## Publications

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**ABSTRACT:**We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the permutation operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that the centre has one non-constant generator and we identify it with the angular part of the Calogero-Moser Hamiltonian. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.09/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement Delta is essentially equivalent to the classification of V-systems associated to Delta. The flat sections of the corresponding V-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any V-system is free in Saito's sense and show this for a special class of V-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations.07/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of R^n to the `angular Calogero-Moser model' on the sphere S^{n-1}. We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a `relative angular Calogero-Moser model', which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.Journal of High Energy Physics 05/2013; 2013(7). · 5.62 Impact Factor - [Show abstract] [Hide abstract]

**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 quasi-invariants 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.Communications in Mathematical Physics 12/2012; · 1.97 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For the rational Baker-Akhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity, which may be viewed as a generalisation of the self-duality property of the usual Gaussian function with respect to the Fourier transformation. We show that the value of properly normalised Baker-Akhiezer function at the origin can be given by an integral of Macdonald-Mehta type and explicitly compute these integrals for all known Baker-Akhiezer arrangements. We use the Dotsenko-Fateev integrals to extend this calculation to all deformed root systems, related to the non-exceptional basic classical Lie superalgebras.Journal of Mathematical Physics 10/2012; 54(5). · 1.30 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d-1+hm, where h is the Coxeter number of W; these polynomials generate an H_c(W) submodule with the parameter c=(d-1)/h+m. We express these singular polynomials through the Saito polynomials that are flat coordinates of the Saito metric on the orbit space V/W. We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c.Compositio Mathematica 10/2011; · 1.02 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in types A_n, (C_n^\vee,C_n). We obtain commutative algebras of difference operators given by the action of invariant combinations of Cherednik-Dunkl operators in the corresponding quotient modules of the polynomial representation. This gives known and new generalized Macdonald-Ruijsenaars systems. Thus in the cases of DAHAs of types A_n and (C_n^\vee,C_n) we derive Chalykh-Sergeev-Veselov operators and a generalization of the Koornwinder operator respectively, together with complete sets of quantum integrals in the explicit form.Advances in Mathematics 02/2011; · 1.37 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider representations of rational Cherednik algebras which are particular ideals in the ring of polynomials. We investigate convergence of the integrals which express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types B and D for the singular values suggested by Cherednik with at most one exception, hence the corresponding modules are unitary. The analogous result on unitarity of the minimal submodules in type A was obtained by Etingof and Stoica, we give a different proof of convergence of the Gaussian product in this case. We also obtain partial results on unitarity of the minimal submodule in the case of exceptional Coxeter groups and group B with unequal parameters.International Mathematics Research Notices 10/2010; · 1.12 Impact Factor - Glasgow Mathematical Journal 01/2009; · 0.44 Impact Factor
- Glasgow Mathematical Journal 01/2009; 51. · 0.44 Impact Factor
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**ABSTRACT:**We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential.Selecta Mathematica 09/2008; · 0.72 Impact Factor -
##### Article: Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

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**ABSTRACT:**We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system ($\vee$-system) and we determine all trigonometric $\vee$-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric $\vee$-system; this inverts a one-way implication observed by Veselov for the rational solutions.Symmetry Integrability and Geometry Methods and Applications 02/2008; · 1.24 Impact Factor -
##### Article: On the geometry of V-systems

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**ABSTRACT:**We consider a complex version of the $\vee$-systems, which appeared in the theory of the WDVV equation. We show that the class of these systems is closed under the natural operations of restriction and taking the subsystems and study a special class of the $\vee$-systems related to generalized root systems and basic classical Lie superalgebras.11/2007; - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.Advances in Mathematics 06/2007; · 1.37 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable. Comment: 17 pagesJournal of Physics A General Physics 06/2006; - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider restrictions and subsystems in the ∨-systems corresponding to the logarithmic solutions of the WDVV equations. We present certain solutions through restrictions of the Coxeter systems.Czechoslovak Journal of Physics 01/2006; · 0.42 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We prove bispectral duality for the generalized Calogero-Moser-Sutherland systems related to configurations $A_{n,2}(m), C_n(l,m)$. The trigonometric axiomatics of Baker-Akhiezer function is modified, the dual difference operators of rational Macdonald type and the Baker-Akhiezer functions related to both series are explicitly constructed.04/2005; -
##### Article: Quasi-invariants of dihedral systems

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**ABSTRACT:**A basis of quasi-invariant module over invariants is explicitly constructed for the two-dimensional Coxeter systems with arbitrary multiplicities. It is proved that this basis consists of $m$-harmonic polynomials, thus the earlier results of Veselov and the author for the case of constant multiplicity are generalized.Mathematical Notes 12/2003; · 0.24 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The rings of quantum integrals of the generalized Calogero-Moser systems related to the deformed root systems ${\cal A}_n(m)$ and ${\cal C}_n(m,l)$ with integer multiplicities and corresponding algebras of quasi-invariants are investigated. In particular, it is shown that these algebras are finitely generated and free as the modules over certain polynomial subalgebras (Cohen-Macaulay property). The proof follows the scheme proposed by Etingof and Ginzburg in the Coxeter case. For two-dimensional systems the corresponding Poincare series and the deformed $m$-harmonic polynomials are explicitly computed.04/2003; - [Show abstract] [Hide abstract]

**ABSTRACT:**We construct the shift operators and the intertwining operators for the spherical parts of generalized Calogero operators related to classical Coxeter systems.Theoretical and Mathematical Physics 01/2003; · 0.67 Impact Factor

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