164 research outputs found
Intersection cohomology of Drinfeld's compactifications
Let be a smooth complete curve, be a reductive group and
a parabolic.
Following Drinfeld, one defines a compactification \widetilde{\on{Bun}}_P
of the moduli stack of -bundles on .
The present paper is concerned with the explicit description of the
Intersection Cohomology sheaf of \widetilde{\on{Bun}}_P. The description is
given in terms of the combinatorics of the Langlands dual Lie algebra
.Comment: An erratum adde
Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps
from the projective line to the flag variety of GL_n. We construct the action
of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon
spaces by certain natural correspondences. Also we construct the action of the
quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the
affine version of Laumon spaces. We write down explicit formulae for this
action in the affine Gelfand-Tsetlin base, corresponding to the fixed point
base in the localized equivariant K-theory.Comment: v2: multiple typos fixed, proofs of Theorems 4.13 and 4.19 expanded,
23 pages. v3: formulas of Theorems 4.9 and 4.13 corrected, resulting minor
changes added. arXiv admin note: text overlap with arXiv:0812.4656,
arXiv:math/0503456, arXiv:0806.0072 by other author
SEMIINFINITE FLAGS. I. CASE OF GLOBAL CURVE P1.
The Semiinfinite Flag Space appeared in the works of B.Feigin and E.Frenkel,
and under different disguises was found by V.Drinfeld and G.Lusztig in the
early 80-s. Another recent discovery (Beilinson-Drinfeld Grassmannian) turned
out to conceal a new incarnation of Semiinfinite Flags. We write down these and
other results scattered in folklore. We define the local semiinfinite flag
space attached to a semisimple group as the quotient (an
ind-scheme), where and are a Cartan subgroup and the unipotent radical
of a Borel subgroup of . The global semiinfinite flag space attached to a
smooth complete curve is a union of Quasimaps from to the flag variety
of . In the present work we use to construct the category of
certain collections of perverse sheaves on Quasimaps spaces, with factorization
isomorphisms. We construct an exact convolution functor from the category of
perverse sheaves on affine Grassmannian, constant along Iwahori orbits, to the
category . Conjecturally, this functor should correspond to the restriction
functor from modules over quantum group with divided powers to modules over the
small quantum group.Comment: References update
Recommended from our members
Intersection cohomology of Drinfeld‚s compactifications
Let X be a smooth complete curve, G be a reductive group and a parabolic. Following Drinfeld, one defines a (relative) compactification of the moduli stack of P-bundles on X. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of . The description is given in terms of the combinatorics of the Langlands dual Lie algebra
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating
4-dimensional super-symmetric gauge theory for a gauge group G with certain
2-dimensional conformal field theory. This conjecture implies the existence of
certain structures on the (equivariant) intersection cohomology of the
Uhlenbeck partial compactification of the moduli space of framed G-bundles on
P^2. More precisely, it predicts the existence of an action of the
corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.
Namely, we replace the Uhlenbeck space with the space of based quasi-maps from
P^1 to any partial flag variety G/P of G and conjecture that its equivariant
intersection cohomology carries an action of the finite W-algebra U(g,e)
associated with the principal nilpotent element in the Lie algebra of the Levi
subgroup of P; this action is expected to satisfy some list of natural
properties. This conjecture generalizes the main result of arXiv:math/0401409
when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the
works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of
certain shifted Yangians.Comment: minor change
Recommended from our members
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
- …