We show that the Whittaker functor on a regular block of the BGG-category $\mathcal{O}$ of a semisimple complex Lie algebra can be obtained by composing a translation to the wall functor with Soergel and Miličić's equivalence between the category of Whittaker modules and a singular block of $\mathcal{O}$. We show that the Whittaker functor is a quotient functor that commutes with all projective functors and endomorphisms between them.
In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type $A_n$ to the category of finite-dimensional modules of the graded affine Hecke algebra of type $A_\ell$. Using results of Backelin and of Arakawa-Suzuki, we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category …
We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary Affine Kac-Moody algebra. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the Affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the Affine Lie algebra modules. Both fun…
In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.
We show that, for an arbitrary quasi-reductive Lie superalgebra with a triangular decomposition and a character $ζ$ of the nilpotent radical, the associated Backelin functor $Γ_ζ$ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category $\mathcal O$. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $Γ_ζ$ and its target category, respect…
Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. In this paper we study the {\em Whittaker coinvariants functor}, which is an exact functor from category $\mathcal O$ for $\mathfrak{gl}_{m|n}(\mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergel's functor $\mathbb V$ in the setting of category $\mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and …
For Iwahori-spherical representations of non-Archimedean general linear groups, Chan-Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori-Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan's formula for the Harish-Chandra--Howe character as a Grothendieck group expansion of the same module. Comparing the two approaches through the lens of Zelevinsky's PSH-algebras, we obtain an explicit unitriangular transition matrix between coefficients of the character ex…
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are t-exact on the category of tempered D-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for GL_n. We also show that Whittaker coefficient functors are t…
In the setting of global geometric Langlands, we show that miraculous duality on the stack of principal bundles on a curve intertwines the functor of Poincare series with the dual functor to Whittaker coefficients. We construct, for arbitrary parabolic subgroups, Jacquet functors controlling constant terms of Poincare series and Whittaker coefficients of Eisenstein series in local terms.
Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Let Bun_{Mp_2} be the stack of metaplectic bundles on X of rank 2. In this paper we study the derived category of genuine l-adic sheaves on Bun_{Mp_2} in the framework of the quantum geometric Langlands. We describe the corresponding Whittaker category, develop the theory of geometric Eisenstein series and calculate the most non-degenerate Fourier coefficients of these Eisenstein series. The existing constructions of automorphic sheaves for GL_n are based on using Whittaker sheaves. Our calculations lea…
We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker-Fourier coefficients of au…
Let $G$ be a connected reductive group, with connected center, and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $\operatorname{Bun}_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a "Fourier transform" functor, called $\mathsf{coeff}_{G,\mathsf{ext}}$, from the DG category of $\mathfrak{D}$-modules on $\operatorname{Bun}_G$ to a certain DG category $\mathcal{W}h(G,\mathsf{ext})$, called the \emph{extended Whittaker category}. Combined wit…
We show that every simple transitive $2$-representation of the $2$-category of projective functors for a certain quotient of the quadratic dual of the preprojective algebra associated with a tree is equivalent to a cell $2$-representation.
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…
In the paper, we mainly connect the Gorenstein derived equivalence and stable functors of Gorenstein projective modules. Specially, we prove that a Gorenstein derived equivalence between CM-finite algebras A and B can induce a stable functor between the factor categories A-mod/A-Gproj and B-mod\B-Gproj. Furthermore, the above stable functor is an equivalence when A and B are Gorenstein.
We classify projective functors on the regular block of Rocha-Caridi's parabolic version of the BGG category $\mathcal{O}$ in type $A$. In fact, we show that, in type $A$, the restriction of an indecomposable projective functor from $\mathcal{O}$ to the parabolic category is either indecomposable or zero. As a consequence, we obtain that projective functors on the parabolic category $\mathcal{O}$ in type $A$ are completely determined, up to isomorphism, by the linear transformations they induce on the level of the Grothendieck group, which was conjectured by Stroppel in \cite{St}.
We describe Serre functors for (generalisations of) the category O associated with a semi-simple complex Lie algebra. In our approach, projective-injective modules play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser property with respect to a symmetric algebra. As an application of the double centraliser property and our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O for sl_n.
We show that any simple transitive $2$-representation of the $2$-ca\-te\-go\-ry of projective endofunctors for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]&\bullet})$ and for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]\ar@/^/@{.}[rr]&\bullet\ar[r]&\bullet})$ is equivalent to a cell $2$-representation.
We develop some aspects of the theory of $D$-modules on ind-schemes of pro-finite type. These notions are used to define $D$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $χ: N(\!(t)\!) \to \mathbb{G}_a$ and a category $\mathcal{C}$ acted upon by $N(\!(t)\!) $, we define the category $\mathcal{C}^{N(\!(t)\!), χ}$ of $(N(\!(t)\!), χ)$-invariant objects, along with the coinvariant category $\mathcal{C}_{N(\!(t)\!), χ}$. These are the Whittaker catego…
In this paper we study simple transitive $2$-representations of certain $2$-subcategories of the $2$-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type $A_2$, simple transitive $2$-representations are classified by cell $2$-representations. In the general case we conjecture that there exist many simple transitive $2$-representations which are not cell $2$-representations and provide some evidence for our conjecture.
The purpose of this paper is to study a categorification of the $n$-th tensor power of the spin representation of $U(\mf{so}(7,\C))$ by using certain singular blocks and projective functors of the BGG category of the complex Lie algebra $\mf{gl}_n$.
We study the projective dimensions of the restriction of functors Hom(-,X) to a contravariantly finite rigid subcategory T of a triangulated category C. We show that the projective dimension of Hom(-,X)|T is at most one if and only if there are no non-zero morphisms between objects in T[1] factoring through X, when the object X belongs to a suitable subcategory of C. As a consequence, we obtain a characterisation of the objects of infinite projective dimension in the category of finitely presented contravariant functors on a cluster-tilting subcategory of C.
We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic $p$ that cohomological Mackey functors are Gorenstein if and only if Sylow $p$-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedeki…
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…
A correspondence functor is a functor from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring. We determine exactly which simple correspondence functors are projective. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor $F$ associated with a finite lattice and we deduce a direct sum decomposition of $F$.
It is important for applications of Homological Algebra in Representation Theory to have control over the behaviour of (minimal) projective resolutions under various functors. In this article we describe three broad families of functors that preserve such resolutions. We will use these results in our work on Representation Theory of Schur algebras.
We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gor…
From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, …
We study projective functors (i.e. direct summands of compositions of translations through walls) for parabolic versions of $\cO$ as well as for integral regular blocks outside the critical hyperplanes in the symmetrizable Kac-Moody case. It turns out that in both situations the functors are completely determined by their restriction to the additive category generated by (the limit of) a `full projective tilting' object. We describe how projective functors in the parabolic setup give rise to an invariant of tangle cobordisms and formulate a conjectural direct connection to Khovanov homology. O…
In this paper, we describe the induction functor from the category of native Mackey functors to the category of biset functors for a finite group $G$ over an algebraically closed field $k$ of characteristic zero. We prove two applications of this description. As the first application, we exhibit that any projective biset functor over $k$ is induced from a (rational) virtual native Mackey functor. The second application is the explicit description of the projective indecomposable biset functors parameterized by simple groups.
We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ 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 $\mathcal{O}$ 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 $\mathcal{O}$ for c…
We study the algebra of regular functions on the big cell of the Gauss decomposition of a simple complex Lie group G. We prove that it is spanned by the matrix elements of big projective modules in the BGG category O, and admits a decomposition of Peter-Weyl type. The subspace of Whittaker vectors in the regular representation on the big cell gives a realization of the big projective modules in O, similar to the Borel-Weil theorem. These results can be extended to the quantum groups.
The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group which admits a tensor product coming from its Hopf algebra structure. It is classical that there exists a functor F from the category of strict polynomial functors to the category of representations of the symmetric group. Our main result is that this functor F is monoidal. In addition we study the relations under F between projective strict polynomial functors and permutati…
Using the projective oscillator representation of sl(n+1) and Shen's mixed product for Witt algebras, Zhao and the second author (2011) constructed a new functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and use the functor successively to obtain a full projective oscillator realization of any finite-dimensional irreducible representation of sl(n+1). The representation formulas of all the root vectors of sl(n+1) are given in terms of first-order differential operators in n(n+1)/2 variables. One can use the result to study tensor decompositions of finite-dimensional irre…
This paper is the first in a series of papers in which we define and study a category of "sheaves of $\mathcal Z$-modules on the set of alcoves" that carries important information on the category of representations of semisimple Lie algebras in positive characteristics. Here, we define this category and study the structure of its projective objects using a new version of wall crossing functors.
In this paper, we construct recollements and ladders for exceptional curves by using reduction/insertion functors due to $p$-cycle construction. As applications to weighted projective lines, we classify recollements for the category of coherent sheaves over a weighted projective line, and give an explicit description of ladders in two different levels: the bounded derived category of coherent sheaves and the stable category of vector bundles.
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, …
In the previous article "Hearts of twin cotorsion pairs on exact categories", we introduced the notion of the heart for any cotorsion pair on an exact category with enough projectives and injectives, and showed that it is an abelian category. In this paper, we construct a half exact functor from the exact category to the heart. This is analog of the construction of Abe and Nakaoka for triangulated categories. We will also use this half exact functor to find out a sufficient condition when two different hearts are equivalent.
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.
We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley--Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and…
We prove a localization theorem for affine $W$-algebras in the spirit of Beilinson--Bernstein and Kashiwara--Tanisaki. More precisely, for any non-critical regular weight $λ$, we identify $λ$-monodromic Whittaker $D$-modules on the enhanced affine flag variety with a full subcategory of Category $\mathscr{O}$ for the $W$-algebra. To identify the essential image of our functor, we provide a new realization of Category $\mathscr{O}$ for affine $W$-algebras using Iwahori--Whittaker modules for the corresponding Kac--Moody algebra. Using these methods, we also obtain a new proof of Arakawa's chara…
We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expec…
Let $\rm{coh}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\rm{coh}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\rm{coh}\mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver $Q$ associated with $\mathbb{X}$. By further dealing with the Ringel--Hall algebra of $\mathbb{X}$, we show that these functors provide a re…
Given an abelian category A with enough projectives, we can form its stable category _A_ := A/Proj(A)$. The Heller operator Omega : _A_ -> _A_ is characterised on an object X by a choice of a short exact sequence Omega X -> P -> X in A with P projective. If A is Frobenius, then Omega is an equivalence, hence has a left and a right adjoint. If A is hereditary, then Omega is zero, hence has a left and a right adjoint. In general, Omega is neither an equivalence nor zero. In the examples we have calculated via Magma, it has a left adjoint, but in general not a right adjoint. If A has projective c…
We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.
Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent b…
