Hilbert-Poincare series for spaces of commuting elements in Lie groups

In this article we study the homology of spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${\rm Hom}(F_n/\Gamma_n^m,G)$, where the subgroups $\Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${\rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.Comment: 20 pages, journal versio

A survey on symplectic singularities and resolutions

This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions, which could also serve as an introduction to this subject

ad-Nilpotent ideals of a Borel subalgebra II

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q} under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.Comment: AMS-TeX file, 9 pages; revised version. To appear in Journal of Algebr

A quantum homogeneous space of nilpotent matrices

A quantum deformation of the adjoint action of the special linear group on the variety of nilpotent matrices is introduced. New non-embedded quantum homogeneous spaces are obtained related to certain maximal coadjoint orbits, and known quantum homogeneous spaces are revisited.Comment: 12 page

Any flat bundle on a punctured disc has an oper structure

We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.Comment: 12 page