263 research outputs found
Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group
In math.RT/0201073 we constructed an equivalence between the derived category
of equivariant coherent sheaves on the cotangent bundle to the flag variety of
a simple algebraic group and a (quotient of) the category of constructible
sheaves on the affine flag variety of the Langlands dual group. Below we prove
certain properties of this equivalence; provide a similar ``Langlands dual''
description for the category of equivariant coherent sheaves on the nilpotent
cone; and deduce some conjectures by Lusztig and Ostrik.Comment: This is a continuation of math.RT/0201073. 13 pages; some correction
On semi-infinite cohomology of finite dimensional algebras
We show that semi-infinite cohomology of a finite dimensional graded algebra
(satisfying some additional requirements) are a particular case of a general
categorical construction. The motivating example is provided by small quantum
groups at a root of unity.Comment: 8 pages. Some expositional changes made; few remarks added, one of
them concerning semi-infinite cohomology of small quantum group
On tensor categories attached to cells in affine Weyl groups
We prove a conjecture by Lusztig, which describes the tensor categories of
perverse sheaves on affine flag manifolds, with tensor structure provided by
truncated convolution, in terms of the Langlands dual group. We also give a
geometric (categorical) description of Lusztig's bijection between two-sided
cells in an affine Weyl group, and unipotent conjugacy classes in the Langlands
dual group. The main tool is the sheaf-theoretic construction of the center of
the affine Hecke algebra due to Gaitsgory (based on ideas of Beilinson and
Kottwitz), see math.AG/9912074.Comment: The published version of this paper misquoted a result of Lusztig,
this version corrects this mistake (thanks to Xinwen Zhu for pointing it out
Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone
In math.AG/0005152 a certain -structure on the derived category of
equivariant coherent sheaves on the nil-cone of a simple complex algebraic
group was introduced (the so-called perverse -structure corresponding to the
middle perversity). In the present note we show that the same -structure can
be obtained from a natural quasi-exceptional set generating this derived
category. As a consequence we obtain a bijection between the sets of dominant
weights and pairs consisting of a nilpotent orbit, and an irreducible
representation of the centralizer of this element, conjectured by Lusztig and
Vogan (and obtained by other means in math.RT/0010089).Comment: 17 page
On two geometric realizations of an affine Hecke algebra
The article is a contribution to the local theory of geometric Langlands
correspondence. The main result is a categorification of the isomorphism
between the (extended) affine Hecke algebra, thought of as an algebra of
Iwahori bi-invariant functions on a semi-simple group over a local
non-Archimedian field, and Grothendieck group of equivariant coherent sheaves
on Steinberg variety of the Langlands dual group; this isomorphism due to
Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified
local Langlands conjectures.
The paper is a continuation of an earlier joint work with S. Arkhipov, it
relies on technical material developed in a paper with Z. Yun.Comment: Further corrections in auxiliary statements, last section expanded.
58p
Noncommutative Counterparts of the Springer Resolution
Springer resolution of the set of nilpotent elements in a semisimple Lie
algebra plays a central role in geometric representation theory. A new
structure on this variety has arisen in several representation theoretic
constructions, such as the (local) geometric Langlands duality and modular
representation theory. It is also related to some algebro-geometric problems,
such as the derived equivalence conjecture and description of T. Bridgeland's
space of stability conditions. The structure can be described as a
noncommutative counterpart of the resolution, or as a -structure on the
derived category of the resolution. The intriguing fact that the same
-structure appears in these seemingly disparate subjects has strong
technical consequences for modular representation theory.Comment: ICM talk; 23 page
Homological properties of representations of p-adic groups related to geometry of the group at infinity
Geometry of buildings is used to prove some homological properties of the
category of smooth representations of a reductive p-adic group (Kazhdan's
"pairing conjecture", Bernstein's description of homological duality in terms
of Deligne-Lusztig duality). A different proof had been obtained a little
earlier by Schneider and Stuhler.Comment: This is my PhD thesis (written in 1998, unchanged since 1999
Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case
Let X be a smooth projective curve over an algebraically closed field k of
characteristic p>0. In this paper we explore the relation between algebraic
D-modules on the moduli space of vector bundles of rank n on X and
coherent sheaves on the moduli space of vector bundles endowed with a
connection (in the way predicted by Beilinson and Drinfeld for k of
characteristic 0). The main technical tools used in the paper are the geometry
of the Hitchin system and the Azumaya property of the algebra of differential
operators in characteristic p.Comment: Dedicated to R.MacPherson on the occasion of his 60th birthda
Cohomology of tilting modules over quantum groups and -structures on derived categories of coherent sheaves
The paper is concerned with cohomology of the small quantum group at a root
of unity, and of its upper triangular subalgebra, with coefficients in a
tilting module. We relate it to a certain t-structure on the derived category
of equivariant coherent sheaves on the Springer resolution, and to equivariant
coherent IC sheaves on the nil-cone. The support of these cohomology is
described in terms of cells in affine Weyl groups, and the basis in the
Grothendieck group they provide is related to the Kazhdan-Lusztig basis, as
predicted by J. Humphreys and V. Ostrik.
The proof is based on earlier results of Arkhipov, Ginzburg and the author
which allow us to reduce the question to purity of IC sheaves on affine flag
varieties.Comment: 26 pages; more lemmas corrected and expande
Perverse coherent sheaves (after Deligne)
This note is mostly an exposition of an unpublished result of Deligne, which
introduces an analogue of perverse -structure on the derived category of
coherent sheaves on a Noetherian scheme with a dualizing complex. Construction
extends to the category of coherent sheaves equivariant under an action of an
algebraic group; though proof of the general statement in this case does not
require new ideas, it provides examples (such as sheaves on the nilpotent cone
of a semi-simple group equivariant under the adjoint action) where construction
of coherent "intersection cohomology" sheaves works.Comment: This preprint is superseded by arxiv:0902.0349, joint with D. Arinki
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