Langlands duality for finite-dimensional representations of quantum affine algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.Comment: 40 pages; several results and comments added. Accepted for publication in Letters in Mathematical Physic

Just married: the synergy between feminist criminology and the Tripartite Cybercrime Framework

This article is a theoretical treatment of feminist epistemology of crime, which advocates the centrality of gender as a theoretical starting point for the investigating of digital crimes. It does so by exploring the synergy between the feminist perspectives and the Tripartite Cybercrime Framework (TCF) (which argues that three possible factors motivate cybercrimes – socioeconomic, psychosocial, and geopolitical) to critique mainstream criminology and the meaning of the term “cybercrime”. Additionally, the article examines gender gaps in online harassment, cyber‐bullying, cyber‐fraud, revenge porn, and cyber‐stalking to demonstrate that who is victimised, why, and to what effect are the critical starting points for the analysis of the connections between gender and crimes. In turn, it uses the lens of intersectionality to acknowledge that, while conceptions of gender and crime interact, they intersect with other categories (e.g., sexuality) to provide additional layers of explanation. To nuance the utilitarian value of the synergy between the TCF and the feminist perspectives, the focus shifts to a recent case study (which compared socioeconomic and psychosocial cybercrimes). The article concludes that, while online and offline lives are inextricably intertwined, the victimisations in psychosocial cybercrimes may be more gendered than in socioeconomic cybercrimes. These contributions align the TCF to the feminist epistemology of crime in their attempt to move gender analysis of digital crimes “from margin to centre”

Morse decomposition for D-module categories on stacks

Let Y be a smooth algebraic stack exhausted by quotient stacks. Given a Kirwan-Ness stratification of the cotangent stack T^*Y, we establish a recollement package for twisted D-modules on Y, gluing the category from subquotients described via modules microsupported on the Kirwan-Ness strata of T^*Y. The package includes unusual existence and "preservation-of-finiteness" properties for functors of the full category of twisted D-modules, extending the standard functorialities for holonomic modules. In the case that Y = X/G is a quotient stack, our results provide a higher categorical analogue of the Atiyah-Bott--Kirwan--Ness "equivariant perfection of Morse theory" for the norm-squared of a real moment map. As a consequence, we deduce a modified form of Kirwan surjectivity for the cohomology of hyperkaehler/algebraic symplectic quotients of cotangent bundles

Compatibility of t-structures for quantum symplectic resolutions

Let W be a smooth complex quasiprojective variety with the action of a connected reductive group G. Adapting the stratification approach of Teleman to a microlocal context, we prove a vanishing theorem for the functor of G-invariant sections---i.e., of quantum Hamiltonian reduction---for G-equivariant twisted D-modules on W. As a consequence, when W is affine we establish an effective combinatorial criterion for exactness of the global sections functors of microlocalization theory. When combined with our earlier derived equivalence results, this gives precise criteria for "microlocalization of representation categories.

Derived equivalence for quantum symplectic resolutions

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson-Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon-Stafford and Kashiwara-Rouquier as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities

Derived equivalence for quantum symplectic resolutions

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson-Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon-Stafford and Kashiwara-Rouquier as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities