Abstract We present a combinatorial procedure (based on the W-graph of the Coxeter group) which s... more Abstract We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan–Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves.
We re-examine some topics in representation theory of Lie algebras and Springer theory in a more ... more 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.
We describe a method of computing equivariant and ordinary intersection cohomology of certain var... more We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero-and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.
Kazhdan-Lusztig polynomials P x,w (q) play an important role in the study of Schubert varieties a... more Kazhdan-Lusztig polynomials P x,w (q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values P x,w (1) in terms of "patterns". A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G. Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties.
For a normal variety X defined over an algebraically closed field with an action of the multiplic... more For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = Gm, we consider the "hyperbolic localization" functor D b (X) → D b (X T ), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.
We compute the category of perverse sheaves on Hermitian symmetric spaces in types A and D, const... more We compute 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 Λ.
We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasse... more 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.
For affine toric varieties X and X ∨ defined by dual cones, we define an equivalence of categorie... more For affine toric varieties X and X ∨ defined by dual cones, we define an equivalence of categories between mixed versions of the equivariant derived category D b T (X) and the derived category of sheaves on X ∨ which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel.
We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smoot... more 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.
Given a hyperplane arrangement in an affine space equipped with a linear functional, we define tw... more Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.
We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf... more We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf imply that other Morse local systems for smaller strata do not vanish. This result is then used to explain the examples of reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in type A and by Boe and Fu for the Lagrangian Grassmannian.
We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in ma... more We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center of A acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation of A, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O for gl n is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category O" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.
We show that certain categories of perverse sheaves on affine toric varieties X σ and X σ ∨ defin... more We show that certain categories of perverse sheaves on affine toric varieties X σ and X σ ∨ defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel [BGS]. The functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties.
We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus ... more 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.
We present a functorial computation of the equivariant intersection cohomology of a hypertoric va... more We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
Abstract We present a combinatorial procedure (based on the W-graph of the Coxeter group) which s... more Abstract We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan–Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves.
We re-examine some topics in representation theory of Lie algebras and Springer theory in a more ... more 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.
We describe a method of computing equivariant and ordinary intersection cohomology of certain var... more We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero-and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.
Kazhdan-Lusztig polynomials P x,w (q) play an important role in the study of Schubert varieties a... more Kazhdan-Lusztig polynomials P x,w (q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values P x,w (1) in terms of "patterns". A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G. Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties.
For a normal variety X defined over an algebraically closed field with an action of the multiplic... more For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = Gm, we consider the "hyperbolic localization" functor D b (X) → D b (X T ), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.
We compute the category of perverse sheaves on Hermitian symmetric spaces in types A and D, const... more We compute 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 Λ.
We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasse... more 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.
For affine toric varieties X and X ∨ defined by dual cones, we define an equivalence of categorie... more For affine toric varieties X and X ∨ defined by dual cones, we define an equivalence of categories between mixed versions of the equivariant derived category D b T (X) and the derived category of sheaves on X ∨ which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel.
We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smoot... more 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.
Given a hyperplane arrangement in an affine space equipped with a linear functional, we define tw... more Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.
We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf... more We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf imply that other Morse local systems for smaller strata do not vanish. This result is then used to explain the examples of reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in type A and by Boe and Fu for the Lagrangian Grassmannian.
We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in ma... more We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center of A acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation of A, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O for gl n is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category O" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.
We show that certain categories of perverse sheaves on affine toric varieties X σ and X σ ∨ defin... more We show that certain categories of perverse sheaves on affine toric varieties X σ and X σ ∨ defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel [BGS]. The functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties.
We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus ... more 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.
We present a functorial computation of the equivariant intersection cohomology of a hypertoric va... more We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
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Papers by Tom Braden