Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

Feigin, Misha; Hallnäs, Martin; Веселов, А. П.

doi:10.1063/1.4804615

Cited by 8 publications

(14 citation statements)

References 35 publications

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“…A Gaussian bilinear form on the space of quasi-invariants when a configuration satisfies the conditions (α j (k)) can be defined (cf. [10]), it might be relevant to the analysis of the Gorenstein property. Furthermore, it would be important to clarify in the two-dimensional case whether the configurations A q (m,m,1 n ) that we considered in this paper exhaust the set BA.…”

Section: Discussionmentioning

confidence: 99%

“…We show that these rings have nice algebraic properties that do not hold in case of general locus configurations. Another nice feature of this subclass of planar locus configurations is explored in [10] where the integrals of Macdonald-Mehta type associated with these configurations are explicitly computed.…”

Section: Definition 11 a Function φ(λ X) λ X ∈ C N Is Called Thementioning

confidence: 99%

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A Class of Baker–Akhiezer Arrangements

Feigin

Johnston

2014

Commun. Math. Phys.

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Abstract:We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh-Veselov Baker-Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker-Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasiinvariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero-Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.

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Section: Discussionmentioning

confidence: 99%

Section: Definition 11 a Function φ(λ X) λ X ∈ C N Is Called Thementioning

confidence: 99%

A Class of Baker–Akhiezer Arrangements

Feigin

Johnston

2014

Commun. Math. Phys.

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“…4). The same statement and proof hold true in the general case provided that φ(0, 0) = 0, which is satisfied in all the known cases (see [FV03,FHV13]). Multiplication of the left-hand side of formula (8) by the Gaussian factor exp(−λ 2 /2) remarkably leads to a very interesting quasi-invariant version of Lassalle's multivariable Hermite polynomials, see formula (4) above.…”

Section: Baker-akhiezer Function Related To Configurations Of Hyperplmentioning

confidence: 54%

Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

Feigin

Hallnäs

Веселов

2021

Commun. Math. Phys.

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Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${\mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${\mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${\mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

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“…In particular,

H = Ξ (ω)

for

ω = z_{1}^{2} + z_{2}^{2}

. (3)The Baker–Akhieser function

normalΦ

has the following expansion:

Φ false(x_{1}, x_{2}; z_{1}, z_{2} false) = (δ (z_{1}, z_{2}) + \sum_{i_{1} + i_{2} < μ} c_{i_{1}, i_{2}} (x_{1}, x_{2}) z_{1}^{i_{1}} z_{2}^{i_{2}}) \cdot \exp false(x_{1} z_{1} + x_{2} z_{2} false),

where

c_{i_{1}, i_{2}} false(x_{1}, x_{2} false) \in C false(x_{1}, x_{2} false)

for all

(i_{1}, i_{2})

. Moreover,

c_{0, 0} false(x_{1}, x_{2} false) = c \prod_{α \in Π} \frac{1}{l_{α} {(x_{1} - ξ_{1}, x_{2} - ξ_{2})}^{μ_{α}}},

where

c \in C

is a certain explicit constant, whose value can be found in [17]. (4)Let

z_{1} = ρ \cos false(φ false)

…”

Section: Spectral Module Of a Rational Calogero–moser System Of Dihedmentioning

confidence: 99%

Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

Burban

Zheglov

2020

Proc. London Math. Soc.

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In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen-Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen-Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero-Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, providing explicit computations of some 'isospectral deformations' of the planar rational Calogero-Moser system in the case of the split rational potential.

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Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

Cited by 8 publications

References 35 publications

A Class of Baker–Akhiezer Arrangements

A Class of Baker–Akhiezer Arrangements

Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

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