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 operators 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 ) e kx = 1 k ( k − 1 x ) e kx It can be defined uniquely as the function of the form ψ ( k , x ) = k − a ( x ) e kx k with the property that ∂ ∂ k ( k ψ ( 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 ) e kx = 1 k ( k − 1 x ) e kx It can be defined uniquely as the function of the form ψ ( k , x ) = k − a ( x ) e kx k with the property that ∂ ∂ k ( k ψ ( k , x )) = 0 when k = 0 for all x . odinger equation L ψ = − k 2 ψ, where It satisfies the Schr¨ L = − D 2 + 2 x 2 .

Multi-dimensional case: configurations of hyperplanes Theorem [Berest-V.] Suppose that the Schr¨ odinger operator x ∈ C n L = ∆ + u ( x ) , 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¨ odinger operator x ∈ C n L = ∆ + u ( x ) , 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 N m i ( m i + 1)( α i , α i ) X u ( x ) = , (( α i , x ) + c i ) 2 i =1 where m i ∈ Z + .

Reflections and quasi-invariance Let A be a finite set of non-isotropic vectors α in complex Euclidean space C n with multiplicities m α ∈ N , Σ be the corresponding linear configuration of hyperplanes Π α : ( α, k ) = 0 .

Reflections and quasi-invariance Let A be a finite set of non-isotropic vectors α in complex Euclidean space C n with multiplicities m α ∈ N , Σ be the corresponding linear configuration of hyperplanes Π α : ( α, k ) = 0 . Let s α be the reflection with respect to Π α . We say that a function f ( k ) , k ∈ C n is quasi-invariant under s α if f ( s α ( k )) − f ( k ) = O (( α, k ) 2 m α ) . Equivalently, all first m α odd normal derivatives ∂ α f ( k ) = ∂ 3 α f ( k ) = . . . = ∂ 2 m α − 1 f ( k ) ≡ 0 α vanish on the hyperplane Π α .

Rational BA function related to configuration of hyperplanes Definition. A function ψ ( k , x ) , k , x ∈ C n 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 ) , α ∈A ( k , α ) m α where P ( k , x ) is a polynomial in k with highest term A ( k ) = Q 2) for all α ∈ A the function ψ ( k , x )( k , α ) m α is quasi-invariant under s α .

Rational BA function related to configuration of hyperplanes Definition. A function ψ ( k , x ) , k , x ∈ C n 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 ) , α ∈A ( k , α ) m α where P ( k , x ) is a polynomial in k with highest term A ( k ) = Q 2) for all α ∈ A the function ψ ( k , x )( k , α ) m α is quasi-invariant under s α . Theorem [CFV] . If BA function ψ exists then it is unique, symmetric with odinger equation L ψ = − k 2 ψ, where respect to x and k and satisfies the Schr¨ m α ( m α + 1)( α, α ) X L = − ∆ + ( α, x ) 2 α ∈A is a generalised Calogero-Moser operator . Conversely, if the Schr¨ odinger equation L ψ = − k 2 ψ 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 Q m be the algebra of quasi-invariants , consisting of polynomials f ( k ) satisfying ∂ α f ( k ) = ∂ 3 α f ( k ) = . . . = ∂ 2 m α − 1 f ( k ) ≡ 0 α on the hyperplane ( α, k ) = 0 for any α ∈ A .

Quasi-invariants and Harish-Chandra homomorphism Let Q m be the algebra of quasi-invariants , consisting of polynomials f ( k ) satisfying ∂ α f ( k ) = ∂ 3 α f ( k ) = . . . = ∂ 2 m α − 1 f ( k ) ≡ 0 α on 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 ) ∈ Q m there exists some differential operator L f ( x , ∂ ∂ x ) such that L f ψ ( k , x ) = f ( k ) ψ ( k , x ) . The corresponding commuting operators L f for f ∈ Q m can be given by L f = c N ( ad L ) N [ˆ f ( x )] , where c N = ( − 1) N / 2 N N ! , N = degf , ˆ f is the operator of multiplication by f ( x ) and ad A B = AB − BA .

Quasi-invariants and m -harmonic polynomials We have Q ∞ = S G ⊂ ... ⊂ Q 2 ⊂ Q 1 ⊂ Q 0 = S ( V ) . Let I m ⊂ Q m be the ideal generated by Casimirs σ 1 , . . . , σ n . The joint kernel of Calogero-Moser integrals L i = L σ i is | G | -dimensional space H m of polynomials called m -harmonics . When m = 0 they satisfy σ 1 ( ∂ ) f = σ 2 ( ∂ ) f = · · · = σ n ( ∂ ) f = 0 .

Quasi-invariants and m -harmonic polynomials We have Q ∞ = S G ⊂ ... ⊂ Q 2 ⊂ Q 1 ⊂ Q 0 = S ( V ) . Let I m ⊂ Q m be the ideal generated by Casimirs σ 1 , . . . , σ n . The joint kernel of Calogero-Moser integrals L i = L σ i is | G | -dimensional space H m of polynomials called m -harmonics . When m = 0 they satisfy σ 1 ( ∂ ) f = σ 2 ( ∂ ) f = · · · = σ n ( ∂ ) f = 0 . Feigin-V, Etingof-Ginzburg : Q m is free module over S G of rank | G | . The action of G on H m = Q m / I m is regular.

Quasi-invariants and m -harmonic polynomials We have Q ∞ = S G ⊂ ... ⊂ Q 2 ⊂ Q 1 ⊂ Q 0 = S ( V ) . Let I m ⊂ Q m be the ideal generated by Casimirs σ 1 , . . . , σ n . The joint kernel of Calogero-Moser integrals L i = L σ i is | G | -dimensional space H m of polynomials called m -harmonics . When m = 0 they satisfy σ 1 ( ∂ ) f = σ 2 ( ∂ ) f = · · · = σ n ( ∂ ) f = 0 . Feigin-V, Etingof-Ginzburg : Q m is free module over S G of rank | G | . The action of G on H m = Q m / I m is regular. When m = 0 and Weyl group G the quotient H = Q 0 / I 0 = S ( V ) / I 0 can be interpreted as the cohomology ring of the corresponding flag manifold. Question. Is there natural topological interpretation of H m = Q m / I m ? Partial results: M. Feigin-Feldman (2004)

Quasi-invariants and m -harmonic polynomials We have Q ∞ = S G ⊂ ... ⊂ Q 2 ⊂ Q 1 ⊂ Q 0 = S ( V ) . Let I m ⊂ Q m be the ideal generated by Casimirs σ 1 , . . . , σ n . The joint kernel of Calogero-Moser integrals L i = L σ i is | G | -dimensional space H m of polynomials called m -harmonics . When m = 0 they satisfy σ 1 ( ∂ ) f = σ 2 ( ∂ ) f = · · · = σ n ( ∂ ) f = 0 . Feigin-V, Etingof-Ginzburg : Q m is free module over S G of rank | G | . The action of G on H m = Q m / I m is regular. When m = 0 and Weyl group G the quotient H = Q 0 / I 0 = S ( V ) / I 0 can be interpreted as the cohomology ring of the corresponding flag manifold. Question. Is there natural topological interpretation of H m = Q m / I m ? Partial results: M. Feigin-Feldman (2004) Hilbert-Poincare series ( Felder-V , 2003): for G = S n n 1 mn ( n − 1) X Y t m ( ℓ k − a k )+ ℓ k P ( Q m , t ) = n ! t h k (1 − t h k ) . 2 λ ∈ Y n k =1 Idea: relation with KZ equation .

Monodromy-free Schr¨ odinger operators Consider first 1 D Schr¨ odinger operator D = d L = − D 2 + u ( z ) , d z , where u = u ( z ) is a meromorphic function of z ∈ C .

Monodromy-free Schr¨ odinger operators Consider first 1 D Schr¨ odinger operator D = d L = − D 2 + u ( z ) , d z , 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¨ odinger equation in the complex domain − ψ ′′ + u ( z ) ψ = k ψ are meromorphic (and hence single-valued) for all k ∈ C .