Related papers: Singularities in Calogero--Moser Varieties

Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name…

We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module.…

To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic…

We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with…

Let $K$ be field of characteristic 2 and let $G$ be a finite non-abelian 2-group with the cyclic derived subgroup $G'$, and there exists a central element $z$ of order 2 in $Z(G) \backslash G'$. We prove that the unit group of the group…

Let $(M,\omega)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion…

We consider the generating series of appropriately completed 0-dimensional special cycles on a toroidal compactification of an orthogonal or unitary Shimura variety with values in the Chow group. We prove that it is a holomorphic Siegel,…

We classify certain cases when the wreath products of distinct pairs of groups generate the same variety. This allows us to investigate the subvarieties of some nilpotent-by-abelian product varieties ${\mathfrak U}{\mathfrak V}$ with the…

Normal subgroups and there properties for finite and infinite iterated wreath products $S_{n_1}\wr \ldots \wr S_{n_m}$, $n, m \in \mathbb{N}$ are founded. The special classes of normal subgroups and there orders are investigated. Special…

We study the local symplectic algebra of the 1-dimensional isolated complete intersection singularity of type S{\mu}. We use the method of algebraic restrictions to classify symplectic S{\mu} singularities. We distinguish these symplectic…

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck…

In this article, we realize the subquotient based modules of certain tensor products or restricted modules via Lusztig's perverse sheaves on multi-framed quivers, and provide a construction of their canonical bases. As an application, we…

We give a complete classification of all $d$-representation-finite symmetric Nakayama algebras and of all $d$-representation-finite trivial extensions of path algebras of quivers, over an arbitrary field. As a consequence we get a…

We give a proof of the cyclicity conjecture of Akasaka-Kashiwara, for simply laced types, via quiver varieties. We get also an algebraic characterization of the standard modules.

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results: (1) The…

Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are…

We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also…

We determine explicitly the irreducible components of the singular locus of any Schubert variety for GL_n(K), K being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along these…

We determine the complete structure of the symmetry algebras associated with the N-body Calogero-Moser system and its maximally superintegrable discretization. We prove that the discretization naturally leads to a nontrivial deformation of…

We prove that any symplectic resolution of the closure of a nilpotent orbit in a semi-simple complex Lie algebra is isomorphic to the collapsing of the cotangent bundle of a projective homogenous variety. Then we give a complete…