Macdonald polynomials and algebraic integrability

We construct explicitly non-polynomial eigenfunctions of the difference operators by Macdonald in case $t=q^k$, $k\in{\mathbb Z}$. This leads to a new, more elementary proof of several Macdonald conjectures, first proved by Cherednik. We also establish the algebraic integrability of Macdonald operators at $t=q^k$ ($k\in {\mathbb Z}$), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including $BC_n$ case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of $A_n$ root system where the previously known methods do not work.Comment: 54 page

Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

The aim of this paper is to clarify the relation between the following objects: $(a)$ rank 1 projective modules (ideals) over the first Weyl algebra A_1(\C); $(b)$ simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and $(c)$ simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on \A-modules over $A_1$ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities \C^2/\Gamma , where $\Gamma$ is a finite cyclic subgroup of \SL(2, \C) .Comment: 16 pp., LaTex, to appear in Moscow Math. J.(2007

Generalized Lame operators

We introduce a class of multidimensional Schr\"odinger operators with elliptic potential which generalize the classical Lam\'e operator to higher dimensions. One natural example is the Calogero--Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lam\'e operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh--Veselov conjecture for the elliptic Calogero--Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to B_2 case, another one is a certain deformation of the A_2 case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald--Ruijsenaars type.Comment: 38 pages, latex; in the new version a reference was adde

Orthogonality Relations and Cherednik Identities for Multivariable Baker-Akhiezer Functions

Author Manuscript 27 Feb 2013We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik–Macdonald–Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald–Ruijsenaars type.National Science Foundation (U.S.) (Grant DMS-1000113

Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case

We consider generalisations of the elliptic Calogero--Moser systems associated to complex crystallographic groups in accordance to \cite{EFMV11ecm}. In our previous work \cite{Argyres:2021iws}, we proposed these systems as candidates for Seiberg--Witten integrable systems of certain SCFTs. Here we examine that proposal for complex crystallographic groups of rank one. Geometrically, this means considering elliptic curves $T^2$ with $\Z_m$-symmetries, $m=2,3,4,6$, and Poisson deformations of the orbifolds $(T^2\times\mathbb{C})/\Z_m$. The $m=2$ case was studied in \cite{Argyres:2021iws}, while $m=3,4,6$ correspond to Seiberg--Witten integrable systems for the rank 1 Minahan--Nemeshansky SCFTs of type $E_{6,7,8}$. This allows us to describe the corresponding elliptic fibrations and the Seiberg--Witten differential in a compact elegant form. This approach also produces quantum spectral curves for these SCFTs, which are given by Fuchsian ODEs with special properties

Deformed Calogero--Moser operators and ideals of rational Cherednik algebras

We consider a class of hyperplane arrangements $\mathcal{A}$ in $\mathbb{C}^n$ which generalise the locus configurations of Chalykh, Feigin and Veselov. To each generalised locus configuration we associate a second order partial differential operator of Calogero--Moser type, and prove that this operator is completely integrable (in the sense that its centraliser in $\mathcal{D}(\mathbb{C}^n\setminus\mathcal{A})$ contains a commutative subalgebra of dimension $n$). The proof is based on the study of certain ideals of (the spherical subalgebra of) the rational Cherednik algebra; these ideals have some rather special properties which may be of independent interest. The class of the deformed Calogero--Moser systems we consider includes those introduced by Sergeev and Veselov in the context of Lie superalgebras. Also, it includes systems constructed by M. Feigin using representation theory of Cherednik algebras at singular values of parameters. Our approach is entirely different, and our results appear to be stronger and more general.Comment: 25 page