q t)-laguerre polynomials ole warnaarmotivationlaguerre polynomials q-laguerre...

Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA

(q, t)-Laguerre polynomials

Ole Warnaar

School of Mathematics and Physics

The University of Queensland

Motivation: The Mukhin–Varchenko conjecture

It is well-known that the class of hypergeometric integrals known asbeta-type integrals is inextricably linked to the theory of orthogonalpolynomials.

For example, modulo a translation in α and β and the changet 7→ (1− t)/2, Euler’s beta integral∫ 1

tα−1(1− t)β−1dt =Γ(α)Γ(β)

Γ(α + β)

is nothing but the integrated orthogonality measure of the Jacobi

polynomials P(α,β)n (t):∫ 1

(1− t)α(1 + t)βP(α,β)m (t)P(α,β)

n (t) dt = 0 m 6= n

Perhaps less well-known is that beta integrals have recently arisen invarious other areas of mathematics.

For example, solutions of the Knizhnik–Zamolodchikov (KZ) equationsfrom conformal field theory are expressible in terms of beta integrals.

The KZ equations are PDEs related to Lie algebras and some elementarynotation from Lie theory is needed.

Simple Lie algebra g of rank n.

Chevalley generators ei , fi , hi , i ∈ [n] := 1, . . . , n.Simple roots αi , i ∈ [n].

Fundamental weights Λi , i ∈ [n].

The Casimir element Ω ∈ g⊗ g.

Highest weight modules Vλ and Vµ of highest weight λ and µ.

The KZ equation for a function u(z ,w) taking values in Vλ ⊗ Vµ is thesystem of partial differential equations

κ∂u

z − wu

κ∂u

w − zu

Simple Lie algebra g of rank n.

Chevalley generators ei , fi , hi , i ∈ [n] := 1, . . . , n.Simple roots αi , i ∈ [n].

Fundamental weights Λi , i ∈ [n].

The Casimir element Ω ∈ g⊗ g.

Highest weight modules Vλ and Vµ of highest weight λ and µ.

The KZ equation for a function u(z ,w) taking values in Vλ ⊗ Vµ is thesystem of partial differential equations

κ∂u

z − wu

κ∂u

w − zu

Let (·, ·) the standard bilinear symmetric form on the dual of the Cartansubalgebra. Then

(αi ,Λj) = δij

and ((αi , αj)

i,j=1= Cartan matrix of g

Example: g = An

((αi , αj)

i,j=1=

2 −1−1 2 −1

−1 2 −1−1 2

The Dynkin diagram of An.

(αi ,Λj) = δij

and ((αi , αj)

Example: g = An

((αi , αj)

i,j=1=

2 −1−1 2 −1

−1 2 −1−1 2

(αi ,Λj) = δij

and ((αi , αj)

Example: g = An

((αi , αj)

i,j=1=

2 −1−1 2 −1

−1 2 −1−1 2

To the simple root αi attach a set of ki integration variables

tjk1+···+ki

j=k1+···+ki−1+1

and writeαtj = αi

if the variable tj is attached to the root αi .

Then the master function is defined as

Φ(z ,w ; t) = (z − w)(λ,µ)k∏

(ti − z)−(λ,αti)(ti − w)−(µ,αti

1≤i<j≤k

(ti − tj)(αti

,αtj)

where k = k1 + · · ·+ kn.

To the simple root αi attach a set of ki integration variables

tjk1+···+ki

j=k1+···+ki−1+1

and writeαtj = αi

if the variable tj is attached to the root αi .

Then the master function is defined as

Φ(z ,w ; t) = (z − w)(λ,µ)k∏

(ti − z)−(λ,αti)(ti − w)−(µ,αti

1≤i<j≤k

(ti − tj)(αti

,αtj)

where k = k1 + · · ·+ kn.

Schechtman and Varchenko proved that in the subspace ofsingular vectors of weight λ+ µ−

∑i kiαi

u(z ,w) =∑∗

uIJ(z ,w)(f I vλ ⊗ f Jvµ

)where

uIJ(z ,w) =

Φ1/κ(z ,w ; t)AIJ(z ,w ; t) dt

The sum is over multisets I and J with elements taken from 1, . . . , nsuch that their union contains the number i exactly ki times, vλ and vµare the highest weight vectors of Vλ and Vµ, and

f I v =(∏

Schechtman and Varchenko proved that in the subspace ofsingular vectors of weight λ+ µ−

∑i kiαi

u(z ,w) =∑∗

uIJ(z ,w)(f I vλ ⊗ f Jvµ

)where

uIJ(z ,w) =

Φ1/κ(z ,w ; t)AIJ(z ,w ; t) dt

The sum is over multisets I and J with elements taken from 1, . . . , nsuch that their union contains the number i exactly ki times, vλ and vµare the highest weight vectors of Vλ and Vµ, and

f I v =(∏

The case g = sl2 = A1, k = 1

Chevalley generators e, f , h,

[e, f ] = h, [h, e] = 2e, [h, f ] = −2f

(0 10 0

), f =

(0 01 0

), h =

(1 00 −1

)Casimir element

Ω = e ⊗ f + f ⊗ e +1

2h ⊗ h

Highest weights λ = m1Λ1 and µ = m2Λ1.

The case g = sl2 = A1, k = 1

Chevalley generators e, f , h,

[e, f ] = h, [h, e] = 2e, [h, f ] = −2f

(0 10 0

), f =

(0 01 0

), h =

(1 00 −1

)Casimir element

Ω = e ⊗ f + f ⊗ e +1

2h ⊗ h

Highest weights λ = m1Λ1 and µ = m2Λ1.

u(z ,w) = u0(z ,w) (v1 ⊗ f v2) + u1(z ,w) (f v1 ⊗ v2)

u0(z ,w) = (z − w)m1m2

(z − t)−m1/κ(t − w)−m2/κ−1dt

u1(z ,w) = (z − w)m1m2

(z − t)−m1/κ−1(t − w)−m2/κdt

with γ a Pochhammer double loop around w and z .

If w = 0 and z = 1 one can deform γ to

γ = t ∈ R, 0 < t < 1

Both u0 and u1 become Euler beta integrals and can therefore beevaluated in terms of gamma functions.

u(z ,w) = u0(z ,w) (v1 ⊗ f v2) + u1(z ,w) (f v1 ⊗ v2)

u0(z ,w) = (z − w)m1m2

(z − t)−m1/κ(t − w)−m2/κ−1dt

u1(z ,w) = (z − w)m1m2

(z − t)−m1/κ−1(t − w)−m2/κdt

with γ a Pochhammer double loop around w and z .

If w = 0 and z = 1 one can deform γ to

γ = t ∈ R, 0 < t < 1

Both u0 and u1 become Euler beta integrals and can therefore beevaluated in terms of gamma functions.

Define the specialised master function (i.e., w = 0, z = 1)

Φ(t) =k∏

t−(λ,αti

i (1− ti )−(µ,αti

1≤i<j≤k

(ti − tj)(αti

,αtj)

Then the example of sl2, k = 1 “justifies” the following conjecture:

The Mukhin–Varchenko conjecture. If the space of singularvectors of weight λ+µ−

∑i kiαi is one-dimensional then there

exists a real domain of integration D such that∫D

|Φ(t)|1/κdt

evaluates as a product of gamma functions.

The case g = sl2 = A1, general k

Highest weights λ = m1Λ1 and µ = m2Λ1,

(α, β, γ) := (1−m1/κ, 1−m2/κ, 1/κ)

andD = t ∈ Rk , 0 ≤ tk ≤ · · · ≤ t1 ≤ 1

k∏i=1

tα−1i (1− ti )

β−1∏

1≤i<j≤k

(ti − tj)2γ dt

=k−1∏i=0

Γ(α + iγ)Γ(β + iγ)Γ(γ + iγ)

Γ(α + β + (k + i − 1)γ

)Γ(γ)

This is the Selberg integral, one of the most famous of all beta integralsand a major player in multivariable orthogonal polynomial theory.

The case g = sl2 = A1, general k

Highest weights λ = m1Λ1 and µ = m2Λ1,

(α, β, γ) := (1−m1/κ, 1−m2/κ, 1/κ)

andD = t ∈ Rk , 0 ≤ tk ≤ · · · ≤ t1 ≤ 1

k∏i=1

tα−1i (1− ti )

β−1∏

1≤i<j≤k

(ti − tj)2γ dt

=k−1∏i=0

Γ(α + iγ)Γ(β + iγ)Γ(γ + iγ)

Γ(α + β + (k + i − 1)γ

)Γ(γ)

This is the Selberg integral, one of the most famous of all beta integralsand a major player in multivariable orthogonal polynomial theory.

After this long introduction we are ready for the following questions:

If (conjecturally) beta-type integrals exist for all simple Lie algebras,what are the corresponding orthogonal polynomials?

So far we only know the answer for A1.

Can we make progress on the Mukhin–Varchenko conjecture usingthe theory of orthogonal polynomials?

The answer is affirmative and using orthogonal polynomials theSelberg integrals has now been generalised to An.

In the remainder of this talk I will focus on multivariable orthogonalpolynomials related to the first question. For simplicity of presentation Iwill only consider A1.

After this long introduction we are ready for the following questions:

If (conjecturally) beta-type integrals exist for all simple Lie algebras,what are the corresponding orthogonal polynomials?

So far we only know the answer for A1.

Can we make progress on the Mukhin–Varchenko conjecture usingthe theory of orthogonal polynomials?

The answer is affirmative and using orthogonal polynomials theSelberg integrals has now been generalised to An.

In the remainder of this talk I will focus on multivariable orthogonalpolynomials related to the first question. For simplicity of presentation Iwill only consider A1.

Although there is nothing q or quantum about the Mukhin–Varchenkoconjecture, it turns out that it is much easier to think about the problemin the q-world.

One of the reasons is that the domain D in . . . there exists a real domainof integration D such that . . . turns out to have the structure of a chainin the algebraic topology sense of the word.

In other words D is a difficult object . . .

But not in the quantum world!

Although there is nothing q or quantum about the Mukhin–Varchenkoconjecture, it turns out that it is much easier to think about the problemin the q-world.

One of the reasons is that the domain D in . . . there exists a real domainof integration D such that . . . turns out to have the structure of a chainin the algebraic topology sense of the word.

In other words D is a difficult object . . . But not in the quantum world!

The Laguerre polynomials

For α > −1 and m a nonnegative integer, Laguerre’s equation is thelinear second order ODE

x y ′′(x) + (α + 1− x)y ′(x) + m y(x) = 0

Its polynomial solutions, denoted by L(α)m (x), are known as the

(generalised) Laguerre polynomials.

Explicitly,

L(α)m (x) =

m∑i=0

(m + α

m − i

)(−x)i

For the last part of this talk it is important to remember the occurrenceof binomial coefficients.

The Laguerre polynomials form a family of orthogonal poynomials withorthogonality measure µ given by

dµ(x) = xαe−xdx

supported on the positive half-line:∫ ∞0

L(α)m (x)L(α)

n (x) dµ(x) = δmnΓ(m + α + 1)

Explicitly,

L(α)m (x) =

m∑i=0

(m + α

m − i

)(−x)i

For the last part of this talk it is important to remember the occurrenceof binomial coefficients.

The Laguerre polynomials form a family of orthogonal poynomials withorthogonality measure µ given by

dµ(x) = xαe−xdx

supported on the positive half-line:∫ ∞0

L(α)m (x)L(α)

n (x) dµ(x) = δmnΓ(m + α + 1)

The orthogonality with respect to the Laguerre measure µ may be provedas follows:

Laguerre’s equation is equivalent to the statement that L(α)m (x) is

the eigenfunction with eigenvalue m of the second order differentialoperator

L = −xd2

dx2+ (x − α− 1)

The operator L is self-adjoint with respect to the inner product

⟨f , g⟩

∫ ∞0

f (x)g(x)dµ(x)

i.e., ⟨Lf , g

⟩=⟨f ,Lg

The orthogonality with respect to the Laguerre measure µ may be provedas follows:

Laguerre’s equation is equivalent to the statement that L(α)m (x) is

the eigenfunction with eigenvalue m of the second order differentialoperator

L = −xd2

dx2+ (x − α− 1)

⟨f , g⟩

∫ ∞0

f (x)g(x)dµ(x)

i.e., ⟨Lf , g

⟩=⟨f ,Lg

The q-Laguerre polynomials

The q-Laguerre polynomials are a discrete or “quantum” variant of theclassical Laguerre polynomials. They were first studied by Hahn aroundthe 1950s.

Differential equation q-difference equation

Integration Jackson-integration

Let Dq be the q-derivative operator

(Dqf )(x) =f (xq)− f (x)

(q − 1)x

The q-Laguerre polynomial L(α)m (x) = L

(α)m (x ; q) is the eigenfunction,

with eigenvalue [m] = (1− qm)/(1− q), of the q-difference operator

L = (x + 1)Dq − q−α−1Dq−1

To describe the discrete orthogonality measure of the q-Laguerrepolynomials we need the Jackson integral∫ ∞

f (x)dqx = (1− q)∞∑

n=−∞f (qi )qi

Because of its discreteness something is lost, and there is no q-analogueof

∫ ∞0

f (cx)dx =

∫ ∞0

f (x)dx c > 0

Hence we also define∫ c·∞

f (x)dqx = (1− q)∞∑

n=−∞f (cqi ) cqi

Then the q-Laguerre polynomials satisfy∫ c·∞

L(α)m (x)L(α)

n (x) dµ(x)

= δmn[c(1− q)]α+1q−m Γq(m + α + 1)

Γq(m + 1)

θ(−acq)

θ(−c)

wheredµ(x) = xα eq(−x) dqx

eq(x) =1

(1− x)(1− xq)(1− xq2) · · ·

is the q-exponential function.

The previous results can again be restated as follows:

The q-Laguerre polynomial L(α)m (x) is the eigenfunction with

eigenvalue [m] of

L = (x + 1)Dq − q−α−1Dq−1

⟨f , g⟩

∫ c·∞

f (x)g(x) dµ(x)

The previous results can again be restated as follows:

The q-Laguerre polynomial L(α)m (x) is the eigenfunction with

eigenvalue [m] of

L = (x + 1)Dq − q−α−1Dq−1

⟨f , g⟩

∫ c·∞

f (x)g(x) dµ(x)

The (q, t)-Laguerre polynomials

For x = (x1, . . . , xn) let ∂q,i be the partial difference operator

(∂q,i f )(x) =f (. . . , qxi , . . . )− f (x)

(q − 1)xi

Define the operators Er (q, t) acting on Λn = Q(q, t)[x1, . . . , xn]Sn by

Er (q, t) =n∑

j=1j 6=i

txi − xj

xi − xj

)∂q,i

Note that for n = 1

E0(q, t) = Dq and E1(q, t) = x Dq

Lemma. For r ≥ 0, Er : Λn → Λn

(∂q,i f )(x) =f (. . . , qxi , . . . )− f (x)

(q − 1)xi

Er (q, t) =n∑

j=1j 6=i

txi − xj

xi − xj

)∂q,i

Note that for n = 1

(∂q,i f )(x) =f (. . . , qxi , . . . )− f (x)

(q − 1)xi

Er (q, t) =n∑

j=1j 6=i

txi − xj

xi − xj

)∂q,i

Note that for n = 1

Let λ = (λ1, . . . , λn) be a partition (a weakly decreasing sequence ofnonnegative integers) and L : Λn → Λn the operator

L = E1(q, t) + E0(q, t)− q−α−1E0(q−1, t−1)

The eigenfunction of L with eigenvalue

tn−1[λ1] + · · ·+ t[λn−1] + [λn]

defines the (q, t)-Laguerre polynomial L(α)λ (x) = L

(α)λ (x ; q, t).

Lemma. The L(α)λ (x) form a basis of Λn.

Let λ = (λ1, . . . , λn) be a partition (a weakly decreasing sequence ofnonnegative integers) and L : Λn → Λn the operator

L = E1(q, t) + E0(q, t)− q−α−1E0(q−1, t−1)

The eigenfunction of L with eigenvalue

tn−1[λ1] + · · ·+ t[λn−1] + [λn]

defines the (q, t)-Laguerre polynomial L(α)λ (x) = L

(α)λ (x ; q, t).

Lemma. The L(α)λ (x) form a basis of Λn.

Let t = qγ and define an inner product on Λn by

⟨f , g⟩

∫ c·∞

∫ tx1

· · ·∫ txn−1

f (x)g(x) dµ(x)

dµ(x) =∏

1≤i<j≤n

x2γi (q1−γxj/xi )γ(xj/xi )γ

n∏i=1

xαi eq(−xi ) dqxi

(a)z =(1− a)(1− aq)(1− aq2) · · ·

(1− aqz)(1− aqz+1)(1− aqz+2) · · ·

Theorem. The operator L is self-adjoint with respectto the inner product on Λn.

“Corollary.”⟨L

(α)λ , L(α)

⟩= cλ δλµ

Theorem. cλ can explicitly be computed in terms of“nice” special functions, such as theta functions and

q-gamma functions.

(α)λ , L(α)

⟩= cλ δλµ

q-gamma functions.

(α)λ , L(α)

⟩= cλ δλµ

q-gamma functions.

Note that if we let q → 1− then the previous results combine to indeedgive a beta-integral evaluation (for g = A1)

⟨1, 1⟩

∫0<xn<···<x1<∞

n∏i=1

xαi e−xi

∏1≤i<j≤n

(xi − xj)2γ dx1 · · · dxn

n−1∏i=0

Γ(α + iγ)Γ(γ + iγ)

Γ(γ)

Ingredients in proofs (including more general integrals than the previousexample) are:

Macdonald polynomials

Knop–Okounkov–Sahi interpolation polynomials

The double affine Hecke algebra

Ingredients in proofs (including more general integrals than the previousexample) are:

Macdonald polynomials

Knop–Okounkov–Sahi interpolation polynomials

The double affine Hecke algebra

Macdonald interpolation polynomials

Recall that the Laguerre polynomials are defined as a sum over binomialcoefficients.

q-Laguerre polynomials can similarly be defined as sums over theq-binomial coefficients [

k∏i=1

1− qi+n−k

1− qi

Note that [n

Mk(〈n〉)Mk(〈k〉)

whereMk(x) = q−(k

2)(x − 1)(x − q) · · · (x − qk−1)

is the Newton interpolation polynomial and

〈m〉 = qm

is the spectral vector.

Macdonald interpolation polynomials

Recall that the Laguerre polynomials are defined as a sum over binomialcoefficients.

q-Laguerre polynomials can similarly be defined as sums over theq-binomial coefficients [

k∏i=1

1− qi+n−k

1− qi

Note that [n

Mk(〈n〉)Mk(〈k〉)

whereMk(x) = q−(k

2)(x − 1)(x − q) · · · (x − qk−1)

is the Newton interpolation polynomial and

〈m〉 = qm

is the spectral vector.(q, t)-Laguerre polynomials

In much the same way, the (q, t)-Laguerre polynomials are defined assums over generalised binomial coefficients, which arise by specialisinghigher-dimensional analogues of the Newton interpolation polynomials.

To understand these generalised or (q, t)-interpolation polynomialsobserve that in the classical Newton case we have

M0(x) = 1 initial condition

Mk+1(x) = (x − 1)Mk(x/q) recurrence relation

In much the same way, the (q, t)-Laguerre polynomials are defined assums over generalised binomial coefficients, which arise by specialisinghigher-dimensional analogues of the Newton interpolation polynomials.

To understand these generalised or (q, t)-interpolation polynomialsobserve that in the classical Newton case we have

M0(x) = 1 initial condition

Mk+1(x) = (x − 1)Mk(x/q) recurrence relation

The previous recursive construction has been generalised by Knop andSahi.

This leads to polynomials, known as the interpolation Macdonaldpolynomials, which are labelled by (weak) compositions u = (u1, . . . , un).

Let si ∈ Sn be the elementary transposition interchanging the variablesxi and xi+1. Then Ti is the operator (acting on polynomials in x1, . . . , xn)defined by

Ti = t +txi+1 − xi

xi+1 − xi(si − 1)

Ti is the unique operator that commutes with functions symmetric in xi

and xi+1 such that

Ti1 = t

Tixi+1 = xi

Let si ∈ Sn be the elementary transposition interchanging the variablesxi and xi+1. Then Ti is the operator (acting on polynomials in x1, . . . , xn)defined by

Ti = t +txi+1 − xi

xi+1 − xi(si − 1)

Ti is the unique operator that commutes with functions symmetric in xi

and xi+1 such that

Ti1 = t

Tixi+1 = xi

One may verify that the Ti for i = 1, . . . , n − 1 satisfy the definingrelations of the Hecke algebra of the symmetric group:

TiTi+1Ti = Ti+1TiTi+1

TiTj = TjTi for |i − j | 6= 1

(Ti + 1)(Ti − t) = 0

The Ti are degree preserving operators. To be able to generate theinterpolation Macdonald polynomials we also need to be able to increasethe degree (like in the recurrence for the Newton interpolationpolynomials).

One may verify that the Ti for i = 1, . . . , n − 1 satisfy the definingrelations of the Hecke algebra of the symmetric group:

(Ti + 1)(Ti − t) = 0

The Ti are degree preserving operators. To be able to generate theinterpolation Macdonald polynomials we also need to be able to increasethe degree (like in the recurrence for the Newton interpolationpolynomials).

This requires the extension of the Hecke algebra to the affine Heckealgebra.

Letτx = (xn/q, x1, . . . , xn−1)

Then the raising operator φ is defined as

(φf )(x) := (xn − 1)f (τx)

Note that for n = 1 this exactly generates the recursion for the Newtoninterpolation polynomials:

φMu(x) = (x − 1)Mu(x/q) = Mu+1(x)

This requires the extension of the Hecke algebra to the affine Heckealgebra.

Letτx = (xn/q, x1, . . . , xn−1)

Then the raising operator φ is defined as

(φf )(x) := (xn − 1)f (τx)

Note that for n = 1 this exactly generates the recursion for the Newtoninterpolation polynomials:

φMu(x) = (x − 1)Mu(x/q) = Mu+1(x)

The algebraic construction of the interpolation Macdonald polynomialscan now be described as follows.

Initial conditionM(0,...,0)(x) = 1

Affine operation=degree raising

M(u2,...,un−1,u1+1)(x) = φMu(x)

Hecke operation=permuting the uIf ui < ui+1

Msiu(x) =

t − 1

〈u〉i+1/〈u〉i − 1

)Mu(x)

The above construction is analogous to that of the Schubert andGrotendieck polynomials.

The algebraic construction of the interpolation Macdonald polynomialscan now be described as follows.

Initial conditionM(0,...,0)(x) = 1

Affine operation=degree raising

M(u2,...,un−1,u1+1)(x) = φMu(x)

Hecke operation=permuting the uIf ui < ui+1

Msiu(x) =

t − 1

〈u〉i+1/〈u〉i − 1

)Mu(x)

The above construction is analogous to that of the Schubert andGrotendieck polynomials.

200 020 002 110 101 011

100 010 001

φφφ

For this to be consistent (for arbitrary n) we must have

φTi+1 = Ti φ

200 020 002 110 101 011

100 010 001

φφφ

φTi+1 = Ti φ

021 012

002 101

100 010

φ2T1 = Tn−1φ2

021 012

002 101

100 010

φ2T1 = Tn−1φ2

In summary, the generators T1, . . . ,Tn−1, φ satisfy the affine Heckealgebra

(Ti + 1)(Ti − t) = 0

φTi+1 = Ti φ

φ2T1 = Tn−1φ2

There is a further extension of the Hecke algebra that plays a central rolein the theory. Let Xi denote the operator “multiplication by xi”:

(Xi f )(x) = xi f (x)

A little lemma shows that

XiXj = XjXi

TiXi+1Ti = tXi

(Xi + Xi+1)Ti = Ti (Xi + Xi+1)

XjTi = TiXj for j 6= i , i + 1

There is a further extension of the Hecke algebra that plays a central rolein the theory. Let Xi denote the operator “multiplication by xi”:

(Xi f )(x) = xi f (x)

A little lemma shows that

XiXj = XjXi

TiXi+1Ti = tXi

(Xi + Xi+1)Ti = Ti (Xi + Xi+1)

XjTi = TiXj for j 6= i , i + 1

Let Yi be the Cherednik operator

Yi = t i−1T−1i−1 · · ·T

−11 τ−1Tn−1 · · ·Ti

for 1 ≤ i ≤ n.

A little-less-little lemma shows that

YiMu(x) = 〈u〉iMu(x) + lower order terms

That is, the top-degree component of the interpolation Macdonaldpolynomials are the eigenfunctions of the Yi .

With a bit of pain one checks the following amazing facts

YiYj = YjYi

TiYi+1Ti = tYi

(Yi + Yi+1)Ti = Ti (Yi + Yi+1)

YjTi = TiYj for j 6= i , i + 1

In other words, at the level of the algebra the “difficult” operators Yi arenot at all harder than the “easy” operators Xi .

Finally one can check that

YiXi+1T2i = tXi+1Yi

The algebra generated by the Ti ,Xi ,Yi subject to all the is

known as the double affine Hecke algebra (DAHA) (of type An−1), andwas discovered by Cherednik.

Finally one can check that

YiXi+1T2i = tXi+1Yi

The algebra generated by the Ti ,Xi ,Yi subject to all the is

known as the double affine Hecke algebra (DAHA) (of type An−1), andwas discovered by Cherednik.

The DAHA plays an essential role in proving facts about the interpolationMacdonald polynomials.

Moreover, the key ingredient in the definition of the (q, t)-Laguerrepolynomials, are the symmetrised versions of the (q, t)-binomialcoefficients [

Mu(〈v〉)Mu(〈u〉)

The End

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