Calogero-Moser space and Kostka polynomials

On the Calogero-Moser space associated with dihedral groups II. The equal parameter case

We continue the study of Calogero-Moser spaces associated with dihedral groups by investigating in more details the equal parameter case: we obtain explicit equations, some informations about the Poisson bracket, the structure of the Lie algebra associated with the cuspidal point and the action of $SL_2(\mathbb{C})$.

Automorphisms and symplectic leaves of Calogero-Moser spaces

We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by an element of finite order of the normalizer of the associated complex reflection group $W$. We give a parametrization {\it à la Harish-Chandra} of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.

Regular automorphisms and Calogero-Moser families

We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.

Intersection cohomology of the Uhlenbeck compactification of the Calogero-Moser space

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero-Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. This allows computing the IC stalks of the Uhlenbeck compactification.

关于非本原复反射群的 Calogero-Moser 胞腔特征标

On Calogero-Moser cellular characters for imprimitive complex reflection groups

本文研究了 Calogero-Moser 胞腔特征标与 $A_{\infty}$ 型 Fock 空间向量定义的特征标的关系，证明了 $B$ 型 Weyl 群的 Lusztig 可构造特征标为其之和，并显式构造了复反射群 $G(l,1,n)$ 的极小 $b$-不变量特征标。

On the cohomology of Calogero-Moser spaces

We compute the equivariant cohomology of smooth Calogero-Moser spaces and some associated symplectic resolutions of symplectic quotient singularities.

Cuspidal Calogero-Moser and Lusztig families for Coxeter groups

The goal of this paper is to compute the cuspidal Calogero-Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero-Moser space and then by classifying certain "rigid" modules. Numerical evidence suggests that there is a very close relationship between Calogero-Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero-Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.

Generalized Calogero-Moser systems from rational Cherednik algebras

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 quadra…

$A_{\infty}$-modules and Calogero-Moser Spaces

We re-examine the bijective correspondence between the set of isomorphism classes of ideals of the first Weyl algebra and associated quiver varieties (Calogero-Moser spaces) \cite{BW1, BW2}. We give a new explicit construction of this correspondence based on the notion of $\A$-envelope of a rank one torsion-free $A_1$-module. Though perhaps less geometric than other methods, our approach is much simpler and seems more natural from the point of view of deformation theory.

On the Calogero-Moser space associated with dihedral groups

Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions coincide with the analogous notions defined using the Hecke algebra by Kazhdan and Lusztig (or Lusztig in the unequal parameters case). In the present paper, we aim to investigate these conjectures whenever $W$ is a …

Cherednik algebras and Calogero-Moser cells

Using the representation theory of Cherednik algebras at $t=0$ and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-s…

The real loci of the Calogero-Moser spaces and the Shapiro-Shapiro conjecture

This paper has been withdrawn by the authors; it is superceeded by the paper: The real loci of Calogero--Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture, by Iain Gordon, Emil Horozov, and Milen Yakimov, arXiv:0711.4336.

The Calogero-Moser partition for G(m,d,n)

We show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m,d,n) from the corresponding partition for G(m,1,n). This confirms, in the case W = G(m,d,n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

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

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, computing a new unexpected explicit example of a deformed Calogero-Moser system.

The Calogero-Moser partition for wreath products

This paper has been withdrawn - the results (with updated proofs) now appear in arXiv:0801.1627.

Calogero--Moser簇中的奇点

Singularities in Calogero--Moser Varieties

本文研究圈积辛反射群对应的Calogero--Moser簇奇点，通过参数化辛叶、描述闭包关系、计算横截截面完整刻画奇点，证实Bonnafé猜想，且相关结果可推广至任意箭图簇。

Gaudin Algebras, RSK and Calogero-Moser Cells in Type A

We study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the $n$-fold tensor representation $\mathbb{C}[x_1, \ldots, x_r]^{\otimes n}$ of the Lie algebra $\mathfrak{gl}_r$. We use the work of Halacheva-Kamnitzer-Rybnikov-Weekes to demonstrate that the Robinson-Schensted-Knuth correspondence describes the behaviour of the spectrum as we move along special paths in the family. We apply the work of Mukhin-Tarasov-Varchenko, which proves that the rational Calogero-Moser phase space can be realised as a part of this spectrum, to relate this to behaviour at $t=0$ o…

论 Calogero–Moser 空间的坐标环与一对矩阵的不变交换簇

On the coordinate rings of Calogero-Moser spaces and the invariant commuting variety of a pair of matrices

本文研究第四个 Calogero–Moser 空间及四阶矩阵不变交换对的坐标环与 Poisson 括号，并据此计算其在复簇 Grothendieck 环中的类，进而揭示仿射平面点 Hilbert 概形几何的新认识。

Calogero-Moser space, reduced rational Cherednik algebras, and two-sided cells

We conjecture that the "nilpotent points" of Calogero-Moser space for reflection groups are parametrised naturally by the two-sided cells of the group with unequal parameters. The nilpotent points correspond to blocks of restricted Cherednik algebras and we describe these blocks in the case G(m,1,n) and show that in type B our description produces an existing conjectural description of two-sided cells.

Orderings on Calogero-Moser partition of imprimitive groups

We extend to all parameters the constructions of the geometric and combinatorial orders on Irr G(l,1,n) due to I. Gordon, as well as the relations with the a and c-functions. This allows us to generalize these properties for the group G(l,e,n), at least when n is not divided by e.

Calogero-Moser versus Kazhdan-Lusztig cells

In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups. They defined in particular partitions of the group in left and two-sided cells. In 1983, Lusztig generalized this theory to Hecke algebras of Coxeter groups with unequal parameters. We propose a definition of left cells and two-sided cells for complex reflection groups, based on ramification theory for Calogero-Moser spaces. These spaces have been defined via rational Cherednik algebras by Etingof and Ginzburg. We conjecture that these coincide with Kazhdan-Lusztig cells, for real reflection groups. C…

Symmetric Lie superalgebras and deformed quantum Calogero-Moser problems

The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g=gl(n,2m), k=osp(n,2m) we establish a natural bijection between projective covers of spherically typical irreducible g-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero-Moser integrals D_{n,m} acting on the corresponding Laurent quasi-invariants A_{n,m}.

The Calogero-Moser partition and Rouquier families for complex reflection groups

Let $W$ be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that $W$ is a wreath product of a symmetric group with a cyclic group of order $l$.

A counter-example to Martino's conjecture about generic Calogero-Moser families

The Calogero-Moser families are partitions of the irreducible characters of a complex reflection group derived from the block structure of the corresponding restricted rational Cherednik algebra. It was conjectured by Martino in 2009 that the generic Calogero-Moser families coincide with the generic Rouquier families, which are derived from the corresponding Hecke algebra. This conjecture is already proven for the whole infinite series G(m,p,n) and for the exceptional group G4. A combination of theoretical facts with explicit computations enables us to determine the generic Calogero-Moser fami…

Calogero-Moser 本征函数模 $p^s$

Calogero-Moser eigenfunctions modulo $p^s$

本文利用 Matsuo-Cherednik 对偶性和 KZ 方程超几何解的积分表示，构造了有理耦合常数下 Calogero-Moser 本征函数的多项式 $p^s$-截断，并分析了 $n=2$ 时 $s\rightarrow\infty$ 趋于纯 $p$-adic 情形的极限。