Karine Kuyumzhiyan
We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces $C_{n_i}$, where the $n_i$ are pairwise distinct, acts $m$-transitively for each $m$.
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We give a generic spectral decomposition of the derived category of twisted D-modules on a moduli stack of mirabolic vector bundles on a curve X in characteristic p: that is, we construct an equivalence with the derived category of quasi-coherent sheaves on a moduli stack of mirabolic local systems on X. This equivalence may be understood as a tamely ramified form of the geometric Langlands equivalence. When X has genus 1, this equivalence generically solves (in the sense of noncommutative geometry) the quantum Calogero-Moser system.
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-sid…
We prove that Calogero-Moser cells coincide with Kazhdan-Lusztig cells for dihedral groups, in the equal parameter case.
We study symplectic leaves of Calogero-Moser spaces of type $G(\ell,1,n)$. We prove that the normalization of the closure of each symplectic leaf is isomorphic to some Calogero-Moser space. We also give a nice combinatorial parameterization of the symplectic leaves.
We prove the algebraic density property for the Calogero--Moser spaces $\mathcal{C}_{n}$, and give a description of the identity component of the group of holomorphic automorphisms of $\mathcal{C}_{n}$.
These are lecture notes of a course on Calogero-Moser systems and their connections with representation theory and geometry, given by the author in Zurich in May-June 2005.
Using a recent construction of Bezrukavnikov and Etingof we prove that there is a factorization of the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space associated to a complex reflection group. In the case W = S_n, this confirms a conjecture of Etingof and Ginzburg.
We prove that every irreducible component of the fixed point variety under the action of $d$-th roots of unity in a smooth Caloger-Moser space is isomorphic to a Calogero-Moser space associated with another reflection group.
We study the natural $GL_2$-action on the Hilbert scheme of points in the plane, resp. $SL_2$-action on the Calogero-Moser space. We describe the closure of the $GL_2$-orbit, resp. $SL_2$-orbit, of each point fixed by the corresponding diagonal torus. We also find the character of the representation of the group $GL_2$ in the fiber of the Procesi bundle, and its Calogero-Moser analogue, over the $SL_2$-fixed point.
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 $SL_2(\mathbb{C})$.
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.
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.
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.
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.
We consider the canonical map from the Calogero-Moser space to symmetric powers of the affine line, sending conjugacy classes of pairs of n by n matrices to their eigenvalues. We show that the character of a natural C^*-action on the scheme-theoretic zero fiber of this map is given by Kostka polynomials.
In our recent works we have used meromorphic differentials on Riemann surfaces all of whose periods are real to study the geometry of the moduli spaces of Riemann surfaces. In this paper we survey the relevant constructions and show how they are related to and motivated by the spectral theory of the elliptic Calogero-Moser integrable system.
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 symplectic holomorphic density property for Calogero-Moser spaces
本文引入辛全纯密度性质等概念及对应理论，证明n粒子Calogero-Moser空间$\mathcal{C}_n$具有该性质并刻画其全纯辛自同构群，为相关空间的自同构研究提供基础。
Sklyanin's formula provides a set of canonical spectral coordinates on the standard Calogero-Moser space associated with the quiver consisting of a vertex and a loop. We generalize this result to Calogero-Moser spaces attached to cyclic quivers by constructing rational functions that relate spectral coordinates to conjugate variables. These canonical coordinates turn out to be well-defined on the corresponding simple singularity of type $A$, and the rational functions we construct define interpolating polynomials between them.
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.
We exhibit the elliptic Calogero-Moser system as a Hitchin system of G-principal Higgs pairs. The group G, though naturally associated to any root system, is not semi-simple. We then interpret the Lax pairs with spectral parameter of [dP1] and [BSC1] in terms of equivariant embeddings of the Hitchin system of G into that of GL(N).
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。
The Density Property for Generalized Calogero--Moser Spaces with Inner Degrees of Freedom
本文证明了具有内部自由度的广义Calogero--Moser空间的密度性质，从而描述了其全纯自同构群，并揭示了其与非交换$\mathbb{R}^4$上$\mathrm{SU}(2)$瞬子模空间的联系。
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…
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。
Elliptic Calogero-Moser system, crossed and folded instantons, and bilinear identities
本研究利用折叠瞬子配分函数构造Q函数仿射类似物，证明其为N体椭圆Calogero-Moser系统量子谱曲线等的解，还指出其与交叉瞬子、穿孔环面opers的关联，并在配套论文给出对应Lax算子量子形式。
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