79 research outputs found
0-cycles on Grassmannians as representations of projective groups
Let be an infinite division ring, be a left -vector space,
be an integer. We study the structure of the representation of the linear group
in the vector space of formal finite linear combinations of
-dimensional vector subspaces of with coefficients in a field . This
gives a series of natural examples of irreducible infinite-dimensional
representations of projective groups. These representations are non-smooth if
is locally compact and non-discrete.Comment: v2: the results are generalized to the case of Grassmannian of
infinite-dimensional subspaces; v3: Assumptions on the coefficient field are
remove
Tilting exercises
This is an application of the theory of tilting objects to the geometric
setting of perverse sheaves. We show that this theory is a natural framework
for Beilinson's gluing of perverse sheaves construction. In the special case of
Schubert stratification of a flag variety we get a short proof of Soergel's
"Struktursatz", and describe (following a conjecture of Kapranov) Serre functor
for category O. Some of our results were obtained independently by Rouquier.Comment: This final version to appear in Moscow Math Journal differs very
slightly from the previous on
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REPRESENTATIONS OF SEMI-SIMPLE LIE ALGEBRAS IN PRIME CHARACTERISTIC AND NONCOMMUTATIVE SPRINGER RESOLUTION
Modules over the small quantum group and semi-infinite flag manifold
We develop a theory of perverse sheaves on the semi-infinite flag manifold
, and show that the subcategory of Iwahori-monodromy
perverse sheaves is equivalent to the regular block of the category of
representations of the small quantum group at an even root of unity
Some results about geometric Whittaker model
Let G be an algebraic reductive group over a field of positive characteristic. Choose a parabolic subgroup P in G and denote by U its unipotent radical. Let X be a G-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of ℓ-adic sheaves on X with respect to a generic character commutes with Verdier duality. Namely, in the first example we take X to be an arbitrary G-variety and we prove the above property for all -equivariant sheaves on X where is the unipotent radical of an opposite parabolic subgroup; in the second example we take X=G and we prove the corresponding result for sheaves which are equivariant under the adjoint action (the latter result was conjectured by B. C. Ngo who proved it for G=GL(n)). As an application of the proof of the first statement we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier–Deligne transform
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Equivariant homology and K-theory of affine Grassmannians and Toda lattices
For an almost simple complex algebraic group G with affine Grassmannian , we consider the equivariant homology and K-theory . They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group , and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of . If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of -equivariant homology of the point gives rise to a polarization which is related to Kostant\u27s Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of -modules
Singular localization and intertwining functors for reductive Lie algebras in prime characteristic
In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) {\em regular} central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.
The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character as sheaves on the partial flag variety corresponding to the singularity of . These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss {\em translation functors} and {\em intertwining functors}. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of -modules and coherent sheaves
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