For a complex reflection group *W* with reflection representation hh, we define and study a natural filtration by Serre subcategories of the category O_*c*(*W*,h) of representations of the rational Cherednik algebra *H_**c*(*W*,h). This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When *W* is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in O_*c*(*W*,h) of given support. We apply these techniques to count the number of irreducible representations in O_*c*(*W*,h) of given support for all exceptional Coxeter groups *W* and all parameters *c*, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of O_*c*(*W*,h) for exceptional Coxeter groups *W* in many new cases.

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.

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a “web of flops” provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g = sl3 we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

We study plane partitions satisfying condition *a**_{n*+1,*m*+1}=0 (this condition is called “pit”) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal gl1 algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra gl*_{m*|*n*}. We discuss representation theoretic interpretation of our formulas using *q*-deformed *W*-algebra gl*_{m*|*n*}.

We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism.

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O. We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier's deformation approach as well as categorical actions on highest weight categories and related combinatorics.

We apply the technique of formal geometry to give a necessary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed deformation quantization of the structure sheaf of an algebraic symplectic variety.

Let $K$ be a field and $G$ be a group of its automorphisms endowed with the compact-open topology, cf. section 1.1. If $G$ is precompact then $K$ is a generator of the category of {\sl smooth} (i.e. with open stabilizers) $K$-{\sl semilinear} representations of $G$, cf. Proposition 1.1. There are non-semisimple smooth semilinear representations of $G$ over $K$ if $G$ is not precompact. In this note the smooth semilinear representations of the group $\Sy_{\Psi}$ of all permutations of an infinite set $\Psi$ are studied. Let $k$ be a field and $k(\Psi)$ be the field freely generated over $k$ by the set $\Psi$ (endowed with the natural $\Sy_{\Psi}$-action). One of principal results describes the Gabriel spectrum of the category of smooth $k(\Psi)$-semilinear representations of $\Sy_{\Psi}$. It is also shown, in particular, that (i) for any smooth $\Sy_{\Psi}$-field $K$ any smooth finitely generated $K$-semilinear representation of $\Sy_{\Psi}$ is noetherian, (ii) for any $\Sy_{\Psi}$-invariant subfield $K$ in the field $k(\Psi)$, the object $k(\Psi)$ is an injective cogenerator of the category of smooth $K$-semilinear representations of $\Sy_{\Psi}$, (iii) if $K\subset k(\Psi)$ is the subfield of rational homogeneous functions of degree 0 then there is a one-dimensional $K$-semilinear representation of $\Sy_{\Psi}$, whose integral tensor powers form a system of injective cogenerators of the category of smooth $K$-semilinear representations of $\Sy_{\Psi}$, (iv) if $K\subset k(\Psi)$ is the subfield generated over $k$ by $x-y$ for all $x,y\in\Psi$ then there is a unique isomorphism class of indecomposable smooth $K$-semilinear representations of $\Sy_{\Psi}$ of each given finite length. Appendix collects some results on smooth {\sl linear} representations of symmetric groups and of the automorphism group of an infinite-dimensional vector space over a finite field.

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker–Maruyama moduli space *M*(*e*, *n*) of rank 2 stable vector bundles with the first Chern class 𝑒=0e=0 or −1−1 and all possible values of the second Chern class *n* on the projective space ℙ3P3. We show that, in a wide range of cases, the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces *M*(*e*, *n*) over all 𝑛≥1n≥1 contains new series of rational components in the case 𝑒=0e=0, exteding and improving previously known results of Vedernikov (Math USSSR-Izv 25:301–313, 1985) on series of rational families of bundles, and a first known infinite series of rational components in the case 𝑒=−1e=−1. Explicit constructions of rationality (stable rationality) of Ein components are given. Our approach is based on the study of a correspondence between generalized null correlation bundles constituting open subsets of Ein components and certain rank 2 reflexive sheaves on ℙ3P3. This correspondence is obtained via elementary transformations along surfaces. We apply the technique of Quot-schemes and universal spaces of extensions of sheaves to relate the parameter spaces of these two types of sheaves. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for 𝑐1=0c1=0 and *n* even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition “*n* is odd”, which is a usual sufficient condition for fineness.

In this paper we classify symplectic leaves of the regular part of the projectivization of the space of meromorphic endomorphisms of a stable vector bundle on an elliptic curve, using the study of shifted Poisson structures on the moduli of complexes from our previous work (Hua and Polishchuk in Adv Math 338:991–1037, 2018). This Poisson ind-scheme is closely related to the ind Poisson–Lie group associated to Belavin’s elliptic *r*-matrix, studied by Sklyanin, Cherednik and Reyman and Semenov-Tian-Shansky. Our result leads to a classification of symplectic leaves on the regular part of meromorphic matrix algebras over an elliptic curve, which can be viewed as the Lie algebra of the above-mentioned ind Poisson–Lie group. We also describe the decomposition of the product of leaves under the multiplication morphism and show the invariance of Poisson structures under autoequivalences of the derived category of coherent sheaves on an elliptic curve.

In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), thel-Kronecker quiver, is biholomorphic toC×Hfor l ≥ 3. In the present paper we define a new notion of norm, which distinguishes {Db(K(l))}l≥2. More precisely, to a triangulated category T which has property of a phase gap we attach a non-negative real number T ε. Natural assumptions on a SOD T = T1, T2 imply T1, T2ε ≤ min{T1ε , T2ε}. Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that [Db(point)] ∈ Cl([Db(S)]). Categories in a neighborhood of Db(K(l)) have the property that Db(K(l)) is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to newcategorical structures and connections to number theory and classical geometry.We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi–Yau curvecounting, where the entities we count are a Calabi–Yau modification of Db(K(l)). In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts stating that isogenous K3 surfaces are L-equivalent. Moreover, we give examples of derived equivalent twisted K3 surfaces, such that the underlying K3 surfaces are not L-equivalent. We also give examples showing that D-equivalent abelian varieties can be non-L-equivalent. This disproves the original version of a conjecture of Kuznetsov and Shinder.

We deduce the statements on (non-)L-equivalence from the very general results on the Grothendieck group of an additive category, whose morphisms are finitely generated abelian groups. In particular, we show that in such a category each stable isomorphism class of objects contains only finitely many isomorphism classes. We also show that a stable isomorphism between two objects *X* and *Y* such that the endomorphism ring of X is Z, implies that *X* and *Y* are isomorphic.

The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the homology groups of the transversal slices of the corresponding strata.

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna’s general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna’s 1993 classification of strongly solvable wonderful subgroups, and the author’s 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the three classifications and exhibit interrelations between the corresponding combinatorial invariants, which enables one to pass from one of these classifications to any other.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in pth cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkähler manifold. As an application, we obtain a theorem about dimension of a compact torus T admitting a holomorphic symplectic embedding to a hyperkähler manifold M. If M is generic in a d-dimensional family of deformations, then dimT≥2^[(d+1)/2].