In this paper the authors investigate infinite-dimensional representations $L$ in blocks of the relative (parabolic) category ${\mathcal O}_S$ for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in $L$ ``looks like'' the cohomology with coefficients in a finite-dimensional module, as in Kostant's theorem. A complete classification of these ``Kostant modules'' in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories.
Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak b=\mathfrak t\oplus\mathfrak u^+$ a fixed Borel subalgebra. Let $Δ^+$ be the set of positive roots associated with $\mathfrak u^+$ and $\mathcal K\subsetΔ^+$ the Kostant cascade. We elaborate on some constructions related to $\mathcal K$ and applications of $\mathcal K$. This includes the cascade element $x_{\mathcal K}$ in the Cartan subalgebra $\mathfrak t$ and properties of certain objects naturally associated with $\mathcal K$: an abelian ideal of $\mathfrak b$, a nilpotent $G$-orbit in $\mathfrak g$, and an involution of $\m…
We give a complete combinatorial classification of those parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated to a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant's problem is positive.
Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus \big(\oplus_{ν\in \mathcal{R}} \mathfrak{g}^ν\big)\] where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\mathfra…
To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant-Toda lattice, while the second is an integrable system defined on the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of $\mathfrak{g}$. These systems are similar in that each exploits and closely reflects the invariant theory of $\mathfrak{g}$, as developed by Chevalley, Kostant, and others. One also has Kostant's description of level sets in the Kostant-Toda lattice, which turns out to suggest deeper similarities betwee…
Let $A_n(K)$ be the Kostant form of $\mathfrak{U}(sl_n^+)$ and $Γ$ the monoid generated by the positive roots of $sl_n$. For each $λ\in Λ(n,r)$ we construct a functor $F_λ$ from the category of finitely generated $Γ$-graded $A_n(K)$-modules to the category of finite dimensional $S^+(n,r)$-modules, with the property that $F_λ$ maps (minimal) projective resolutions of the one-dimensional $A_n(K)$-module $K_{A}$ to (minimal) projective resolutions of the simple $S^+(n,r)$-module $K_λ$.
For any Hermitian Lie group $G$ of tube type we give a geometric quantization procedure of certain $K_{\mathbb{C}}$-orbits in $\mathfrak{p}_{\mathbb{C}}^*$ to obtain all scalar type highest weight representations. Here $K_{\mathbb{C}}$ is the complexification of a maximal compact subgroup $K\subseteq G$ with corresponding Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ of the Lie algebra of $G$. We explicitly realize every such representation $π$ on a Fock space consisting of square integrable holomorphic functions on its associated variety $Ass(π)\subseteq\mathfrak{p}_{\mathbb{C…
We compute projective dimension of translated simple modules in the regular block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan Kåhrström. We also derive some inequalities, in terms of Lusztig's $\mathbf{a}$-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we relate Kostant's problem with decomposability and isomorphism of translated …
Deligne's category $\underline{\rm Rep}(S_t)$ is a tensor category depending on a parameter $t$ "interpolating" the categories of representations of the symmetric groups $S_n$. We construct a family of categories $\mathcal{C}_λ$ (depending on a vector of variables $λ= (λ_1, λ_2, \ldots, λ_l)$, that may be specialised to values in the ground ring) which are module categories over $\underline{\rm Rep}(S_t)$. The categories $\mathcal{C}_λ$ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to $S_n$-mod which are …
We discuss several topics of homological algebra for the Lie superalgebra osp(1|2n). First we focus on Bott-Kostant cohomology, which yields classical results although the cohomology is not given by the kernel of the Kostant quabla operator. Based on this cohomology we can derive strong Bernstein-Gelfand-Gelfand resolutions for finite dimensional osp(1|2n)-modules. Then we state the Bott-Borel-Weil theorem which follows immediately from the Bott-Kostant cohomology by using the Peter-Weyl theorem for osp(1|2n). Finally we calculate the projective dimension of irreducible and Verma modules in th…
Given a reductive group G, Kostant and Kumar defined a nil Hecke algebra that may be viewed as a degenerate version of the double affine nil Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical subalgebra of the nil Hecke algebra with a Whittaker type quantum Hamiltonian reduction of the algebra of differential operators on G. This result has an interpretation in terms of the geometric Satake and the Langlands dual group. Specifically, the isomorphism provides a bridge between very differently looking descriptions of equivariant Borel-Moore homolog…
Modules for the Lie algebra of Hamiltonian vector fields on a torus, which admit a compatible action for the commutative algebra of multivariate Laurent polynomials are called category $\mathcal{J}$. This paper classifies the indecomposable and the irreducible modules in category $\mathcal{J}$.
Given a reductive Lie algebra over the complex numbers, we introduce a family of category which generalises the BGG category $\mathcal{O}$. We also classify the simple modules for some of these categories and prove a semisimplicity result.
The BGG category $\mathcal{O}$ and its generalization $\mathcal{O}^\mathfrak{p}$ play essential roles in representation theory and have led to far-reaching work. Some elementary problems remain open for several decades, such as the block decomposition of category $\mathcal{O}^\mathfrak{p}$. In this paper, we solve the problem of blocks by applying the theory of Jantzen coefficients.
We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture \cite[Conjecture 5.10]{CKW} for the BGG category $\mathcal{O}_{k,ζ}$ of $\mathfrak{q}(n)$-modules of "$\pm ζ$-weights", where $k\leq n$ and $ζ\in\mathbb{C} \setminus \frac{1}{2} \mathbb{Z}$. As a consequence, the irreducible characters of these $\mathfrak{q}(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $\mathfrak{q}(n)$-modules resembling polynomial and Kostant modules of the general linea…
We show that the standard graded cover of the well-known category $\mathcal O$ of Bernstein-Gelfand-Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.
We classify blocks of category $\mathcal{O}$ for rational Cherednik algebras and of cyclotomic Hecke algebras of type G(r,p,n) by using the "residue equivalence" for multi-partitions.
We classify the blocks, compute the Verma flags of tilting and projective modules in the BGG category $\mathcal O$ for the exceptional Lie superalgebra $G(3)$. The projective injective modules in $\mathcal O$ are classified. We also compute the Jordan-Hölder multiplicities of the Verma modules in $\mathcal O$.
We prove that the parabolic induction functor on BGG-category $\mathcal{O}$ associated to a complex reductive Lie algebra is gradable, that is, lifts to graded category $\mathcal{O}$ as constructed by Beilinson-Ginzburg-Soergel. Graded category $\mathcal{O}$ is equivalent to a category of stratified mixed Tate motives on a corresponding flag variety as recently defined by Soergel-Wendt. The graded version of parabolic induction is induced by a geometric parabolic induction functor we construct on the level of stratified mixed Tate motives. We also describe the effect of parabolic induction on …
This paper explores various homological regularity phenomena (in the sense of Auslander) in category $\mathcal{O}$ and its several variations and generalizations. Additionally, we address the problem of determining projective dimension of twisted and shuffled projective and tilting modules.
We give an introduction to the $\mathbb{Z}$-graded representation theory of the BGG category $\mathcal{O}$ of a complex semisimple Lie algebras, with an emphasis on Soergel's combinatorial $\mathbb{V}$ functor, definitions of $\mathbb{Z}$-graded duality functors and definitions of $\mathbb{Z}$-graded translation functors.
Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}Λ$, where $Λ$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(Λ)$, the submodule category of $Λ$, consisting of all monomorphisms in $\mathcal{M}$. We construct two functors from $\mathcal{S}(\mathcal{M})$ to ${\rm mod}\mbox{-}\underline{\mathcal{M}}$, the category of finitely presented (coherent) additive contravariant functors on the stable category of $\mathcal{M}$. We show that these functors are full, dense and objective. So they induce equivalences from the quotie…
In this paper we study the category $\mathcal{O}$ over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset $\mathcal{K}$ of weights we define a subquotient $\mathcal{O}_{[\mathcal{K}]}$ of $\mathcal{O}$. It has the property that its simple objects are parametrized by elements in $\mathcal{K}$. We then show that $\mathcal{O}_{[\mathcal{K}]}$ is equivalent to $\mathcal{O}_{[\mathcal{K}+p^lγ]}$ for any dominant weight $γ$ if $l>0$ is an integer such that $\mathcal{K}\cap (\mathcal{K}+p^lη)=\emptyset$ for all dominant weights $η$. This allows …
We study the category $\mathcal{O}$ 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 Verma's result about homomorphisms between Verma modules.
We provide a linkage principle in an arbitrary parabolic category $\mathcal O^{\mathfrak p}$ for the periplectic Lie superalgebras $\mathfrak{pe}(n)$. As an application, we classify indecomposable blocks in $\mathcal O^{\mathfrak p}$. We classify indecomposable tilting modules in $\mathcal O^{\mathfrak p}$ whose characters are controlled by the Kazhdan-Lusztig polynomials of type $\bf A$ Lie algebras. We establish the complete list of characters of indecomposable tilting modules in $\mathcal O^{\mathfrak p}$ for $\mathfrak{pe}(3)$.
We consider a category of modules that admit compatible actions of the commutative algebra of Laurent polynomials and the Lie algebra of divergence zero vector fields on a torus and have a weight decomposition with finite dimensional weight spaces. We classify indecomposable and irreducible modules in this category.
Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}Λ$, the category of finitely generated right $Λ$-modules. We associate to $\mathcal{X}$ the subcategory $\mathcal{S}_{\mathcal{X}}(Λ)$ of the morphism category $\rm{H}(Λ)$ consisting of all monomorphisms $(A\stackrel{f}\rightarrow B)$ with $A, B$ and $\rm{Cok} \ f$ in $\mathcal{\mathcal{X}}$. Since $\mathcal{S}_{\mathcal{X}}(Λ)$ is closed under extensions then it inherits naturally an exact structure from $\rm{H}(Λ)$. We will define two other different exact structures else than the canonical one on $\m…
Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for $U_q(\widehat{\mathfrak{sl}(2)})$. In this paper we show the existence of imaginary crystal basis for any object in the category $\mathcal O^q_{\text{red,im}}$.
We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $\mathcal{O}$ for the orthosymplectic Lie superalgebras $\mathfrak{osp}(3|4)$ by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.
Given a skeletally small category $\mathcal{C}$, we show that any locally finite endo-length $\mathcal{C}$-module is the direct sum of indecomposable $\mathcal{C}$-modules, whose endomorphism algebra is local.
We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism $[T]:\mathcal A\to \mathcal B$ is the equivalence class of a rigid object $T$ in the cluster category of $\mathcal A$ so that $\mathcal B$ is the right hom-ext perpendicular category of the underlying object $|T|\in \mathcal A$. Factorizations of a morphism $[T]$ are given by total orderings of the components of $T$. This is equivalent to a "signed exceptional sequence." For an a…
We establish character formulae for representations of the one-parameter family of simple Lie superalgebras $D(2|1;ζ)$. We provide a complete description of the Verma flag multiplicities of the tilting modules and the projective modules in the BGG category $\mathcal O$ of $D(2|1;ζ)$-modules of integral weights, for any complex parameter $ζ$. The composition factors of all Verma modules in $\mathcal O$ are then obtained.
We show, in full generality, that Lusztig's $\mathbf{a}$-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category $\mathcal{O}$, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category $\mathcal{O}$. Finally, …
We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.
In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
A Lie algebra is said to be generalised reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. In this paper we extend the BGG category $\mathcal{O}$ over complex semisimple Lie algebras to the category $\mathcal {O'}$ over complex generalised reductive Lie algebras. Then we make a preliminary research on the highest weight modules and the projective modules in this new category $\mathcal {O'}$, and generalize some conclusions in the classical case. As a critical difference from the complex semisimple Lie algebra case, we prove that there is no projective module…
We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ be a full subcategory of $\mathcal{C}$ containing $\mathcal{P}$. We show that certain quotient category of $\mathfrak{s}\textup{-def}(\mathcal{M})$, the category of $\mathfrak{s}$-deflations $f:M_{1}\rightarrow M_2$ with $M_1,M_2\in\mathcal{M}$, is abelian. Our main theorem has two applic…
We extend Kostant's result on annihilator ideals of non-singular simple Whittaker modules over Lie algebras to (possibly singular) simple Whittaker modules over Lie superalgebras. We describe these annihilator ideals in terms of certain primitive ideals coming from the category $\mathcal O$ for quasireductive Lie superalgebras. To determine these annihilator ideals, we develop annihilator-preserving equivalences between certain full subcategories of the Whittaker category $\tilde{\mathcal N}$ and the categories of certain projectively presentable modules in the category $\mathcal O$. These equ…
We construct a new analogue of the BGG category $\mathcal O$ for the infinite-dimensional Lie algebras $\fg=\mathfrak{sl}(\infty),\mathfrak{o}(\infty), \mathfrak{sp}(\infty)$. A main difference with the categories studied in \cite{Nam} and \cite{CP} is that all objects of our category satisfy the large annihilator condition introduced in \cite{DPS}. Despite the fact that the splitting Borel subalgebras $\fb$ of $\fg$ are not conjugate, one can eliminate the dependency on the choice of $\fb$ and introduce a universal highest weight category $\mathcal {OLA}$ of $\fg$-modules, the letters $\mathc…
Khovanov-Lauda-Rouquier algebras $R_θ$ of finite Lie type are affine quasihereditary with standard modules $Δ(π)$ labeled by Kostant partitions of $θ$. Let $Δ$ be the direct sum of all standard modules. It is known that the Yoneda algebra $\mathcal{E}_θ:=\operatorname{Ext}_{R_θ}^*(Δ, Δ)$ carries a structure of an $A_\infty$-algebra which can be used to reconstruct the category of standardly filtered $R_θ$-modules. In this paper, we explicitly describe $\mathcal{E}_θ$ in two special cases: (1) when $θ$ is a positive root in type $\mathtt{A}$, and (2) when $θ$ is of Lie type $\mathtt{A_2}$. In t…
We investigate blocks of the Category $\mathcal O$ for the Virasoro algebra over the complex numbers. We demonstrate that the blocks have Kazhdan-Lusztig theories, and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.
Let $\mathcal{A}$ be an essentially small abelian category. We prove that if $\mathcal{A}$ admits a generator $M$ with ${\rm End}_{\mathcal{A}}(M)$ right artinian, then $\mathcal{A}$ admits a projective generator. If $\mathcal{A}$ is further assumed to be Grothendieck, then this implies that $\mathcal{A}$ is equivalent to a module category. When $\mathcal{A}$ is Hom-finite over a field $k$, the existence of a generator is the same as the existence of a projective generator, and in case there is such a generator, $\mathcal{A}$ has to be equivalent to the category of finite dimensional right mod…
For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives h…
Let $\mathcal O_p$ denote the characteristic $p>0$ version of the ordinary category $\mathcal O$ for a semisimple complex Lie algebra. In this paper we give some (formal) character formulas in $\mathcal O_p$. First we concentrate on the irreducible characters. Here we give explicit formulas for how to obtain all irreducible characters from the characters of the finitely many restricted simple modules as well as the characters of a small number of infinite dimensional simple modules in $\mathcal O_p$ with specified highest weights. We next prove a strong linkage principle for Verma modules whic…
For a Frobenius abelian category $\mathcal{A}$, we show that the category ${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$ modulo projective objects is called the stable monomorphism category of $\mathcal{A}$. We show that a tilting object in the stable category $\underline{\mathcal{A}}$ of $\mathcal{A}$ modulo projective objects induces naturally a tilting object in $\underline{\rm Mon}(\mathcal{A})$. We show that if $\mathcal{A}$ is the category of (graded) modules over a (graded) self-…
The center $Z(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category $\mathcal{C}$ is said to be stable if it is stable under essential extensions. The set $\mathbf{L}^{st}(\mathcal{C})$ of stable localizing subcategories of $\mathcal{C}$ is partially ordered under reverse inclusion. We show $\mathcal{L} \mapsto Z(\mathcal{C}/\mathcal{L})$ defines a sheaf of commutative rings on $\mathbf{L}^{st}(\mathcal{C})$ with respect to finite coverings. When $\mathcal{C}$ is assumed to be lo…
