Tilting sheaves for real groups and Koszul duality

For a certain class of real analytic varieties with Lie group actions we develop a theory of (free-monodromic) tilting sheaves, and apply it to flag varieties stratified by real group orbits. For quasi-split real groups, we construct a fully faithful embedding of the category of tilting sheaves to a real analog of the category of Soergel bimodules, establishing real group analogs of Soergel's Structure Theorem and Endomorphism Theorem. We apply these results to give a purely geometric proof of the theorem of Bezrukavnikov and Vilonen which proves Soergel's conjecture for quasi-split groups

Hodge diamonds of the Landau--Ginzburg orbifolds

Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of $\mathrm{SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessarily abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi-Yau condition. We prove that the nonvanishing bi-graded pieces of the B--model state space of $(f,G)$ form a diamond. We identify its top-most, bottom-most, left-most and right-most entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.Comment: 24 pages, 5 figure

Equivariant derived category of a reductive group as a categorical center

We prove that the adjoint equivariant derived category of a reductive group $G$ is equivalent to the appropriately defined monoidal center of the torus-equivariant version of the Hecke category. We use this to give new proofs, independent of sheaf-theoretic set up, of the fact that the Drinfeld center of the abelian Hecke category is equivalent to the abelian category of unipotent character sheaves; and of a characterization of strongly-central sheaves on the torus

Unravelling the atomic and electronic structure of nanocrystals on superconducting Nb(110): Impact of the oxygen monolayer

The Niobium surface is almost always covered by a native oxide layer which greatly influences the performance of superconducting devices. Here we investigate the highly stable Niobium oxide overlayer of Nb(110), which is characterised by its distinctive nanocrystal structure as observed by scanning tunnelling microscopy (STM). Our ab-initio density functional theory (DFT) calculations show that a subtle reconstruction in the surface Niobium atoms gives rise to rows of 4-fold coordinated oxygen separated by regions of 3-fold coordinated oxygen. The 4-fold oxygen rows are determined to be the source of the nanocrystal pattern observed in STM, and the two chemical states of oxygen observed in core-level X-ray photoelectron spectroscopy (XPS) are ascribed to the 3-fold and 4-fold oxygens. Furthermore, we find excellent agreement between the DFT calculated electronic structure with scanning tunnelling spectroscopy and valence XPS measurements.Comment: 8 pages, 4 figures, plus 3 pages of Supporting Informatio

Tilting sheaves for real groups and Koszul duality

For a certain class of real analytic varieties with the real Lie group action we define a tstructure on the category of equivariant-monodromic sheaves and develop the theory of tilting sheaves. In case of a quasi-split real form of an algebraic group acting on the flag variety we construct an analog of a Soergel functor, which fully-faithfully embeds the subcategory of tilting objects to the category of coherent sheaves on a block variety. We apply the results to give a new, purely geometric, proof of the Soergel’s conjecture for quasi-split groups. The thesis is based on a joint work with Zhiwei Yun.Ph.D