arXiv subjects
Publications and source records attributed to Ivan Losev.
The goal of this paper is to construct an action of the cactus group of a Weyl group W on W that is nicely compatible with Kazhdan-Lusztig cells. The action is realized by the wall-crossing bijections that are combinatorial shadows of wall-crossing functors on the category O.
We study the quantizations of the algebras of regular functions on nilpotent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quantization has integral central character in all cases but four (one orbit in E_7 and three orbits in E_8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E_8). Our main ingredient is results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa.
In this survey we describe an interplay between Procesi bundles on symplectic resolutions of quotient singularities and Symplectic reflection algebras. Procesi bundles were constructed by Haiman and, in a greater generality, by Bezrukavnikov and Kaledin. Symplectic reflection algebras are deformations of skew-group algebras defined in complete generality by Etingof and Ginzburg. We construct and classify Procesi bundles, prove an isomorphism between spherical Symplectic reflection algebras, give a proof of wreath Macdonald positivity and of localization theorems for cyclotomic Rational Cherednik algebras.
This paper continues the study of highest weight categorical sl_2-actions started in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the refined sense. Then we prove that any highest weight sl_2-categorification can be filtered in such a way that the successive quotients are so called basic highest weight sl_2-categorifications. For a basic highest weight categorification we determine minimal projective resolutions of standard objects. We use this, in particular, to examine the structure of tilting objects in basic categorifications and to show that the Ringel duality is given by the Rickard complex. We finish by discussing open problems.
We introduce the notion of a totally aspherical parameter for a Rational Cherednik algebra. We get an explicit construction of the projective object defining the KZ functor for such parameters. We establish the existence of sufficiently many totally aspherical parameters for the groups G(l,1,n).
The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite W-algebra. We concentrate on the integral central character case. We prove, modulo a conjecture, that in this case the two are equal. Also, modulo the same conjecture, we compute certain scale factors introduced by Joseph. Our conjecture asserts that there is a one-dimensional module over the W-algebra with certain additional properties. The conjecture is proved for the classical types. This completes a program of computing Goldie ranks proposed by Joseph in the 80's. We also provide an essentially Kazhdan-Lusztig type formula for computing the characters of the irreducibles in the Brundan-Goodwin-Kleshchev category O for a W-algebra again under the assumption that the central character is integral. The formula is based on a certain functor (a generalized Soegel functor) from an appropriate parabolic category O to the W-algebra category O. We prove a number of properties of this functor including the quotient property and the double centralizer property. We develop several topics related to our generalized Soergel functor. For example, we discuss its analog for the category of Harish-Chandra bimodules. We also discuss generalizations to the case of categories O over Dixmier algebras. The most interesting example of this situation comes from the theory of quantum groups: we prove that an algebra that is basically Luszitg's form of a quantum group at a root of unity is a Dixmier algebra. For this we check that the quantum Frobenius epimorphism splits.
Let W be a complex reflection group. We prove that there is the maximal finite dimensional quotient of the Hecke algebra H_q(W) of W and that the dimension of this quotient coincides with |W|. This is a weak version of a Broué-Malle-Rouquier conjecture from 1998. The proof is based on categories O for Rational Cherednik algebras.
In this article we establish an isomorphism between universal infinitesimal Cherednik algebras and W-algebras for Lie algebras of the same type and 1-block nilpotent elements. As a consequence we obtain some fundamental results about infinitesimal Cherednik algebras.
In this paper we prove the abelian localization theorem for modules over cyclotomic Rational Cherednik algebras.
Procesi bundles are certain vector bundles on symplectic resolutions of symplectic quotient singularities for wreath-products of the symmetric groups with Kleinian groups. Roughly speaking, we can define Procesi bundles as bundles on resolutions that provide derived McKay equivalence. In this paper we classify Procesi bundles on resolutions obtained by Hamiltonian reduction and relate the Procesi bundles to the tautological bundles on the resolutions. Our proofs are based on deformation arguments and a connection of Procesi bundles with Symplectic reflection algebras.
We study equivalences for category O_p of the rational Cherednik algebras H_p of type G_l(n) = μ_l^n\rtimes S_n: a highest weight equivalence between O_p and O_{σ(p)} for σ\in S_l and an action of S_l on a non-empty Zariski open set of parameters p; a derived equivalence between O_p and O_{p'} whenever p and p' have integral difference; a highest weight equivalence between O_p and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.
We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SL_m. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution by Berest-Etingof-Ginzburg and Gordon, and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras H_{m/n}(S_n) and H_{n/m}(S_m). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry has a natural interpretation in terms of invariants of torus knots.
Varagnolo and Vasserot conjectured an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a truncation of an affine parabolic category O of type A. In this paper we reduce their conjecture to some purely combinatorial conjecture. Our principal tool for the reduction is the theory of categorical sl_2-actions on highest weight categories. Then we prove our combinatorial conjecture in some cases.
Finite W-algebras are certain associative algebras arising in Lie theory. Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our base field is algebraically closed and of characteristic 0) and its nilpotent element e. In this paper we classify finite dimensional irreducible modules with integral central character over W-algebras. In more detail, in a previous paper the first author proved that the component group A(e) of the centralizer of the nilpotent element under consideration acts on the set of finite dimensional irreducible modules over the W-algebra and the quotient set is naturally identified with the set of primitive ideals in U(g) whose associated variety is the closure of the adjoint orbit of e. In this paper for a given primitive ideal with integral central character we compute the corresponding A(e)-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of A(e) introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules of semisimple Lie algebras, the representation theory of W-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.
This is an expanded version of the lectures given by the author on the 3rd school "Lie algebras, algebraic groups and invariant theory" in Togliatti, Russia. In these notes we explain the concept of a categorical Kac-Moody action by studying an example of the category of rational representations of a general linear group in positive characteristic. We also deal with some more advanced topics: a categorical action on the polynomial representations and crystals of categorical actions.
We define highest weight categorical actions of sl_2 on highest weight categories and show that basically all known examples of categorical sl_2-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories O of type $A$, categories O for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Rational Cherednik algebras starting from their labels.
In this note we classify the primitive ideals in finite W-algebras of type A.
In this paper we study the structure of completions of symplectic reflection algebras. Our results provides a reduction to smaller algebras. We apply this reduction to the study of two-sided ideals and Harish-Chandra bimodules.