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Ivan Losev

Publications and source records attributed to Ivan Losev.

At least 19 recordsLinked to original sources
arXiv2026

Affine Harish-Chandra center in positive characteristic

Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish--Chandra centers of the associated Kac--Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $κ$ for $G$, consider the associated Kac--Moody vertex algebra $V_κ(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish--Chandra center of arc group invariants $$V_κ(\mathfrak{g})^{\mathscr{J}G} \subset V_κ(\mathfrak{g}).$$ Write $\checkκ$ for the dual level for $\check{G}$, $\checkκ^p - \checkκ$ for its image under the Artin--Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}$ for the moduli space of $(\checkκ^p - \checkκ)$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_κ(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}.$$ There is a similar identification for the Harish--Chandra center of the filtered complete enveloping algebra at level $κ$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. The presence of these Harish--Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish--Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.

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arXiv2026

A bar operation on the double affine Hecke algebra

We construct a bar-type operation for the affine Hecke algebra $\Ha$ attached to a symmetrizable Kac--Moody group. Our main tool is a geometric bimodule defined using the equivariant $K$-theory of a suitable version of the Steinberg variety for this Kac--Moody group. Using elements of this bimodule corresponding to standard and costandard Hodge $D$-modules on the Kashiwara flag variety, we define a bar involution on $\mathcal{H}_{W}^a$ by relating the left and right actions of $\mathcal{H}_{W}^a$ on this bimodule. A central issue is that the resulting formulas for the bar operation do not preserve the algebra itself and naturally produce infinite expansions. To address this, we introduce certain completions of $\mathcal{H}_{W}^a$ on which the bar operation is well-defined. For this construction, we prove a number of combinatorial finiteness results for products in $\mathcal{H}_{W}^a$. In affine type, we further analyze the level-zero part of the algebra and obtain an explicit combinatorial formula for a bar operation on a suitable localization of Cherednik's double affine Hecke algebra. In this case the bar operation itself is not an involution but we also show that this bar operation becomes an involution when the lattice parameter $q$ is set to $1$.

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arXiv2026

Unipotent Ideals and Harish-Chandra Bimodules

Let $G$ be a complex reductive algebraic group. In this paper, we give a geometric definition of a unipotent representation of $G$. Our definition generalizes the notion of a special unipotent representation, due to Barbasch-Vogan and Arthur. The representations we define arise from finite equivariant covers of nilpotent co-adjoint $G$-orbits. To each such cover $\tilde{\mathbb{O}}$, we attach a distinguished filtered algebra $\mathcal{A}_0$ equipped with a graded Poisson isomorphism $\mathrm{gr}(\mathcal{A}_0)\simeq \mathbb{C}[\tilde{\mathbb{O}}]$. The algebra $\mathcal{A}_0$ receives a distinguished homomorphism from the universal enveloping algebra $U(\mathfrak{g})$, and the kernel of this homomorphism is a completely prime primitive ideal in $U(\mathfrak{g})$ with associated variety $\overline{\mathbb{O}}$. A unipotent ideal is any ideal in $U(\mathfrak{g})$ which arises in this fashion. A unipotent representation is an irreducible Harish-Chandra bimodule which is annihilated (on both sides) by such an ideal. Our unipotent ideals and representations have all of the expected properties: the unipotent representations attached to $\tilde{\mathbb{O}}$ are parameterized by irreducible representations of a certain finite group (generalizing Lusztig's canonical quotient) and, when restricted to $K$, are of the form conjectured by Vogan. In classical types, all unipotent ideals are maximal, and all unipotent representations are unitary (we expect these properties to hold for arbitrary groups). Finally, all special unipotent representations are unipotent. To prove the last assertion, we introduce a refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality of Braden, Licata, Proudfoot, and Webster.

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arXiv2026

On De Concini-Kac forms of quantum groups

Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable modification to the De Concini-Kac form, namely the even part algebra, which has some appealing features. Notably, it behaves uniformly with respect to the order of the roots of unity and admits an adjoint action of the Lusztig form. We revisit several results due to De Concini-Kac-Procesi and Tanisaki for the even part algebra. Namely, we give conceptual definitions of the Frobenius and Harish-Chandra centers and describe the entire center in terms of these two subalgebras getting a complete quantum analog of the Veldkamp theorem on the center of the universal enveloping algebras in positive characteristic. We investigate the Azumaya locus of the even part algebra over its center. We also show that the locally finite part of the even part algebra under the adjoint action of the Lusztig form is isomorphic to the reflection equation algebra, which is the quantized coordinate algebra with the product twisted by $R$-matrix. Some results on Lusztig forms at roots of unity are revisited and proved in greater generality including Kempf vanishing theorem and good filtrations on the quantized coordinate algebra.

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arXiv2025

On Harish-Chandra modules over quantizations of nilpotent orbits

Let $G$ be a semisimple algebraic group over the complex numbers and $K$ be a connected reductive group mapping to $G$ so that the Lie algebra of $K$ gets identified with a symmetric subalgebra of $\mathfrak{g}$. So we can talk about Harish-Chandra $(\mathfrak{g},K)$-modules, where $\mathfrak{g}$ is the Lie algebra of $G$. The goal of this paper is to give a geometric classification of irreducible Harish-Chandra modules with full support over the filtered quantizations of the algebras of the form $\mathbb{C}[\mathbb{O}]$, where $\mathbb{O}$ is a nilpotent orbit in $\mathfrak{g}$ with codimension of the boundary at least $4$. Namely, we embed the set of isomorphism classes of irreducible Harish-Chandra modules into the set of isomorphism classes of irreducible $K$-equivariant suitably twisted local systems on $\mathbb{O}\cap \mathfrak{k}^\perp$. We show that under certain conditions, for example when $K\subset G$ or when $\mathfrak{g}\cong \mathfrak{so}_n,\mathfrak{sp}_{2n}$, this embedding is in fact a bijection. On the other hand, for $\mathfrak{g}=\mathfrak{sl}_n$ and $K=\operatorname{Spin}_n$, the embedding is not bijective and we give a description of the image. Finally, we perform a partial classification for exceptional Lie algebras.

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arXiv2025

On uniqueness of tensor products of irreducible categorifications

In this paper, we propose an axiomatic definition for a tensor product categorification. A tensor product categorification is an abelian category with a categorical action of a Kac-Moody algebra g in the sense of Rouquier or Khovanov-Lauda whose Grothendieck group is isomorphic to a tensor product of simple modules. However, we require a much stronger structure than a mere isomorphism of representations; most importantly, each such categorical representation must have standardly stratified structure compatible with the categorification functors, and with combinatorics matching those of the tensor product. With these stronger conditions, we recover a uniqueness theorem similar in flavor to that of Rouquier for categorifications of simple modules. Furthermore, we already know of an example of such a categorification: the representations of algebras T^λpreviously defined by the second author using generators and relations. Next, we show that tensor product categorifications give a categorical realization of tensor product crystals analogous to that for simple crystals given by cyclotomic quotients of KLR algebras. Examples of such categories are also readily found in more classical representation theory; for finite and affine type A, tensor product categorifications can be realized as quotients of the representation categories of cyclotomic q-Schur algebras.

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arXiv2025

Bernstein inequality and holonomic modules

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that the generalized Bernstein inequality holds for simple modules and turns into equality for holonomic simples provided the algebraic fundamental groups of all leaves are finite. Under the same assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length or if the algebra in question is a quantum Hamiltonian reduction, then any holonomic module has finite length. This allows to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

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arXiv2024

Categorical Heisenberg action I: rational Cherednik algebras

In this paper we introduce and study a categorical action of the positive part of the Heisenberg Lie algebra on categories of modules over rational Cherednik algebras associated to symmetric groups. We show that the generating functor for this action is exact. We then produce a categorical Heisenberg action on the categories $\mathcal{O}$ and show it is the same as one constructed by Shan and Vasserot. Finally, we reduce modulo a large prime $p$. We show that the functors constituting the action of the positive half of the Heisenberg algebra send simple objects to semisimple ones, and we describe these semisimple objects.

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arXiv2023

Quantum category O vs affine Hecke category

The goal of this paper is to relate the quantum category $\mathcal{O}$ (known also as the category of modules over the mixed quantum group) at an odd root of unity to the affine Hecke category. Namely, we prove equivalences of highest weight categories between integral blocks of the affine category $\mathcal{O}$ and the heart of the so called ``new'' t-structure on the affine Hecke category. In order to do this we deform our categories over the formal neighborhood of $0$ in the dual affine Cartan and show that the categories of standardly filtered objects in the deformations are equivalent. For this, we construct functors from the deformed categories to the category of bimodules over the formal power series on the affine Cartan. Then we use what we call the Rouquier-Soergel theory, also developed in this paper, to show that on the categories of standardly filtered objects, these functors are full embeddings with the same image.

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arXiv2023

Categorical braid group actions and cactus groups

Let $\mathfrak{g}$ be a semisimple simply-laced Lie algebra of finite type. Let $\mathcal{C}$ be an abelian categorical representation of the quantum group $U_q(\mathfrak{g})$ categorifying an integrable representation $V$. The Artin braid group $B$ of $\mathfrak{g}$ acts on $D^b(\mathcal{C})$ by Rickard complexes, providing a triangulated equivalence $Θ_{w_0}:D^b(\mathcal{C}_μ) \to D^b(\mathcal{C}_{w_0(μ)})$, where $μ$ is a weight of $V$ and $Θ_{w_0}$ is a positive lift of the longest element of the Weyl group. We prove that this equivalence is t-exact up to shift when $V$ is isotypic, generalising a fundamental result of Chuang and Rouquier in the case $\mathfrak{g}=\mathfrak{sl}_2$. For general $V$, we prove that $Θ_{w_0}$ is a perverse equivalence with respect to a Jordan-Hölder filtration of $\mathcal{C}$. Using these results we construct, from the action of $B$ on $V$, an action of the cactus group on the crystal of $V$. This recovers the cactus group action on $V$ defined via generalised Schützenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.

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arXiv2023

On modular Soergel bimodules, Harish-Chandra bimodules, and category O

In this paper we continue the study of the category of modular Harish-Chandra bimodules initiated by Bezrukavnikov and Riche and also study the modular version of the BGG category $\mathcal{O}$. We prove a version of the Bezrukavnikov-Mirkovic-Rumynin localization theorem for the Harish-Chandra bimodules and for the category $\mathcal{O}$. We also relate the category of Harish-Chandra bimodules to the affine Hecke category building on the prior work of Bezrukavnikov and Riche.

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arXiv2021

Deformations of symplectic singularities and Orbit method for semisimple Lie algebras

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of adjoint orbits in a semisimple Lie algebra to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras.

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arXiv2021

Affine Springer Fibers, Procesi bundles, and Cherednik algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{t}$ its Cartan subalgebra and $W$ the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus \mathfrak{t}^*)/W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimodules over rational and trigonometric Cherednik algebras.

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arXiv2021

On modular categories O for quantized symplectic resolutions

In this paper we study highest weight and standardly stratified structures on modular analogs of categories $\mathcal{O}$ over quantizations of symplectic resolutions and show how to recover the usual categories $\mathcal{O}$ (reduced mod $p\gg 0$) from our modular categories. More precisely, we consider a conical symplectic resolution that is defined over a finite localization of $\mathbb{Z}$ and is equipped with a Hamiltonian action of a torus $T$ that has finitely many fixed points. We consider algebras $\mathcal{A}_λ$ of global sections of a quantization in characterstic $p\gg 0$, where $λ$ is a parameter. Then we consider a category $\tilde{\mathcal{O}}_λ$ consisting of all finite dimensional $T$-equivariant $\mathcal{A}_λ$-modules. We show that for $λ$ lying in a {\it p-alcove} $\,^p\!A$, the category $\tilde{\mathcal{O}}_λ$ is highest weight (in some generalized sense). Moreover, we show that every face of $\,^p\!A$ that survives in $\,^p\!A/p$ when $p\rightarrow \infty$ defines a standardly stratified structure on $\tilde{\mathcal{O}}_λ$. We identify the associated graded categories for these standardly stratified structures with reductions mod $p$ of the usual categories $\mathcal{O}$ in characteristic $0$. Applications of our construction include computations of wall-crossing bijections in characteristic $p$ and the existence of gradings on categories $\mathcal{O}$ in characteristic $0$.

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arXiv2021

Localization theorems for quantized symplectic resolutions

The goal of this paper is to establish Beilinson-Bernstein type localization theorems for quantizations of some conical symplectic resolutions. We prove the full localization theorems for finite and affine type A Nakajima quiver varieties. The proof is based on two partial results that hold in more general situations. First, we establish an exactness result for global section functor if there is a tilting generator that has a rank 1 summand. Second, we examine when the global section functor restricts to an equivalence between categories $\mathcal{O}$.

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arXiv2021

Representations with minimal support for quantized Gieseker varieties

We study the minimally supported representations of quantizations of Gieseker moduli spaces. We relate them to $\operatorname{SL}_n$-equivariant D-modules on the nilpotent cone of $\mathfrak{sl}_n$ and to minimally supported representations of type A rational Cherednik algebras. Our main result is character formulas for minimally supported representations of quantized Gieseker moduli spaces.

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