Singular localization for Quantum groups at generic qq

We quantize parabolic flag manifolds and describe categories of equivariant quantum \D-modules on them at a singular central character. We compute global sections at any q \in \C^* and we also prove a singular version of Beilinson-Bernstein localization for a quantized enveloping algebra \Uq(\g), when $q$ is generic.Comment: 21 page

Higher Auslander-Reiten sequences and tt-structures

Let $R$ be an artin algebra and $\mathcal{C}$ an additive subcategory of $\operatorname{mod}(R)$. We construct a $t$-structure on the homotopy category $\operatorname{K}^{-}(\mathcal{C})$ whose heart $\mathcal{H}_{\mathcal{C}}$ is a natural domain for higher Auslander-Reiten (AR) theory. The abelian categories $\mathcal{H}_{\operatorname{mod}(R)}$ (which is the natural domain for classical AR theory) and $\mathcal{H}_{\mathcal{C}}$ interact via various functors. If $\mathcal{C}$ is functorially finite then $\mathcal{H}_{\mathcal{C}}$ is a quotient category of $\mathcal{H}_{\operatorname{mod}(R)}$. We illustrate the theory with two examples: Iyama developed a higher AR theory when $\mathcal{C}$ is a maximal $n$-orthogonal subcategory, see \cite{I}. In this case we show that the simple objects of $\mathcal{H}_{\mathcal{C}}$ correspond to Iyama's higher AR sequences and derive his higher AR duality from the existence of a Serre functor on the derived category $\operatorname{D}^b(\mathcal{H}_{\mathcal{C}})$. The category $\mathcal{O}$ of a complex semi-simple Lie algebra $\mathfrak{g}$ fits into higher AR theory by considering $R$ to be the coinvariant algebra of the Weyl group of $\mathfrak{g}$.Comment: 26 pages, accepted for publication in Journal of Algebra 201

Hochschild cohomology and string topology of global quotient orbifolds

Let M be a connected, simply connected, closed and oriented manifold, and G a finite group acting on M by orientation preserving diffeomorphisms. In this paper we show an explicit ring isomorphism between the orbifold string topology of the orbifold [M/G] and the Hochschild cohomology of the dg-ring obtained by performing the smash product between the group G and the singular cochain complex of M.Comment: Revised version. Sections have been reorganized, and section 4.1 is new. Version accepted for publication in the Journal of Topolog

Endomorphisms of quantized Weyl algebras

Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity

Projective and Whittaker functors on category O\mathcal{O}

We show that the Whittaker functor on a regular block of the BGG-category $\mathcal{O}$ of a semisimple complex Lie algebra can be obtained by composing a translation to the wall functor with Soergel and Mili\v{c}i\'{c}'s equivalence between the category of Whittaker modules and a singular block of $\mathcal{O}$. We show that the Whittaker functor is a quotient functor that commutes with all projective functors and endomorphisms between them.Comment: 14 page

The Hom-spaces between projective functors

The category of projective functors on a block of the category O ( g ) \mathcal O(\mathfrak g) of Bernstein, Gelfand and Gelfand, over a complex semisimple Lie algebra g \mathfrak g , embeds to a corresponding block of the category O ( g × g ) \mathcal O(\mathfrak g \times \mathfrak g) . In this paper we give a nice description of the object V V in O ( g × g ) \mathcal O(\mathfrak g \times \mathfrak g) corresponding to the identity functor; we show that V V is isomorphic to the module of invariants, under the diagonal action of the center Z \mathcal Z of the universal enveloping algebra of g \mathfrak g , in the so-called anti-dominant projective. As an application we use Soergel’s theory about modules over the coinvariant algebra C C , of the Weyl group, to describe the space of homomorphisms of two projective functors T T and T ′ T’ . We show that there exists a natural C C -bimodule structure on Hom { Functors } ⁡ ( T , T ′ ) \operatorname {Hom}_{\{\operatorname {Functors}\}}(T, T’) such that this space becomes free as a left (and right) C C -module and that evaluation induces a canonical isomorphism k ⊗ C Hom { Functors } ⁡ ( T , T ′ ) ≅ Hom O ( g ) ⁡ ( T ( M e ) , T ′ ( M e ) ) k \otimes _C \operatorname {Hom}_{\{\operatorname {Functors}\}} (T, T’) \cong \operatorname {Hom}_{\mathcal O(\mathfrak g)}(T(M_e), T’(M_e)) , where M e M_e denotes the dominant Verma module in the block and k k is the complex numbers.</p

Koszul duality for parabolic and singular category 𝒪

This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category O \mathcal O over a complex semi-simple Lie-algebra, established by Beilinson, Ginzburg and Soergel in Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527. In that paper it was proved that any “block” in O \mathcal O , determined by an integral, but possibly singular weight, is Koszul (i.e. equivalent to the category of finitely generated modules over some Koszul ring) and, moreover, that the “Koszul dual” of such a block is isomorphic to a “parabolic subcategory” of the trivial block in O \mathcal O . We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in O \mathcal O is Koszul and we also calculate the Koszul dual of such a category.</p

Koszul duality for parabolic and singular category

Abstract. This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category O over a complex semisimple Lie-algebra, established by Beilinson, Ginzburg and Soergel in Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473– 527. In that paper it was proved that any “block ” in O, determined by an integral, but possibly singular weight, is Koszul (i.e. equivalent to the category of finitely generated modules over some Koszul ring) and, moreover, that the “Koszul dual ” of such a block is isomorphic to a “parabolic subcategory ” of the trivial block in O. We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in O is Koszul and we also calculate the Koszul dual of such a category. 1