91 research outputs found
Remarks on the combinatorial intersection cohomology of fans
We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric and polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study . We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that implies and
.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied
Math Quarterl
On the reducibility of characteristic varieties
We present a result which can be used for stratifications with conical
singularities to deduce that a perverse sheaf (in particular, an intersection
homology sheaf) has reducible characteristic variety, given a hypothesis on the
monodromy of the vanishing cycles local system of a stratum. We apply it to
explain most of the examples currently known where SS(IC(X)) is reducible for X
a Schubert variety in a flag variety.Comment: LaTeX, 7 page
Perverse sheaves on Grassmannians
We give a complete quiver description of the category of perverse sheaves on
Hermitian symmetric spaces in types A and D, constructible with respect to the
Schubert stratification. The calculation is microlocal, and uses the action of
the Borel group to study the geometry of the conormal variety.Comment: AMS-LaTeX, 35 pages, 11 figure
Lower bounds for Kazhdan-Lusztig polynomials from patterns
We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in
a Weyl group W in terms of "patterns''. This is expressed by a "pattern map"
from W to W' for any parabloic subgroup W'. This notion generalizes the concept
of patterns and pattern avoidance for permutations to all Weyl groups. The main
tool of the proof is a "hyperbolic localization" on intersection cohomology;
see the related paper http://front.math.ucdavis.edu/math.AG/0202251Comment: 14 pages; updated references. Final version; will appear in
Transformation Groups vol.8, no.
Quantizations of conical symplectic resolutions I: local and global structure
We re-examine some topics in representation theory of Lie algebras and
Springer theory in a more general context, viewing the universal enveloping
algebra as an example of the section ring of a quantization of a conical
symplectic resolution. While some modification from this classical context is
necessary, many familiar features survive. These include a version of the
Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules
and their relationship to convolution operators on cohomology, and a discrete
group action on the derived category of representations, generalizing the braid
group action on category O via twisting functors.
Our primary goal is to apply these results to other quantized symplectic
resolutions, including quiver varieties and hypertoric varieties. This provides
a new context for known results about Lie algebras, Cherednik algebras, finite
W-algebras, and hypertoric enveloping algebras, while also pointing to the
study of new algebras arising from more general resolutions.Comment: 74 pages; v4: minor changes based on referee comments; v5: minor
adjustment in numbering to match published versio
The Equivariant Chow rings of quot schemes
We give a presentation for the (integral) torus-equivariant Chow ring of the
quot scheme, a smooth compactification of the space of rational curves of
degree d in the Grassmannian. For this presentation, we refine Evain's
extension of the method of Goresky, Kottwitz, and MacPherson to express the
torus-equivariant Chow ring in terms of the torus-fixed points and explicit
relations coming from the geometry of families of torus-invariant curves. As
part of this calculation, we give a complete description of the torus-invariant
curves on the quot scheme and show that each family is a product of projective
spaces.Comment: Revised slightly. Clarifed some statements and remove one
straightforward proof. 26 pages, many .eps figure
Hypertoric category O
We study the representation theory of the invariant subalgebra of the Weyl
algebra under a torus action, which we call a "hypertoric enveloping algebra."
We define an analogue of BGG category O for this algebra, and identify it with
a certain category of sheaves on a hypertoric variety. We prove that a regular
block of this category is highest weight and Koszul, identify its Koszul dual,
compute its center, and study its cell structure. We also consider a collection
of derived auto-equivalences analogous to the shuffling and twisting functors
for BGG category O.Comment: 65 pages, TikZ figures (PDF is recommended; DVI will not display
correctly on all computers); v3: switched terminology for twisting and
shuffling; final version; v4: small correction in definition of standard
module
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