Zuckerman Functors Research Articles

Affine category O, Koszul duality and Zuckerman functors

The parabolic category O for affine glN at level −N−e admits a structure of a categorical representation of sl˜e with respect to some endofunctors E and F. This category contains a smaller category A that categorifies the higher level Fock space. We prove that the functors E and F in the category A are Koszul dual to Zuckerman functors.The key point of the proof is to show that the functor F for the category A at level −N−e can be decomposed in terms of the components of the functor F for the category A at level −N−e−1. To prove this, we use the following fact from [9]: a category with an action of sl˜e+1 contains a (canonically defined) subcategory with an action of sl˜e.We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor.

Dg analogues of the Zuckerman functors and the dual Zuckerman functors I

We study the category of dg Harish-Chandra modules over arbitrary commutative rings, and generalize the induction functor, the production functor, the Zuckerman functor and the dual Zuckerman functor.

ON CATEGORIES OF ADMISSIBLE ( g $$ \mathfrak{g} $$ , sl(2))-MODULES

Let \( \mathfrak{g} \) be a complex finite-dimensional semisimple Lie algebra and \( \mathfrak{k} \) be any sl(2)-subalgebra of \( \mathfrak{g} \). In this paper we prove an earlier conjecture by Penkov and Zuckerman claiming that the first derived Zuckerman functor provides an equivalence between a truncation of a thick parabolic category \( \mathcal{O} \) for \( \mathfrak{g} \) and a truncation of the category of admissible (\( \mathfrak{g} \) , \( \mathfrak{k} \))-modules. This latter truncated category consists of admissible (\( \mathfrak{g} \) , \( \mathfrak{k} \))-modules with sufficiently large minimal \( \mathfrak{k} \)-type. We construct an explicit functor inverse to the Zuckerman functor in this setting. As a corollary we obtain an estimate for the global injective dimension of the inductive completion of the truncated category of admissible (\( \mathfrak{g} \) , \( \mathfrak{k} \))-modules.

Derived functor modules, dual pairs and [formula omitted]-actions

Derived functors (or Zuckerman functors) play a very important role in the study of unitary representations of real reductive groups. These functors are usually applied on highest weight modules in the so-called good range and the theory is well-understood. On the other hand, there were several studies on the irreducibility and unitarizability, in which derived functors are applied to singular modules. See Enright et al. (Acta Math. 1985), for example. In this article, we apply derived functors to certain modules arising from the formalism of local theta lifting, and investigate the irreducible constituent of resulting modules. The key technique is to understand U(g)K-actions in the setting of a see-saw pair. Our results suggest that derived functor constructions are compatible with local theta liftings.

Relating signed and classical Kazhdan–Lusztig polynomials

Motivated by studying the unitary dual problem, a variation of Kazhdan–Lusztig polynomials was defined in a previous publication by the author which encodes signature information at each level of the Jantzen filtration. These so-called signed Kazhdan–Lusztig polynomials may be used to compute the signatures of invariant Hermitian forms on irreducible highest weight modules. The key result of this paper is a simple relationship between signed Kazhdan–Lusztig polynomials and classical Kazhdan–Lusztig polynomials: signed Kahzdan–Lusztig polynomials are shown to equal classical Kazhdan–Lusztig polynomials evaluated at −q rather than q and multiplied by a sign. A simple signature character inversion formula follows from this relationship. These results have applications to finding the unitary dual for real reductive Lie groups since Harish–Chandra modules may be constructed by applying Zuckerman functors to the highest weight modules.

The category [formula omitted] for a general Coxeter system

We study the category O for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Vermaʼs result about homomorphisms between Verma modules.

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.

Signatures of Invariant Hermitian Forms on Irreducible Highest-Weight Modules

Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com- position factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig Conjecture which states that the multiplicities may be computed via Kazhdan-Lusztig polynomials. In this paper, we introduce signed Kazhdan-Lusztig polynomials, a variation of Kazhdan-Lusztig polyno- mials which encodes signature information in addition to composition factor multiplicities and Jantzen filtration level. Careful consideration of Gabber and Joseph's proof of Kazhdan and Lusztig's recursive formula for computing Kazhdan-Lusztig polynomials and an application of Jantzen's determinant for- mula lead to a recursive formula for the signed Kazhdan-Lusztig polynomials. We use these polynomials to compute the signature of an invariant Hermitian form on an irreducible highest weight module. Such a formula has applications to unitarity testing. 1.1. The Unitary Dual Problem. In the 1930s, I.M. Gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif- ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra. Fourier analysis is just one incarnation of this programme. An unresolved component in Gelfand's programme is the classification of the irre- ducible unitary representations of a group, known as the unitary dual problem. In the case of a real reductive Lie group, the problem is equivalent to identifying all irreducible Harish-Chandra modules which admit a positive definite invariant Hermitian form. As Harish-Chandra modules may be constructed via an algebraic method introduced by Zuckerman in 1978 known as cohomological induction, it is of interest to study signatures of invariant Hermitian forms on cohomologically induced modules and to understand how positivity can fail. Cohomological induction is a two-step process in which we compose an induction functor with a Zuckerman functor i . The intermediate module in cohomological induction is a generalized Verma module which admits an invariant Hermitian form if the module to which induction was applied admits an invariant Hermitian form. Formulas for the signatures of invariant Hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo- logically induced modules (eg. (Wal84)). This motivates the study of invariant Hermitian forms on Verma modules (see (Wal84), (Yee05)) and irreducible highest weight modules.

Zuckerman functors between equivariant derived categories

We review the Beilinson-Ginzburg construction of equivariant derived categories of Harish-Chandra modules, and introduce analogues of Zuckerman functors in this setting. They are given by an explicit formula, which works equally well in the case of modules with a given infinitesimal character. This is important if one wants to apply Beilinson-Bernstein localization, as is explained in [MP1]. We also show how to recover the usual Zuckerman functors from the equivariant ones by passing to cohomology.

A categorification of the Temperley-Lieb algebra and Schur quotients of $ U({\frak sl}_2) $ via projective and Zuckerman functors

Author(s): Bernstein, Joseph; Frenkel, Igor; Khovanov, Mikhail | Abstract: We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the wall functors act as generators of the Temperley-Lieb algebra while Zuckerman functors act as generators of U(sl(2)).

On Springer representations and the Zuckerman functor

On Springer representations and the Zuckerman functor

Nilpotent orbits, normality and Hamiltonian group actions

The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncompact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). This paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-homogeneous cover M of 0 is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the same nilpotent orbit may be by more than one simple group. The key result is the determination of all pairs of simple Lie groups having a shared nilpotent orbit. Furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

Nilpotent orbits, normality,\\ and Hamiltonian group actions

The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncompact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). This paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-homogeneous cover M of 0 is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the same nilpotent orbit may be by more than one simple group. The key result is the determination of all pairs of simple Lie groups having a shared nilpotent orbit. Furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

Produced Representations of Lie Algebras and Harish-Chandra Modules

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the $\mathfrak {k}$-finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.

Produced representations of Lie algebras and Harish-Chandra modules

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the k \mathfrak {k} -finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.