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Ivan Losev

Publications and source records attributed to Ivan Losev.

At least 19 recordsLinked to original sources
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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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

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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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

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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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

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 $\Theta_{w_0}:D^b(\mathcal{C}_\mu) \to D^b(\mathcal{C}_{w_0(\mu)})$, where $\mu$ is a weight of $V$ and $\Theta_{w_0}$ 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 $\Theta_{w_0}$ is a perverse equivalence with respect to a Jordan-H\"older 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 Sch\"utzenberger 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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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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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.

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arXiv2020

Supports of simple modules in cyclotomic Cherednik categories O

The goal of this paper is to compute the supports of simple modules in the categories $\mathcal{O}$ for the rational Cherednik algebras associated to groups $G(\ell,1,n)$. For this we compute some combinatorial maps on the set of simples: wall-crossing bijections and a certain $\mathfrak{sl}_\infty$-crystal associated to a Heisenberg algebra action on a Fock space.

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arXiv2020

Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $\mathcal{J}\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms.

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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.

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