the list of generators. Namely, = X1,Y1,T1,,Tn-1 (5.61) or =X1,π,T1,,Tn-1. (5.62)

In terms of {T,π,X}, the list of defining relations of is as follows:

(a) XiXj=XjXi (1i,jn),
(b) the braid relations and quadratic relations for T1,,Tn-1,
(c) πXi=Xi-1π (i=1,,n-1) and πnXi=q-1Xiπn (i=1,,n),
(d) πTi=Ti+1π (i=1,,n-2) and πnTi=Tiπn (i=1,,n-1).

For instance, let us deduce (5.55) from these formulas. Substituting, the left hand side equals: (T1πTn-1-1T2-1)X1 (T2Tn-1π-1T1-1)X1-1 = T1πTn-1-1T2-1 (T2Tn-1)X1π-1 (T1-1X1-1T1-1)T1 = T1(πX1π-1X2-1)T1=T12. This representation, however, is not convenient from the viewpoint of the symmetry between Xi and Yi, which will be discussed next. It is better to use {Y} instead of π.

The algebra contains the following two affine Hecke algebras: Xt= X1,,Xn, T1,,Tn-1 . Yt= Y1,,Yn, T1,,Tn-1 . (5.63) They are isomorphic to each other by the correspondence XiYi-1. This map can be extended to an anti-involution of . It is a general statement which holds for any root systems.

Theorem 5.7. There exists an anti-involution ϕ:q,tq,t such that ϕ(Xi)=Yi-1, ϕ(Yi)=Xi-1 for i=1,,n, and ϕ(Ti)=Ti (i=1,,n-1). It preserves q,t.


We need to check that the relation (5.55) is self-dual with respect to ϕ. The other relations are obviously ϕ-invariant. One has: 1 = T1-2Y2-1 X1Y2X1-1= T1-1 { Y1-1 (T1X1T1-1) Y1T1-1X1-1 } = Y1-1X2T1-2Y1 (T1-1X1-1T1-1) =Y1-1X2T1-2 Y1X2-1. The latter can be rewritten as Y1X2-1Y1-1X2=T12, which is the ϕ-image of (5.55).

Using this involution we can establish the duality theorem for the GLn case in the same way as we did in the A1 case. Generalizing the theory to the case of arbitrary roots we can prove the Macdonald conjectures and much more. It gives a very convincing example of the power of the modern difference-operator methods.

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