Baker-Akhiezer functions and configurations of hyperplanes Alexander - - PowerPoint PPT Presentation

Baker-Akhiezer functions and configurations of hyperplanes

Alexander Veselov, Loughborough University ENIGMA conference on Geometry and Integrability, Obergurgl, December 2008

Plan

◮ BA function related to configuration of hyperplanes ◮ Trivial monodromy and locus configurations ◮ Coxeter configurations and deformed root systems ◮ BA function as iterated residue and Selberg-type integral ◮ Applications to Hadamard’s problem ◮ Some open problems

References

• O. Chalykh, M. Feigin, A.V. Comm. Math. Phys. 206 (1999), 533-566
• O. Chalykh, A.V. Phys. Letters A 285 (2001), 339-349
• M. Feigin, A.V. IMRN 10 (2002), 521-545
• G. Felder, A.V. Moscow Math J. 3 (2003), 1269-1291

A.N. Sergeev, A.V. Comm. Math Phys 245 (2004), 249-278-

• G. Felder, A.V. arXiv (2008)

Brief history

Clebsch, Gordan (1860s): generalisations of the exponential function on a Riemann surface of arbitrary genus Burchnall-Chaundy, Baker (1920s): relation with commuting differential

• perators

Akhiezer (1961): relations with spectral theory Novikov, Dubrovin, Its, Matveev (1974-75): relations with the theory of KdV equation and finite-gap theory Krichever (1976): general notion of BA function Chalykh, M. Feigin, Veselov (1998): BA function related to configuration of hyperplanes Motivation: links with Hadamard’s problem (Berest, Veselov (1993))

Simplest example in dimension 1

The simplest BA function is ψ(k, x) = (1 − 1 kx )ekx = 1 k (k − 1 x )ekx It can be defined uniquely as the function of the form ψ(k, x) = k − a(x) k ekx with the property that ∂ ∂k (kψ(k, x)) = 0 when k = 0 for all x.

Simplest example in dimension 1

The simplest BA function is ψ(k, x) = (1 − 1 kx )ekx = 1 k (k − 1 x )ekx It can be defined uniquely as the function of the form ψ(k, x) = k − a(x) k ekx with the property that ∂ ∂k (kψ(k, x)) = 0 when k = 0 for all x. It satisfies the Schr¨

• dinger equation Lψ = −k2ψ, where

L = −D2 + 2 x2 .

Multi-dimensional case: configurations of hyperplanes

Theorem [Berest-V.] Suppose that the Schr¨

• dinger operator

L = ∆ + u(x), x ∈ Cn with meromorphic potential u(x) has an eigenfunction of the form ϕ(x, k) = P(k, x)e(k,x), where P(k, x) is a polynomial in k with coefficients meromorphic in x, then the singularities of the potential lie on a configuration of hyperplanes (possibly, infinite).

Multi-dimensional case: configurations of hyperplanes

Theorem [Berest-V.] Suppose that the Schr¨

• dinger operator

L = ∆ + u(x), x ∈ Cn with meromorphic potential u(x) has an eigenfunction of the form ϕ(x, k) = P(k, x)e(k,x), where P(k, x) is a polynomial in k with coefficients meromorphic in x, then the singularities of the potential lie on a configuration of hyperplanes (possibly, infinite). In the rational case the potential has a form u(x) =

N

X

i=1

mi(mi + 1)(αi, αi) ((αi, x) + ci)2 , where mi ∈ Z+.

Reflections and quasi-invariance

Let A be a finite set of non-isotropic vectors α in complex Euclidean space Cn with multiplicities mα ∈ N, Σ be the corresponding linear configuration of hyperplanes Πα : (α, k) = 0.

Reflections and quasi-invariance

Let A be a finite set of non-isotropic vectors α in complex Euclidean space Cn with multiplicities mα ∈ N, Σ be the corresponding linear configuration of hyperplanes Πα : (α, k) = 0. Let sα be the reflection with respect to Πα. We say that a function f (k), k ∈ Cn is quasi-invariant under sα if f (sα(k)) − f (k) = O((α, k)2mα). Equivalently, all first mα odd normal derivatives ∂αf (k) = ∂3

αf (k) = . . . = ∂2mα−1 α

f (k) ≡ 0 vanish on the hyperplane Πα.

Rational BA function related to configuration of hyperplanes

• Definition. A function ψ(k, x), k, x ∈ Cn is called rational Baker-Akhiezer

function related to configuration of hyperplanes Σ if 1) ψ(k, x) has a form ψ(k, x) = P(k, x) A(k) e(k,x), where P(k, x) is a polynomial in k with highest term A(k) = Q

α∈A (k, α)mα

2) for all α ∈ A the function ψ(k, x)(k, α)mα is quasi-invariant under sα.

Rational BA function related to configuration of hyperplanes

• Definition. A function ψ(k, x), k, x ∈ Cn is called rational Baker-Akhiezer

function related to configuration of hyperplanes Σ if 1) ψ(k, x) has a form ψ(k, x) = P(k, x) A(k) e(k,x), where P(k, x) is a polynomial in k with highest term A(k) = Q

α∈A (k, α)mα

2) for all α ∈ A the function ψ(k, x)(k, α)mα is quasi-invariant under sα. Theorem [CFV]. If BA function ψ exists then it is unique, symmetric with respect to x and k and satisfies the Schr¨

• dinger equation Lψ = −k2ψ, where

L = −∆ + X

α∈A

mα(mα + 1)(α, α) (α, x)2 is a generalised Calogero-Moser operator. Conversely, if the Schr¨

• dinger

equation Lψ = −k2ψ has a solution ψ(k, x) of the form above, then ψ(k, x) has to be BA function.

Quasi-invariants and Harish-Chandra homomorphism

Let Qm be the algebra of quasi-invariants, consisting of polynomials f (k) satisfying ∂αf (k) = ∂3

αf (k) = . . . = ∂2mα−1 α

f (k) ≡ 0

• n the hyperplane (α, k) = 0 for any α ∈ A.

Quasi-invariants and Harish-Chandra homomorphism

Let Qm be the algebra of quasi-invariants, consisting of polynomials f (k) satisfying ∂αf (k) = ∂3

αf (k) = . . . = ∂2mα−1 α

f (k) ≡ 0

• n the hyperplane (α, k) = 0 for any α ∈ A.

Theorem [CFV, Berest] If the BA function ψ(k, x) exists then for any quasi-invariant f (k) ∈ Qm there exists some differential operator Lf (x, ∂

∂x ) such

that Lf ψ(k, x) = f (k)ψ(k, x). The corresponding commuting operators Lf for f ∈ Qm can be given by Lf = cN(adL)N[ˆ f (x)], where cN = (−1)N/2NN!, N = degf , ˆ f is the operator of multiplication by f (x) and adAB = AB − BA .

Quasi-invariants and m-harmonic polynomials

We have Q∞ = SG ⊂ ... ⊂ Q2 ⊂ Q1 ⊂ Q0 = S(V ). Let Im ⊂ Qm be the ideal generated by Casimirs σ1, . . . , σn. The joint kernel of Calogero-Moser integrals Li = Lσi is |G|-dimensional space Hm of polynomials called m-harmonics. When m = 0 they satisfy σ1(∂)f = σ2(∂)f = · · · = σn(∂)f = 0.

Quasi-invariants and m-harmonic polynomials

We have Q∞ = SG ⊂ ... ⊂ Q2 ⊂ Q1 ⊂ Q0 = S(V ). Let Im ⊂ Qm be the ideal generated by Casimirs σ1, . . . , σn. The joint kernel of Calogero-Moser integrals Li = Lσi is |G|-dimensional space Hm of polynomials called m-harmonics. When m = 0 they satisfy σ1(∂)f = σ2(∂)f = · · · = σn(∂)f = 0. Feigin-V, Etingof-Ginzburg: Qm is free module over SG of rank |G|. The action of G on Hm = Qm/Im is regular.

Quasi-invariants and m-harmonic polynomials

We have Q∞ = SG ⊂ ... ⊂ Q2 ⊂ Q1 ⊂ Q0 = S(V ). Let Im ⊂ Qm be the ideal generated by Casimirs σ1, . . . , σn. The joint kernel of Calogero-Moser integrals Li = Lσi is |G|-dimensional space Hm of polynomials called m-harmonics. When m = 0 they satisfy σ1(∂)f = σ2(∂)f = · · · = σn(∂)f = 0. Feigin-V, Etingof-Ginzburg: Qm is free module over SG of rank |G|. The action of G on Hm = Qm/Im is regular. When m = 0 and Weyl group G the quotient H = Q0/I0 = S(V )/I0 can be interpreted as the cohomology ring of the corresponding flag manifold.

• Question. Is there natural topological interpretation of Hm = Qm/Im ?

Partial results: M. Feigin-Feldman (2004)

Quasi-invariants and m-harmonic polynomials

We have Q∞ = SG ⊂ ... ⊂ Q2 ⊂ Q1 ⊂ Q0 = S(V ). Let Im ⊂ Qm be the ideal generated by Casimirs σ1, . . . , σn. The joint kernel of Calogero-Moser integrals Li = Lσi is |G|-dimensional space Hm of polynomials called m-harmonics. When m = 0 they satisfy σ1(∂)f = σ2(∂)f = · · · = σn(∂)f = 0. Feigin-V, Etingof-Ginzburg: Qm is free module over SG of rank |G|. The action of G on Hm = Qm/Im is regular. When m = 0 and Weyl group G the quotient H = Q0/I0 = S(V )/I0 can be interpreted as the cohomology ring of the corresponding flag manifold.

• Question. Is there natural topological interpretation of Hm = Qm/Im ?

Partial results: M. Feigin-Feldman (2004) Hilbert-Poincare series (Felder-V, 2003): for G = Sn P(Qm, t) = n! t

mn(n−1) 2

X

λ∈Yn n

Y

k=1

tm(ℓk −ak )+ℓk 1 hk(1 − thk ). Idea: relation with KZ equation.

Monodromy-free Schr¨

• dinger operators

Consider first 1D Schr¨

• dinger operator

L = −D2 + u(z), D = d dz , where u = u(z) is a meromorphic function of z ∈ C.

Monodromy-free Schr¨

• dinger operators

Consider first 1D Schr¨

• dinger operator

L = −D2 + u(z), D = d dz , where u = u(z) is a meromorphic function of z ∈ C.

• Definition. Operator L is called monodromy free if all solutions of the

corresponding Schr¨

• dinger equation in the complex domain

−ψ′′ + u(z)ψ = kψ are meromorphic (and hence single-valued) for all k ∈ C.

Monodromy-free Schr¨

• dinger operators

Consider first 1D Schr¨

• dinger operator

L = −D2 + u(z), D = d dz , where u = u(z) is a meromorphic function of z ∈ C.

• Definition. Operator L is called monodromy free if all solutions of the

corresponding Schr¨

• dinger equation in the complex domain

−ψ′′ + u(z)ψ = kψ are meromorphic (and hence single-valued) for all k ∈ C. Theorem [Duistermaat–Gr¨ unbaum]. A Schr¨

• dinger operator L is

monodromy-free iff all poles of u are of second order and the coefficients of its Laurent expansion u(z) = c−2 (z − z0)2 + X

n≥−1

cn(z − z0)n around any pole z0 satisfy the conditions i) c−2 = m(m + 1) for a positive integer m and ii) c2k−1 = 0 for k = 0, . . . , m.

Locus configurations and BA function

A configuration of hyperplanes Σ is called a locus configuration if the corresponding potential uΣ(x) = X

α∈A

mα(mα + 1)(α, α) (α, x)2 satisfies the quasi-invariance conditions uΣ(x) − uΣ(si(x)) = O((α, x)2mα) for all α ∈ A. This is a condition that the Schr¨

• dinger operator L = −△ + uΣ

has trivial monodromy.

Locus configurations and BA function

A configuration of hyperplanes Σ is called a locus configuration if the corresponding potential uΣ(x) = X

α∈A

mα(mα + 1)(α, α) (α, x)2 satisfies the quasi-invariance conditions uΣ(x) − uΣ(si(x)) = O((α, x)2mα) for all α ∈ A. This is a condition that the Schr¨

• dinger operator L = −△ + uΣ

has trivial monodromy. Theorem [CFV]. The BA function exists iff the corresponding Σ is a locus

• configuration. In that case it can be given by the Berest formula

ψ(k, x) = [(−2)MM!A(k)]−1(L + k2)M[ Y

α∈A

(α, x)mαexp(k, x)], where M = P

α∈A mα, A(k) = Q α∈A(α, k)mα

Examples of locus configurations

Coxeter configurations: reflection hyperplanes of a Coxeter group G taken with integer G-invariant multiplicities

Examples of locus configurations

Coxeter configurations: reflection hyperplanes of a Coxeter group G taken with integer G-invariant multiplicities Deformed root systems [CFV] An,1(m) =  ei − ej with multiplicity m ei − √men+1 with multiplicity 1 L(n,1)

m

= −

n

X

i=1

∂2 ∂x2

i

− m ∂2 ∂y 2 +

n

X

i<j

2m(m + 1) (xi − xj)2 +

n

X

i=1

2(m + 1) (xi − y)2

Examples of locus configurations

Coxeter configurations: reflection hyperplanes of a Coxeter group G taken with integer G-invariant multiplicities Deformed root systems [CFV] An,1(m) =  ei − ej with multiplicity m ei − √men+1 with multiplicity 1 L(n,1)

m

= −

n

X

i=1

∂2 ∂x2

i

− m ∂2 ∂y 2 +

n

X

i<j

2m(m + 1) (xi − xj)2 +

n

X

i=1

2(m + 1) (xi − y)2 Cn,1(m, l) = 8 > > < > > : ei ± ej with multiplicity k 2ei with multiplicity m 2 √ ken+1 with multiplicity l ei ± √ ken+1 with multiplicity 1 where l and m are integer parameters such that k = 2m+1

2l+1 ∈ Z, 1 ≤ i < j ≤ n.

2D examples A

2(m)

θ cosθ = m m +1 m 1 1 m ϕ 1 1 C2(m,l) l cos2ϕ = m− l m + l+1

Other locus configurations

Berest-Lutsenko configurations are 2D configurations with the potential u(r, ϕ) = 1 r 2 ∂2 ∂ϕ2 log W (χ1, . . . , χN), where χi(ϕ) = cos(kiϕ + θi), i = 1, . . . , n, where 0 < k1 < · · · < kN are positive integers, θi ∈ C are arbitrary complex parameters and W denotes the Wronskian. They give all linear locus configurations in dimension 2.

Other locus configurations

Berest-Lutsenko configurations are 2D configurations with the potential u(r, ϕ) = 1 r 2 ∂2 ∂ϕ2 log W (χ1, . . . , χN), where χi(ϕ) = cos(kiϕ + θi), i = 1, . . . , n, where 0 < k1 < · · · < kN are positive integers, θi ∈ C are arbitrary complex parameters and W denotes the Wronskian. They give all linear locus configurations in dimension 2. A special complex series [Chalykh-V]: An−1,2(m) = 8 > > < > > : ei − ej, 1 ≤ i < j ≤ n, with multiplicity m , ei − √men+1, i = 1, . . . , n with multiplicity 1 , ei − √−1 − men+2, i = 1, . . . , n with multiplicity 1 , √men+1 − √−1 − men+2 with multiplicity 1 .

BA function as iterated residue

Felder-V: The rational Baker-Akhiezer function for the configuration An(m) can be given by the following iterated residue formula ψ(n)

m (x, k) =

„ m! 2πi « n(n−1)

2

ekn(x1+···+xn)A(x)1+mA(k)−m I

Σ

ωm, where ωm = Y

i≤j,l≤j+1≤n

(ti,j − tl,j+1)−m−1 Y

1≤i<l≤j<n

(ti,j − tl,j)2+2m Y

l≤j<n

e(kj −kj+1)tl,j dtl,j, Σ as the product of circles |tk,j − xk| = ǫ(n − j) with ǫ small enough and tk,n = xk. Based on Stanley identity: cf. Awata et al, Kazarnovski-Krol, Okounkov-Olshanski, Kuznetsov-Mangazeev-Sklyanin, Langmann

BA function as Selberg-type integral

The rational BA function can be given by the following Selberg-type integral ψ(n)

m (x, k) = ((−1)m+1m!)− n(n−1)

2

ekn(x1+···+xn)A(x)−mA(k)m+1 Z

Γ

αm with αm = Y

i≤j,l≤j+1≤n

(ti,j − tl,j+1)m Y

1≤i<l≤j<n

(ti,j − tl,j)−2m Y

l≤j<n

e(kj −kj+1)tl,j dtl,j and the integration contour Γ such that ti,j = ti,j+1 + τi,j with real variables τi,j, 1 ≤ i ≤ j = 1, . . . , n − 1 changing from zero to infinity.

BA function as Selberg-type integral

The rational BA function can be given by the following Selberg-type integral ψ(n)

m (x, k) = ((−1)m+1m!)− n(n−1)

2

ekn(x1+···+xn)A(x)−mA(k)m+1 Z

Γ

αm with αm = Y

i≤j,l≤j+1≤n

(ti,j − tl,j+1)m Y

1≤i<l≤j<n

(ti,j − tl,j)−2m Y

l≤j<n

e(kj −kj+1)tl,j dtl,j and the integration contour Γ such that ti,j = ti,j+1 + τi,j with real variables τi,j, 1 ≤ i ≤ j = 1, . . . , n − 1 changing from zero to infinity. Remark 1. This may be considered as a new case of explicit calculation of Selberg-type integrals.

BA function as Selberg-type integral

The rational BA function can be given by the following Selberg-type integral ψ(n)

m (x, k) = ((−1)m+1m!)− n(n−1)

2

ekn(x1+···+xn)A(x)−mA(k)m+1 Z

Γ

αm with αm = Y

i≤j,l≤j+1≤n

(ti,j − tl,j+1)m Y

1≤i<l≤j<n

(ti,j − tl,j)−2m Y

l≤j<n

e(kj −kj+1)tl,j dtl,j and the integration contour Γ such that ti,j = ti,j+1 + τi,j with real variables τi,j, 1 ≤ i ≤ j = 1, . . . , n − 1 changing from zero to infinity. Remark 1. This may be considered as a new case of explicit calculation of Selberg-type integrals. Remark 2. These two representations can be related by an analytic continuation from m to −m − 1, which is very similar to the Riemann’s proof

• f the reflection property of the Riemann zeta-function.

Example: two-particle case

In that case the rational BA function is known to be Ψ(2)

m = (k1−k2)−m(D12−

2m x1 − x2 )(D12−2(m − 1) x1 − x2 ) . . . (D12− 2 x1 − x2 ) exp(k1x1+k2x2), where D12 = ∂ ∂x1 − ∂ ∂x2 .

Example: two-particle case

In that case the rational BA function is known to be Ψ(2)

m = (k1−k2)−m(D12−

2m x1 − x2 )(D12−2(m − 1) x1 − x2 ) . . . (D12− 2 x1 − x2 ) exp(k1x1+k2x2), where D12 = ∂ ∂x1 − ∂ ∂x2 . We have two different representations for it. The first one is as a residue Ψ(2)

m = m!(x1 − x2)m+1

(k1 − k2)m ek2(x1+x2)Resz=x1 e(k1−k2)z (z − x1)m+1(z − x2)m+1 , the second one is the integral Ψ(2)

m = (k2 − k1)m+1

m!(x1 − x2)m ek2(x1+x2) Z +∞

x1

(z − x1)m(z − x2)me(k1−k2)zdz, which in this case can be effectively computed using the Γ-integral Γ(a) = Z +∞ za−1e−zdz = (a − 1)!

Application: Huygens’ principle and Hadamard’s problem

Huygens’ Principle in the narrow sense: an instantaneous signal remains instantaneous for every observer at each later time. Mathematically: the fundamental solution of the corresponding hyperbolic equation is located on the characteristic conoid.

Application: Huygens’ principle and Hadamard’s problem

Huygens’ Principle in the narrow sense: an instantaneous signal remains instantaneous for every observer at each later time. Mathematically: the fundamental solution of the corresponding hyperbolic equation is located on the characteristic conoid. Example: pure wave equation in Rn: ✷n(φ) = 0, ✷n = ∂2

0 − ∂2 1 − . . . − ∂2 n,

∂i = ∂ ∂xi , x0 = t. Huygens’ Principle holds only in odd dimensions starting from 3. Fundamental solution in that case is Φ = C(n)δ(k)(t2 − x2), k = n − 3 2 .

Hadamard’s problem

Describe all second-order hyperbolic equations for which Huygens’ Principle holds. Special case: hyperbolic equations of the form (✷n + u(x))φ = 0.

Hadamard’s problem

Describe all second-order hyperbolic equations for which Huygens’ Principle holds. Special case: hyperbolic equations of the form (✷n + u(x))φ = 0.

◮ 1923

Hadamard: Dimension n must be odd and larger than 1. ”Hadamard’s Conjecture”: HP holds only for pure wave equations

Hadamard’s problem

Describe all second-order hyperbolic equations for which Huygens’ Principle holds. Special case: hyperbolic equations of the form (✷n + u(x))φ = 0.

◮ 1923

Hadamard: Dimension n must be odd and larger than 1. ”Hadamard’s Conjecture”: HP holds only for pure wave equations

◮ 1939-40

Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero.

Hadamard’s problem

Describe all second-order hyperbolic equations for which Huygens’ Principle holds. Special case: hyperbolic equations of the form (✷n + u(x))φ = 0.

◮ 1923

Hadamard: Dimension n must be odd and larger than 1. ”Hadamard’s Conjecture”: HP holds only for pure wave equations

◮ 1939-40

Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero.

◮ 1953-55

Stellmacher: If u = m(m+1)

x2

1

with integer m then HP holds in any odd dimension starting from 2m + 3.

Hadamard’s problem

Describe all second-order hyperbolic equations for which Huygens’ Principle holds. Special case: hyperbolic equations of the form (✷n + u(x))φ = 0.

◮ 1923

Hadamard: Dimension n must be odd and larger than 1. ”Hadamard’s Conjecture”: HP holds only for pure wave equations

◮ 1939-40

Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero.

◮ 1953-55

Stellmacher: If u = m(m+1)

x2

1

with integer m then HP holds in any odd dimension starting from 2m + 3.

◮ 1967-69

Stellmacher and Lagnese: solution of the Hadamard problem in the class (✷ + u(x1))φ = 0

Hadamard’s problem

Describe all second-order hyperbolic equations for which Huygens’ Principle holds. Special case: hyperbolic equations of the form (✷n + u(x))φ = 0.

◮ 1923

Hadamard: Dimension n must be odd and larger than 1. ”Hadamard’s Conjecture”: HP holds only for pure wave equations

◮ 1939-40

Mathisson, Asgeirsson, Hadamard: If n = 3 then u must be zero.

◮ 1953-55

Stellmacher: If u = m(m+1)

x2

1

with integer m then HP holds in any odd dimension starting from 2m + 3.

◮ 1967-69

Stellmacher and Lagnese: solution of the Hadamard problem in the class (✷ + u(x1))φ = 0

◮ 1993 Berest-V: examples related to Coxeter groups

Main result

Theorem [CFV] Hyperbolic equation (✷ + u(x))φ = 0 with the potential u(x) =

K

X

j=1

mj(mj + 1)(αj, αj) ((αj, x) + cj)2 related to any locus configuration satisfies HP if n is odd and large enough: n ≥ 2M + 3, M = PK

j=1 mj.

Main result

Theorem [CFV] Hyperbolic equation (✷ + u(x))φ = 0 with the potential u(x) =

K

X

j=1

mj(mj + 1)(αj, αj) ((αj, x) + cj)2 related to any locus configuration satisfies HP if n is odd and large enough: n ≥ 2M + 3, M = PK

j=1 mj.

Conversely, if the equation (✷ + u(x))φ = 0 satisfies HP and all the Hadamard’s coefficients are rational functions, then the potential u(x) must be related to locus configuration.

Some open problems

Some open problems

◮ Classification of locus configurations

Partial results: CFV, Sergeev-V

Some open problems

◮ Classification of locus configurations

Partial results: CFV, Sergeev-V

◮ Effective description of quasi-invariants and m-harmonic polynomials

Partial results: Feigin-V, Felder-V, Etingof-Ginzburg

Some open problems

◮ Classification of locus configurations

Partial results: CFV, Sergeev-V

◮ Effective description of quasi-invariants and m-harmonic polynomials

Partial results: Feigin-V, Felder-V, Etingof-Ginzburg

◮ Elliptic case: generalised Lam`

e operators Partial results: Chalykh-Etingof-Oblomkov

Some open problems

◮ Classification of locus configurations

Partial results: CFV, Sergeev-V

◮ Effective description of quasi-invariants and m-harmonic polynomials

Partial results: Feigin-V, Felder-V, Etingof-Ginzburg

◮ Elliptic case: generalised Lam`

e operators Partial results: Chalykh-Etingof-Oblomkov

◮ Spectral theory of the deformed Calogero-Moser systems