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Andrei Ionov

Publications and source records attributed to Andrei Ionov.

14 recordsLinked to original sources
arXiv2026

The Unitarity of Arthur Packets for Real Reductive Groups

Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$. In the 1980s, Arthur conjectured the existence of certain packets of irreducible admissible representations of $G(\mathbb{R})$ satisfying various remarkable properties. These packets were given a precise definition in the book of Adams, Barbasch, and Vogan in terms of microlocal geometry on a space of Langlands parameters. A longstanding conjecture, originally due to Arthur, is that all Arthur packets consist of $\textit{unitary}$ representations. In this paper, we prove this conjecture in general. The main new idea is a `Jordan decomposition' for Arthur packets: a canonical two-step process for realizing an arbitrary Arthur packet via real parabolic and cohomological induction from a unipotent Arthur packet for a certain Levi subgroup. This process is analogous to the decomposition of an element of a complex algebraic group as a (unique) commuting product of elliptic, hyperbolic, and unipotent parts. Using our Jordan decomposition, we reduce the question of unitarity to the case of unipotent Arthur packets, where the answer is already known (by work of Adams-Arancibia-Mezo, Adams-van Leeuwen-Miller-Vogan, Arthur, Barbasch, Barbasch-Ma-Sun-Zhu, and Davis-Mason-Brown). As an application of the same methods, we also give a proof of Jiang's conjecture for real reductive groups, which gives an upper bound on the wavefront sets of the members of an Arthur packet in terms of the Barbasch-Vogan dual of the Arthur $SL_2(\mathbb{C})$.

math.RT↗
arXiv2024

Surface Modification and Subsequent Fermi Density Enhancement of Bi(111)

Defects introduced to the surface of Bi(111) break the translational symmetry and modify the surface states locally. We present a theoretical and experimental study of the 2D defects on the surface of Bi(111) and the states that they induce. Bi crystals cleaved in ultrahigh vacuum (UHV) at low temperature (110 K) and the resulting ion-etched surface are investigated by low-energy electron diffraction (LEED), X-ray photoelectron spectroscopy, ultraviolet photoelectron spectroscopy (UPS), and scanning tunneling microscopy (STM) as well as spectroscopy (STS) techniques in combination with density functional theory (DFT) calculations. STS measurements of cleaved Bi(111) reveal that a commonly observed bilayer step edge has a lower density of states (DOS) around the Fermi level as compared to the atomic-flat terrace. Following ion bombardment, the Bi(111) surface reveals anomalous behavior at both 110 and 300 K: Surface periodicity is observed by LEED, and a significant increase in the number of bilayer step edges and energetically unfavorable monolayer steps is observed by STM. It is suggested that the newly exposed monolayer steps and the type A bilayer step edges result in an increase to the surface Fermi density as evidenced by UPS measurements and the Kohn-Sham DOS. These states appear to be thermodynamically stable under UHV conditions.

cond-mat.mtrl-sci↗
arXiv2024

On a t-exactness property of the Harish-Chandra transform

Using hyperbolic localization, we identify the nearby cycles along the Vinberg degeneration with the composition of Radon and Harish-Chandra functors, both considered for the category of character sheaves. This provides a new, simple proof of the exactness of this composition, extending previously known results to arbitrary monodromy and more general sheaf-theoretic set-ups.

math.RT↗
arXiv2024

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.

math.RT↗
arXiv2024

Fourier transform on the locus of cyclic spectral curves in the Hitchin base

We compute the Fourier transform of some of the summands of the push-forward of the constant sheaf under the Hitchin map for $\mathrm{SL}_n$ restricted to the locus of cyclic spectral curves inside the Hitchin base (for $\mathrm{SL}_2$ all spectral curves are cyclic) and give an estimate on the support of the Fourier transforms of the other summands.

math.AG↗
arXiv2024

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 ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi-Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of $(f,G)$ form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.

math.AG↗
arXiv2023

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a new proof of a McKay type equivalence for the matrix factorization categories.

math.AG↗
arXiv2023

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.

cond-mat.mtrl-sci↗
arXiv2023

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.

math.AG↗
arXiv2021

Hochschild cohomology of Fermat type polynomials with non-abelian symmetries

For a polynomial $f = x_1^n + \dots + x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g \in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x}) = f(g \cdot \mathbf{x})$. For any $G \subseteq G_f$ we compute explicitly Hochschild cohomology of the category of $G$--equivarint matrix factorizations of $f$. We introduce the pairing on it showing that it is a Frobenius algebra.

math.AG↗
arXiv2021

Filtrations on block subalgebras of reduced universal enveloping algebras

We study the interaction between the block decompositions of reduced universal enveloping algebras in positive characteristic, the PBW filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra $A_λ$ of the restricted universal enveloping algebra $\mathcal{U}_χ(\mathfrak{g})$ and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighbourhood of $0$ in the nilpotent cone and the coinvariants algebra corresponding to $λ$. In the case of $\mathfrak{g}=\mathfrak{sl}_2(k)$ in characteristic $p>2$ we determine the associated graded algebras of these filtrations on block subalgebras of $\mathcal{U}_0(\mathfrak{sl}_2)$. We also apply this to determine the structure of the adjoint representation of $\mathcal{U}_0(\mathfrak{sl}_2)$.

math.RT↗
arXiv2019

Primitive forms for Gepner singularities

We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form $F_{k,n}=\sum_{i=1}^n x_i^k$ invariant under the natural $S_n$-action.

math.AG↗
arXiv2017

Kostka-Shoji polynomials and Lusztig's convolution diagram

We propose an $r$-variable version of Kostka-Shoji polynomials $K^-_{λμ}$ for $r$-multipartitions $λ,μ$. Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of a line bundle over Lusztig's iterated convolution diagram for the cyclic quiver $\tilde{A}_{r-1}$.

math.AG↗