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 $g$ and $h$ 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 $g_2$. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that $g_k(P) = 0$ implies $g_k(P^*) = 0$ and $g_{k+1}(P) = 0$.Comment: 34 pages.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