q t)-Laguerre polynomials Ole WarnaarMotivationLaguerre polynomials q-Laguerre...
Transcript of 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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
0
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
(1− t)α(1 + t)βP(α,β)m (t)P(α,β)
n (t) dt = 0 m 6= n
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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=
Ω
z − wu
κ∂u
∂w=
Ω
w − zu
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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=
Ω
z − wu
κ∂u
∂w=
Ω
w − zu
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Let (·, ·) the standard bilinear symmetric form on the dual of the Cartansubalgebra. Then
(αi ,Λj) = δij
and ((αi , αj)
)n
i,j=1= Cartan matrix of g
Example: g = An
((αi , αj)
)n
i,j=1=
2 −1−1 2 −1
. . .
−1 2 −1−1 2
n1 2
The Dynkin diagram of An.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Let (·, ·) the standard bilinear symmetric form on the dual of the Cartansubalgebra. Then
(αi ,Λj) = δij
and ((αi , αj)
)n
i,j=1= Cartan matrix of g
Example: g = An
((αi , αj)
)n
i,j=1=
2 −1−1 2 −1
. . .
−1 2 −1−1 2
n1 2
The Dynkin diagram of An.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Let (·, ·) the standard bilinear symmetric form on the dual of the Cartansubalgebra. Then
(αi ,Λj) = δij
and ((αi , αj)
)n
i,j=1= Cartan matrix of g
Example: g = An
((αi , αj)
)n
i,j=1=
2 −1−1 2 −1
. . .
−1 2 −1−1 2
n1 2
The Dynkin diagram of An.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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∏
i=1
(ti − z)−(λ,αti)(ti − w)−(µ,αti
)
×∏
1≤i<j≤k
(ti − tj)(αti
,αtj)
where k = k1 + · · ·+ kn.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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∏
i=1
(ti − z)−(λ,αti)(ti − w)−(µ,αti
)
×∏
1≤i<j≤k
(ti − tj)(αti
,αtj)
where k = k1 + · · ·+ kn.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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) =
∫γ
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 =(∏
i∈I
fi)v
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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) =
∫γ
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 =(∏
i∈I
fi)v
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
The case g = sl2 = A1, k = 1
Chevalley generators e, f , h,
[e, f ] = h, [h, e] = 2e, [h, f ] = −2f
e =
(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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
The case g = sl2 = A1, k = 1
Chevalley generators e, f , h,
[e, f ] = h, [h, e] = 2e, [h, f ] = −2f
e =
(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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
u(z ,w) = u0(z ,w) (v1 ⊗ f v2) + u1(z ,w) (f v1 ⊗ v2)
where
u0(z ,w) = (z − w)m1m2
2κ
∫γ
(z − t)−m1/κ(t − w)−m2/κ−1dt
u1(z ,w) = (z − w)m1m2
2κ
∫γ
(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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
u(z ,w) = u0(z ,w) (v1 ⊗ f v2) + u1(z ,w) (f v1 ⊗ v2)
where
u0(z ,w) = (z − w)m1m2
2κ
∫γ
(z − t)−m1/κ(t − w)−m2/κ−1dt
u1(z ,w) = (z − w)m1m2
2κ
∫γ
(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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Define the specialised master function (i.e., w = 0, z = 1)
Φ(t) =k∏
i=1
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
∫D
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
∫D
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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!
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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!
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Explicitly,
L(α)m (x) =
m∑i=0
(m + α
m − i
)(−x)i
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)
m!
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Explicitly,
L(α)m (x) =
m∑i=0
(m + α
m − i
)(−x)i
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)
m!
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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)
d
dx
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
⟩
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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)
d
dx
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
⟩
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
To describe the discrete orthogonality measure of the q-Laguerrepolynomials we need the Jackson integral∫ ∞
0
f (x)dqx = (1− q)∞∑
n=−∞f (qi )qi
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Because of its discreteness something is lost, and there is no q-analogueof
c
∫ ∞0
f (cx)dx =
∫ ∞0
f (x)dx c > 0
Hence we also define∫ c·∞
0
f (x)dqx = (1− q)∞∑
n=−∞f (cqi ) cqi
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Then the q-Laguerre polynomials satisfy∫ c·∞
0
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
and
eq(x) =1
(1− x)(1− xq)(1− xq2) · · ·
is the q-exponential function.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
The operator L is self-adjoint with respect to the inner product
⟨f , g⟩
=
∫ c·∞
0
f (x)g(x) dµ(x)
wheredµ(x) = xα eq(−x) dqx
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
The operator L is self-adjoint with respect to the inner product
⟨f , g⟩
=
∫ c·∞
0
f (x)g(x) dµ(x)
wheredµ(x) = xα eq(−x) dqx
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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∑
i=1
x ri
(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, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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∑
i=1
x ri
(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, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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∑
i=1
x ri
(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, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Let t = qγ and define an inner product on Λn by
⟨f , g⟩
=
∫ c·∞
x1=0
∫ tx1
x2=0
· · ·∫ txn−1
xn=0
f (x)g(x) dµ(x)
where
dµ(x) =∏
1≤i<j≤n
x2γi (q1−γxj/xi )γ(xj/xi )γ
n∏i=1
xαi eq(−xi ) dqxi
and
(a)z =(1− a)(1− aq)(1− aq2) · · ·
(1− aqz)(1− aqz+1)(1− aqz+2) · · ·
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Theorem. The operator L is self-adjoint with respectto the inner product on Λn.
“Corollary.”⟨L
(α)λ , L(α)
µ
⟩= cλ δλµ
Theorem. cλ can explicitly be computed in terms of“nice” special functions, such as theta functions and
q-gamma functions.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Theorem. The operator L is self-adjoint with respectto the inner product on Λn.
“Corollary.”⟨L
(α)λ , L(α)
µ
⟩= cλ δλµ
Theorem. cλ can explicitly be computed in terms of“nice” special functions, such as theta functions and
q-gamma functions.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Theorem. The operator L is self-adjoint with respectto the inner product on Λn.
“Corollary.”⟨L
(α)λ , L(α)
µ
⟩= cλ δλµ
Theorem. cλ can explicitly be computed in terms of“nice” special functions, such as theta functions and
q-gamma functions.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
=1
n!
n−1∏i=0
Γ(α + iγ)Γ(γ + iγ)
Γ(γ)
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Ingredients in proofs (including more general integrals than the previousexample) are:
Macdonald polynomials
Knop–Okounkov–Sahi interpolation polynomials
The double affine Hecke algebra
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
Ingredients in proofs (including more general integrals than the previousexample) are:
Macdonald polynomials
Knop–Okounkov–Sahi interpolation polynomials
The double affine Hecke algebra
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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 [
n
k
]=
k∏i=1
1− qi+n−k
1− qi
Note that [n
k
]=
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
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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 [
n
k
]=
k∏i=1
1− qi+n−k
1− qi
Note that [n
k
]=
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
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
and
Mk+1(x) = (x − 1)Mk(x/q) recurrence relation
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
and
Mk+1(x) = (x − 1)Mk(x/q) recurrence relation
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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).
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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).
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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).
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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)
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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)
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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) =
(Ti +
t − 1
〈u〉i+1/〈u〉i − 1
)Mu(x)
The above construction is analogous to that of the Schubert andGrotendieck polynomials.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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) =
(Ti +
t − 1
〈u〉i+1/〈u〉i − 1
)Mu(x)
The above construction is analogous to that of the Schubert andGrotendieck polynomials.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
200 020 002 110 101 011
100 010 001
000
φ
φφφ
2
12
2
21
For this to be consistent (for arbitrary n) we must have
φTi+1 = Ti φ
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
200 020 002 110 101 011
100 010 001
000
φ
φφφ
2
12
2
21
For this to be consistent (for arbitrary n) we must have
φTi+1 = Ti φ
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
021 012
002 101
100 010
φ
φφ
φ
2
1
For this to be consistent (for arbitrary n) we must have
φ2T1 = Tn−1φ2
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
021 012
002 101
100 010
φ
φφ
φ
2
1
For this to be consistent (for arbitrary n) we must have
φ2T1 = Tn−1φ2
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
In summary, the generators T1, . . . ,Tn−1, φ satisfy the affine Heckealgebra
TiTi+1Ti = Ti+1TiTi+1
TiTj = TjTi for |i − j | 6= 1
(Ti + 1)(Ti − t) = 0
φTi+1 = Ti φ
φ2T1 = Tn−1φ2
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
DAHA
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
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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 .
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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 .
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
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 [
v
u
]=
Mu(〈v〉)Mu(〈u〉)
.
(q, t)-Laguerre polynomials
Motivation Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials DAHA
The End
(q, t)-Laguerre polynomials