Volume 12 (2012), Number 3. Abstracts

C. De Concini and A. Maffei. A Generalized Steinberg Section and Branching Rules for Quantum Groups at Roots of 1 [PDF]

In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our result including the construction of a sort of Gelfand–Zeitlin basis for a generic irreducible representation of Uq(GL(n)) when q is a primitive odd root of unity.

Keywords. Algebraic groups, quantum groups.

2010 Mathematics Subject Classification. 20G07, 20G42.

P. Deligne. Finitude de l'extension de ℚ engendrée par des traces de Frobenius, en caractéristique finie [PDF]

Soient Z0 un schéma de type fini sur 𝔽q et ℱ0 un \overline{ℚ}l-faisceau sur Z0. Nous montrons qu'il existe une extension de type fini E⊂\overline{ℚ}l de ℚ telle que les facteurs locaux de la fonction L de ℱ0 soient tous à coefficients dans E. Si Z0 est normal connexe et que ℱ0 est un système local l-adique irréductible, dont le déterminant est d'ordre fini, on peut prendre pour E une extension finie de ℚ.

Keywords. l-adic sheaves, Frobenius traces.

2010 Mathematics Subject Classification. 14F20, 14G15.

V. Drinfeld. On a Conjecture of Deligne [PDF]

Let X be a smooth variety over 𝔽p. Let E be a number field. For each nonarchimedean place λ of E prime to p consider the set of isomorphism classes of irreducible lisse \overline{E}λ-sheaves on X with determinant of finite order such that for every closed point xX the characteristic polynomial of the Frobenius Fx has coefficents in E. We prove that this set does not depend on λ.

The idea is to use a method developed by G. Wiesend to reduce the problem to the case where X is a curve. This case was treated by L. Lafforgue.

Keywords. ℓ-adic representation, independence of ℓ, local system, Langlands conjecture, arithmetic scheme, Hilbert irreducibility, weakly motivic.

2010 Mathematics Subject Classification. 14G15, 11G35.

P. Etingof. Symplectic Reflection Algebras and Affine Lie Algebras [PDF]

The goal of this paper is to present some results and (more importantly) state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory of affine Lie algebras (namely, decompositions of the restriction of the basic representation to finite dimensional and affine subalgebras). These conjectures arose from the insight due to R. Bezrukavnikov and A. Okounkov on the link between quantum connections for Hilbert schemes of resolutions of Kleinian singularities and representations of symplectic reflection algebras.

Keywords. Symplectic reflection algebra, affine Lie algebra, basic representation, root, weight.

2010 Mathematics Subject Classification. 17B67, 33D80, 20C08.

N. Hitchin. Deformations of Holomorphic Poisson Manifolds [PDF]

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deformation is determined by two parameters–an elliptic curve and a translation on it.

Keywords. Poisson manifold, Kodaira–Spencer class, deformation of complex structure, Hilbert scheme, exceptional divisor.

2010 Mathematics Subject Classification. 32G05, 53D17, 53D18.

D. Kaledin. Universal Witt Vectors and the “Japanese Cocycle” [PDF]

We give a direct interpretation of the Witt vector product in terms of tame residue in algebraic K-theory.

2010 Mathematics Subject Classification. 15A54.

B. Kostant. The Cascade of Orthogonal Roots and the Coadjoint Structure of the Nilradical of a Borel Subgroup of a Semisimple Lie Group [PDF]

Let G be a semisimple Lie group and let 𝔤 = 𝔫 + 𝔥 + 𝔫 be a triangular decomposition of 𝔤 = Lie G. Let 𝔟 = 𝔥 + 𝔫 and let H, N, B be Lie subgroups of G corresponding respectively to 𝔥, 𝔫 and 𝔟. We may identify 𝔫 with the dual space to 𝔫. The coadjoint action of N on 𝔫 extends to an action of B on 𝔫. There exists a unique nonempty Zariski open orbit X of B on 𝔫. Any N-orbit in X is a maximal coadjoint orbit of N in 𝔫. The cascade of orthogonal roots defines a cross-section 𝔯× of the set of such orbits leading to a decomposition X = N/R × 𝔯×. This decomposition, among other things, establishes the structure of S(𝔫)𝔫 as a polynomial ring generated by the prime polynomials of H-weight vectors in S(𝔫)𝔫. It also leads to the multiplicity 1 of H weights in S(𝔫)𝔫.

Keywords. Cascade of orthogonal roots, Borel subgroups, nilpotent coadjoint action.

2010 Mathematics Subject Classification. 20C, 14L24.

G. Lusztig. On the Cleanness of Cuspidal Character Sheaves [PDF]

We prove the cleanness of cuspidal character sheaves in arbitrary characteristic in the few cases where it was previously unknown.

Keywords. Character sheaf, unipotent class.

2010 Mathematics Subject Classification. 20G99.

H. Nakajima. Handsaw Quiver Varieties and Finite W-Algebras [PDF]

Following Braverman–Finkelberg–Feigin–Rybnikov (arXiv:1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W-algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang–Mills theory with surface operators. Our new observation is that the ℂ*-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type A, which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the W-algebra are described in terms of IC sheaves of graded quiver varieties of type A, which were known to be related to Kazhdan–Lusztig polynomials. This gives a new proof of a conjecture by Brundan–Kleshchev on composition multiplicities on Verma modules, which was proved by Losev, in a wider context, by a different method.

Keywords. Quiver variety, shifted Yangian, finite W-algebra, quantum affine algebra, Kazhdan–Lusztig polynomial.

2010 Mathematics Subject Classification. Primary: 17B37; Secondary: 14D21.

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