arXiv subjects

Roman Bezrukavnikov

Publications and source records attributed to Roman Bezrukavnikov.

At least 37 records · Page 2Linked to original sources

arXiv2023

Quantization of Hitchin integrable system via positive characteristic

In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where they are deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.

math.RT↗

arXiv2023

On the structure of the affine asymptotic Hecke algebras

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit a description in terms of the Grothedieck group of sheaves on the square of a finite set equivariant under the action of the centralizer of a nilpotent element in the reductive group. A weaker form of this statement, allowing for possible central extensions of stabilizers of that action, has been proved by the first named author with Ostrik. In the present paper we describe an example showing that nontrivial central extensions do arise, thus the above weaker statement is optimal. We also show that Lusztig's homomorphism from the affine Hecke algebra to the asymptotic affine Hecke algebra induces an isomorphism on cocenters and discuss the relation of the above central extensions to the structure of the cocenter.

math.RT↗

arXiv2023

A geometric realization of the center of the small quantum group

We propose a new geometric model for the center of the small quantum group using the cohomology of certain affine Springer fibers. More precisely, we establish an isomorphism between the equivariant cohomology of affine Spaltenstein fibers for a split element and the center of the deformed graded modules for the small quantum group. We also obtain an embedding from the invariant part of the nonequivariant cohomology under the action of the extended affine Weyl group to the invariant part of the center of the small quantum group under Langlands dual group action, which we conjecture to be an isomorphism. Finally, we give a dimension formula for the invariants on the cohomology side, thus providing a lower bound for the dimension of the center.

math.RT↗

arXiv2022

Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers

Starting from a homogeneous affine Springer fiber $Fl_ψ$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $Fl_ψ$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $Fl_ψ$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.

math.AG↗

arXiv2022

Hecke action on the principal block

In this paper we construct an action of the affine Hecke category (in its "Soergel bimodules" incarnation) on the principal block of representations of a simply-connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical $p$-Kazhdan-Lusztig polynomials.

math.RT↗

arXiv2022

Monodromic model for Khovanov-Rozansky homology

We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov-Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.

math.RT↗

arXiv2021

Cherednik algebras and Hilbert schemes in characteristic p (with an appendix by Pavel Etingof)

We prove a localization theorem for the type A rational Cherednik algebra H_c=H_{1,c} over an algebraic closure of the finite field F_p. In the most interesting special case where the parameter c takes values in F_p, we construct an Azumaya algebra A_c on Hilb^n, the Hilbert scheme of n points in the plane, such that the algebra of global sections of A_c is isomorphic to H_c. Our localisation theorem provides an equivalence between the bounded derived categories of H_c-modules and sheaves of coherent A_c-modules on the Hilbert scheme, respectively. Furthermore, we show that the Azumaya algebra splits on the formal completion of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland-King-Reid and Haiman.

math.RT↗

arXiv2021

On two geometric realizations of an affine Hecke algebra

The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.

math.RT↗

arXiv2021

Some results about the geometric Whittaker model

Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on $X$ with respect to a generic character of $U$ commutes with Verdier duality. In the first example we take $χ$ to be an arbitrary $G$-variety and we prove the above property for all $\overline{P}$-equivariant sheaves on $X$ where $\overline{P}$ is an opposite parabolic subgroup assuming $χ$ satisfies a strong nondegeneracy condition (such a $χ$ exists for some but not all choices of $P$). In the case when $P$ is a Borel subgroup it is enough to require that the sheaf in question is $\overline{U}$ equivariant where $\overline{U}$ is the unipotent radical of $\overline{P}$. In the second example we take $X = G$ where $G$ acts by left translations and we prove the corresponding result when $P$ is a Borel subgroup for sheaves equivariant under the adjoint action of $G$ (the latter result was conjectured by B. C. Ngo who proved it for $G = GL(n)$). As an application we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier-Deligne transform.

math.AG↗

arXiv2021

Perverse coherent sheaves

We describe an analogue of the notion of a perverse sheaf in the setting of the derived category of coherent sheaves on an algebraic stack. Under strong additional assumptions the construction of coherent "intersection cohomology" complexes works. This paper supersedes the preprint arXiv:math/0005152 by the second author.

math.AG↗

arXiv2020

Etingof conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof's conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac-Moody actions and wall-crossing functors. We finish the paper outlining some future directions of research.

math.RT↗

arXiv2020

Dimensions of modular irreducible representations of semisimple Lie algebras

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished $p$-characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category $\mathcal{O}$. For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent $p$-character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.

math.RT↗

arXiv2020

A topological approach to Soergel theory

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\subset B$ are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where $T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$.

math.RT↗

arXiv2020

Perverse sheaves on affine flags and Langlands dual group

The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.

math.RT↗

arXiv2019

An Iwahori-Whittaker model for the Satake category

In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.

math.RT↗

arXiv2018

Localization properties of Chern insulators

We study the localization properties of the equal-time electron Green's function in a Chern insulator in an arbitrary dimension and with an arbitrary number of bands. We prove that the Green's function cannot decay super-exponentially if the Hamiltonian is finite-range and the quantum Hall response is nonzero. For a general band Hamiltonian (possibly infinite-range), we prove that the Green's function cannot be finite-range if the quantum Hall response is nonzero. The proofs use methods of algebraic geometry.

cond-mat.mes-hall↗

arXiv2018

Bernstein components via Bernstein center

Let G be a reductive p-adic group. Let $Φ$ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that $Φ$ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modiification of the arguments of J.F.Dat who proved our result in one direction.

math.RT↗

arXiv2018

Deligne-Lusztig duality and wonderful compactification

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $G=GL(n)$ by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct we obtain a description of the Serre functor for representations of a p-adic group.

math.RT↗