Harish-Chandra Bimodules Research Articles

Harish-Chandra bimodules for type A rational Cherednik algebras

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group Sn. In particular, we show that for any parameter c∈C, the category of Harish-Chandra Hc-bimodules admits a fully faithful embedding into the category Oc, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Harish–Chandra Bimodules of Finite K-type in Deligne Categories

Abstract We continue the study of Harish–Chandra bimodules in the setting of the Deligne categories $\operatorname {Rep}(G_{t})$ that we started in [17]. In this work, we construct a family of Harish–Chandra bimodules that generalize simple finite dimensional bimodules in the classical case. It turns out that they have finite $K$-type, which is a non-vacuous condition for the Harish–Chandra bimodules in $\operatorname {Rep}(G_{t})$. The full classification of (simple) finite $K$-type bimodules is yet unknown. This construction also yields some examples of central characters $\chi $ of the universal enveloping algebra $U(\mathfrak {g}_{t})$ for which the quotient $U_{\chi }$ is not simple, and, thereby, it allows us to partially solve Problem 3.23 posed in [10].

HARISH-CHANDRA BIMODULES IN THE DELIGNE CATEGORY REP(GLt)

In this paper, we study the category of Harish-Chandra bimodules HCχ,ψ in the Deligne category Rep(GLt). In particular, we answer Question 3.25 posed in Pavel Etingof’s paper [7] and determine for which central characters χ and ψ this category is not zero.

A categorical approach to dynamical quantum groups

Abstract We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.

Whittaker Modules for Classical Lie Superalgebras

We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Miličić–Soergel equivalence of a category of Whittaker modules and a category of Harish–Chandra bimodules. For classical Lie superalgebras of type I, we reduce the problem of composition factors of standard Whittaker modules to that of Verma modules in their BGG categories $${\mathcal {O}}$$ . As a consequence, the composition series of standard Whittaker modules over the general linear Lie superalgebras $$\mathfrak {gl}(m|n)$$ and the ortho-symplectic Lie superalgebras $$\mathfrak {osp}(2|2n)$$ can be computed via the Kazhdan–Lusztig combinatorics.

Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A

Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlev\'e systems.

HARISH-CHANDRA BIMODULES OVER QUANTIZED SYMPLECTIC SINGULARITIES

In this paper we classify the irreducible Harish-Chandra bimodules with full support over filtered quantizations of conical symplectic singularities under the condition that none of the slices to codimension 2 symplectic leaves has type E8. More precisely, consider the quantization 𝒜⋋ with parameter ⋋. We show that the top quotient \( \overline{\mathrm{HC}}\left(\mathcal{A}\lambda \right) \) of the category of Harish-Chandra 𝒜⋋-bimodules embeds into the category of representations of the algebraic fundamental group, Γ, of the open leaf. The image coincides with the representations of Γ/Γ⋋, where Γ⋋ is a normal subgroup of Γ that can be recovered from the quantization parameter ⋋ combinatorially. As an application of our results, we describe the Lusztig quotient group in terms of the geometry of the normalization of the orbit closure in almost all cases.

The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

AbstractWe solve two problems in representation theory for the periplectic Lie superalgebra$\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

Harish-Chandra bimodules over rational Cherednik algebras

We study Harish-Chandra bimodules over the rational Cherednik algebra Hc(W) associated to a complex reflection group W with parameter c. Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters (c,c′) for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When W is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa–Weyl group on the set of parameters, and the study of partial KZ functors.

Derived equivalences for rational Cherednik algebras

Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra for $W$ depending on a parameter c. One can consider the category O for H_c(W). We prove a conjecture of Rouquier that the categories O for H_c(W) and H_{c'}(W) are derived equivalent provided the parameters c,c' have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analog of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.

Some homological properties of category [formula omitted]. III

We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish–Chandra bimodules, but disprove it for singular blocks.

Classification of finite-dimensional irreducible modules over -algebras

AbstractFinite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (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 $U({\mathfrak{g}})$ 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 for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan–Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones–Wenzl projector, \Theta -networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev–Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.

Serre functors for Lie algebras and superalgebras

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category 𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra 𝔮(2).

Completions of symplectic reflection algebras

In this paper, we study the structure of completions of symplectic reflection algebras. Our results provide a reduction to smaller algebras. We apply this reduction to the study of two-sided ideals and Harish–Chandra bimodules.

Character D-modules via Drinfeld center of Harish-Chandra bimodules

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg (Represent. Theory 3, 1–31, 1999). Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category of Lusztig (Adv. Math. 129, 85–98, 1997) can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category (Bezrukavnikov et al. in Isr. J. Math. 170, 207–234, 2009) this allows us to derive (under a mild technical assumption) a classification of irreducible character sheaves over ℂ obtained by Lusztig by a different method.

Finite-dimensional representations of W-algebras

W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this article, we prove a conjecture of Premet that gives an almost complete classification of finite-dimensional irreducible modules for W-algebras. A key ingredient in our proof is a relationship between Harish-Chandra bimodules and bimodules over W-algebras that is also of independent interest.

Harish-Chandra bimodules for quantized Slodowy slices

The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet’s algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, to define projective functors, and to construct “noncommutative resolutions” of Slodowy slices via translation functors.

Quadratic duals, Koszul dual functors, and applications

This paper studies quadratic and duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also de. ne a Koszul duality functor for not necessarily categories. As an illustration of the techniques we reprove the duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category O in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a Koszul dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category O.

A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for $${\mathfrak{sl}_2}$$ . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra $${\mathfrak{gl}_n}$$ . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.