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

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

Gale duality and Koszul duality

Given an affine hyperplane arrangement with some additional structure, 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.Comment: 55 pages; v2 contains significant revisions to proofs and to some of the results. Section 7 has been deleted; that material will be incorporated into a later paper by the same author

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. 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 $\mathcal{O}$ for $\mathfrak{gl}_n$ is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category $\mathcal{O}$" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.Comment: 39 pages; v3: final versio

A Prototype Toolkit For Evaluating Indoor Environmental Quality In Commercial Buildings

Measurement of building environmental parameters is often complex, expensive, and not easily proceduralized in a manner that covers all commercial buildings. Evaluating building indoor environmental quality performance is therefore not standard practice. This project developed a prototype toolkit that addressed existing barriers to widespread indoor environmental quality performance evaluation. A toolkit with both hardware and software elements was designed for practitioners around the indoor environmental quality requirements of the American Society of Heating, Refrigeration and Air Conditioning Engineers / Chartered Institution of Building Services / United States Green Building Council Performance Measurement Protocols. This unique toolkit was built on a wireless mesh network with a web-based data collection, analysis, and reporting application. The toolkit provided a fast, robust deployment of sensors, real-time data analysis, Performance Measurement Protocol-based analysis methods and a scorecard and report generation tools. A web-enabled Geographic Information System-based metadata collection system also reduced field-study deployment time. The toolkit was evaluated through three case studies, which were discussed in this report

Quantizations of conical symplectic resolutions II: category O\mathcal O and symplectic duality

We define and study category $\mathcal O$ for a symplectic resolution, generalizing the classical BGG category $\mathcal O$, which is associated with the Springer resolution. This includes the development of intrinsic properties parallelling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples. We observe that category $\mathcal O$ is often Koszul, and its Koszul dual is often equivalent to category $\mathcal O$ for a different symplectic resolution. This leads us to define the notion of a symplectic duality between symplectic resolutions, which is a collection of isomorphisms between representation theoretic and geometric structures, including a Koszul duality between the two categories. This duality has various cohomological consequences, including (conjecturally) an identification of two geometric realizations, due to Nakajima and Ginzburg/Mirkovi\'c-Vilonen, of weight spaces of simple representations of simply-laced simple algebraic groups. An appendix by Ivan Losev establishes a key step in the proof that $\mathcal O$ is highest weight.Comment: 118 pages. v3: mostly changes based on new material which has appeared since last version; with an appendix by Ivan Losev; v4: edited in response to referee comment

Iron-Catalyzed H/D Exchange of Primary Silanes, Secondary Silanes and Tertiary Siloxanes

[Image: see text] A synthetic study into the catalytic hydrogen/deuterium (H/D) exchange of 1° silanes, 2° silanes, and 3° siloxanes is presented, facilitated by iron-β-diketiminato complexes (1a and 1b). Near-complete H/D exchange is observed for a variety of aryl- and alkyl-containing hydrosilanes and hydrosiloxanes. The reaction tolerates alternative hydride source pinacolborane (HBpin), with quantitative H/D exchange. A synthetic and density functional theory (DFT) investigation suggests that a monomeric iron-deuteride is responsible for the H/D exchange