Volume 7 (2007), Number 4. Abstracts

V. Baranovsky. BGG Correspondence for Toric Complete Intersections [PDF]

We prove a BGG-type correspondence describing coherent sheaves on complete intersections in toric varieties, and a similar assertion for the stable categories of related complete intersection singularities.

Keywords. Toric varieties, complete intersections, derived category, coherent sheaves, Koszul duality, A-infinity algebras

2000 Mathematics Subject Classification. Primary: 18E30; Secondary: 14F05, 14M25, 14M10.

P. Etingof and Ching-Hwa Eu. Hochschild and Cyclic Homology of Preprojective Algebras of ADE Quivers [PDF]

We compute the additive structure of the Hochschild homology and cohomology and cyclic homology of the preprojective algebra of an ADE quiver over a field of characteristic zero.

Keywords. Preprojective algebra, quiver, Hochscild homology, cyclic homology

2000 Mathematics Subject Classification. 16G20

S. Evens and Jiang-Hua Lu. Poisson Geometry of the Grothendieck Resolution of a Complex Semisimple Group [PDF]

Let G be a complex semi-simple Lie group with a fixed pair of opposite Borel subgroups (B, B). We study a Poisson structure π on G and a Poisson structure Π on the Grothendieck resolution X of G such that the Grothendieck map μ: (X,Π) → (G,π) is Poisson. We show that the orbits of symplectic leaves of π in G under the conjugation action by the Cartan subgroup H= BB are intersections of conjugacy classes and Bruhat cells BwB, while the H-orbits of symplectic leaves of Π on X give desingularizations of intersections of Steinberg fibers and Bruhat cells in G. We also give birational Poisson isomorphisms from quotients by H × H of products of double Bruhat cells in G to intersections of Steinberg fibers and Bruhat cells.

Keywords. Poisson structure, symplectic leaves, Grothendieck resolution, Steinberg fiber, Bruhat cell

2000 Mathematics Subject Classification. Primary 53D17; Secondary 14M17, 20G20

D. Kaledin. Some Remarks on Formality in Families [PDF]

We prove some results on formality for families of DG algebras; in particular, we prove that formality is stable under specialization. The results are more-or-less known, but it seems that there are no published proofs.

Keywords. Formality, DG algebras, Massey products, Hochschild cohomology

2000 Mathematics Subject Classification. 18G50

D. Kaledin and M. Lehn. Local Structure of Hyperkähler Singularities in O'Grady's Examples [PDF]

We study the local structure of the singularity in the moduli space of sheaves on a K3 surface which has been resolved by K. O'Grady in his construction of new examples of hyperkaehler manifolds. In particular, we identify the singularity with the closure of a certain nilpotent orbit in the coadjoint representation of the group Sp(4). We also prove that the moduli spaces for some other sets of numerical parameters do not admit a smooth symplectic resolution of singularities.

Keywords. Hyperkaehler, symplectic, O'Grady examples, nilpotent orbits, formality

2000 Mathematics Subject Classification. 14D20

A. Kuznetsov. Quiver Varieties and Hilbert Schemes [PDF]

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, if X = C2, Γ ⊂ SL(C2) is a finite subgroup, and XΓ is a minimal resolution of X/Γ, we show that XΓ[n] (the Γ-equivariant Hilbert scheme of X), and XΓ[n] (the Hilbert scheme of XΓ) are quiver varieties for the affine Dynkin graph corresponding to Γ via the McKay correspondence with the same dimension vectors but different parameters ζ (for earlier results in this direction see works by M. Haiman, M. Varagnolo and E. Vasserot, and W. Wang). In particular, it follows that the varieties XΓ[n] and XΓ[n] are diffeomorphic. Computing their cohomology (in the case Γ=Z/dZ) via the fixed points of a (C*×C*)-action we deduce the following combinatorial identity: the number UCY(n,d) of Young diagrams consisting of nd boxes and uniformly colored in d colors equals the number CY(n,d) of collections of d Young diagrams with the total number of boxes equal to n.

Keywords. Quiver variety, Hilbert scheme, McKay correspondence, moduli space

2000 Mathematics Subject Classification. Primary 14D21; Secondary 53C26, 16G20

H. Nakajima. Sheaves on ALE Spaces and Quiver Varieties [PDF]

We identify a quiver variety of an affine type with a framed moduli space of torsion free sheaves on an ALE space, a fiber of a simultaneous resolution of the semi-universal deformation of C2/Γ. This result is an analog of a similar identification for a framed moduli space of anti-self-dual connections on an ALE space, given by Kronheimer and the author. It simultaneously generalizes the description on C2 given in Chapter 2 of my ``Lectures on Hilbert schemes of points on surfaces'' to arbitrary ALE space, and also the description of the Hilbert schemes of points on the ALE space by Kuznetsov to arbitrary torsion free sheaves.

Keywords. ALE space, ADHM description, coherent sheaf, framed moduli space

2000 Mathematics Subject Classification. Primary 14D21; Secondary 53C26, 16G20

E. Opdam. The Central Support of the Plancherel Measure of an Affine Hecke Algebra [PDF]

We give conceptual proofs of certain basic properties of the arrangement of shifted root hyperplanes associated to a root system R0 and a W0=W(R0)-invariant real valued parameter function on R0. The method is based on the role of this shifted root hyperplane arrangement for the harmonic analysis of affine Hecke algebras. In addition this yields a conceptual proof of the description of the central support of the Plancherel measure of an affine Hecke algebra given by the author earlier.

Keywords. Affine Hecke algebra, Plancherel measure, positivity, residue calculus, support

2000 Mathematics Subject Classification. Primary 20C08; Secondary 22D25, 22E35, 43A30

A. Premet. Primitive Ideals, Non-Restricted Representations and Finite W-Algebras [PDF]

Let G be a simple algebraic group over C and g=Lie G. Let (e,h,f) be an sl2-triple in g and (⋅,⋅) the G-invariant bilinear form on g such that (e,f)=1. Let χ∈g* be such that χ(x)=(e,x) for all xg and let Hχ denote the enveloping algebra of the Slodowy slice e+Ker ad f. Let I be a primitive ideal of the universal enveloping algebra U(g) whose associated variety is the closure of the coadjoint orbit Oχ. We prove in this note that if I has rational infinitesimal character, then there is a finite dimensional irreducible Hχ-module V such that I = AnnU(g)(QχHχV), where Qχ is the generalised Gelfand–Graev g-module associated with the triple (e,h,f). In conjunction with well-known results of Barbasch and Vogan this implies that all finite W-algebras possess finite dimensional irreducible representations.

Keywords. Primitive ideal, modular representation, finite W-algebra

2000 Mathematics Subject Classification. Primary 17B35. Secondary 17B63, 17B81

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