Misha Feigin

New integrable generalizations of Calogero–Moser quantum problem

A one-parameter deformation of Calogero–Moser quantum problem is introduced. It is shown that corresponding Schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value.

Painlevé IV and degenerate Gaussian unitary ensembles

We consider those Gaussian unitary ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for 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.

Quasi-invariants of Coxeter groups and m-harmonic polynomials

The space of m-harmonic polynomials related to a Coxeter group G and a multiplicity function m on its root system is defined as the joint kernel of the properly gauged invariant integrals of the corresponding generalised quantum Calogero-Moser problem. The relation between this space and the ring of all quantum integrals of this system (which is isomorphic to the ring of corresponding quasiinvariants) is investigated.

Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems

The rings of quantum integrals of the generalized Calogero-Moser systems related to the deformed root systems An(m) and Cn(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.

Intertwining relations for the spherical parts of generalized Calogero operators

We construct the shift operators and the intertwining operators for the spherical parts of generalized Calogero operators related to classical Coxeter systems.

Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

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 (_-system) and we determine all trigonometric _-systems with up to five vectors. We show that generali- zed Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric _-system; this inverts a one-way implication observed by Veselov for the rational solutions.

Bispectrality for deformed Calogero-Moser-Sutherland systems

Abstract We prove bispectral duality for the generalized Calogero–Moser–Sutherland systems related to configurations , . The trigonometric axiomatics of the Baker–Akhiezer function is modified, the dual difference operators of rational Macdonald type and the Baker–Akhiezer functions related to both series are constructed.

Singular polynomials from orbit spaces

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.

A Class of Baker–Akhiezer Arrangements

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.

Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

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.