We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl_2 and of Heisenberg Lie algebras, Smith's generalizations of U(sl_2), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(phi,t) are in bijection with the phi-stable left ideals of R. We determine the annihilator Ann_A(w) of the cyclic generator w of V. We also describe the annihilator ideal Ann_A(V) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U(sl_2).
Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.
We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric $q$-Whittaker functions. In particular, we show that the series part of the nonsymmetric $q$-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.
Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.
In this paper, Whittaker modules for the W-algebra W(2,2) are studied. We obtain analogues to several results from the classical setting and the case of the Virasoro algebra, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.
In this paper, we study Whittaker modules for a Lie algebras of Block type. We define Whittaker modules and under some conditions, obtain a one to one correspondence between the set of isomorphic classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.
In this paper, we study Whittaker modules for graded Lie algebras. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we study some concrete algebras as examples.
Inspired by recent activities on Whittaker modules over various (Lie) algebras we describe some general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case we obtain a very general setup for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We e…
This paper builds on earlier work, where the authors described Whittaker modules for the Virasoro algebra. Using a framework of Batra and Mazorchuk, the current paper investigates a category of Virasoro algebra modules that includes Whittaker modules. Results in this paper include a classification of the simple modules in the category and a description of certain induced modules that are a natural generalization of simple Whittaker modules.
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…
The Whittaker module $M_ψ$ and its quotient Whittaker module $L_{ψ, ξ}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$ are studied. For nonsingular type, it is proved that if $ξ\neq 0$, then $L_{ψ,ξ}$ is irreducible and any irreducible Whittaker $\widehat{H}_{4}[τ]$-module of type $ψ$ with ${\bf k}$ acting as a non-zero scalar $ξ$ is isomorphic to $L_{ψ,ξ}$. Furthermore, for $ξ=0$, all Whittaker vectors of $L_{ψ, 0}$ are completely determined. For singular type, the Whittaker vectors of $L_{ψ, ξ}$ with $ξ\neq 0$ are fully characterized.
We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.
Let $\mathcal{G}$ be the planar Galilean conformal algebra and $\widetilde{\mathcal{G}}$ be its universal central extension. Then $\mathcal{G}$ (resp. $\widetilde{\mathcal{G}}$) admits a triangular decomposition: $\mathcal{G}=\mathcal{G}^{+}\oplus\mathcal{G}^{0}\oplus\mathcal{G}^{-}$ (resp. $\widetilde{\mathcal{G}}=\widetilde{\mathcal{G}}^{+}\oplus\widetilde{\mathcal{G}}^{0}\oplus\widetilde{\mathcal{G}}^{-}$). In this paper, we study universal and generic Whittaker $\mathcal{G}$-modules (resp. $\widetilde{\mathcal{G}}$-modules) of type $ϕ$, where $ϕ:\mathcal{G}^{+}=\widetilde{\mathcal{G}}^{+}\…
We construct weak (i.e. non-graded) modules over the vertex operator algebra $M(1)^+$, which is the fixed-point subalgebra of the higher rank free bosonic (Heisenberg) vertex operator algebra with respect to the $-1$ automorphism. These weak modules are constructed from Whittaker modules for the higher rank Heisenberg algebra. We prove that the modules are simple as weak modules over $M(1)^+$ and calculate their Whittaker type when regarded as modules for the Virasoro Lie algebra. Lastly, we show that any Whittaker module for the Virasoro Lie algebra occurs in this way. These results are a hig…
For a real reductive linear Lie group G, the space of Whittaker functions is the representation space induced from a non-degenerate unitary character of the Iwasawa nilpotent subgroup. Defined are the standard Whittaker (g,K)-modules, which are K-admissible submodules of the space of Whittaker functions. We first determine the structures of them when the infinitesimal characters characterizing them are generic. As an example of the integral case, we determine the composition series of the standard Whittaker (g,K)-module when G is the group U(n,1) and the infinitesimal character is regular inte…
Let $\mathcal{L}$ be the derivation Lie algebra of ${\mathbb C}[t_1^{\pm 1},t_2^{\pm 1}]$. Given a triangle decomposition $\mathcal{L} =\mathcal{L}^{+}\oplus\mathfrak{h}\oplus\mathcal{L}^{-}$, we define a nonsingular Lie algebra homomorphism $ψ:\mathcal{L}^{+}\rightarrow\mathbb{C}$ and the universal Whittaker $\mathcal{L}$-module $W_ψ$ of type $ψ$. We obtain all Whittaker vectors and submodules of $W_ψ$, and all simple Whittaker $\mathcal{L}$-modules of type $ψ$.
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 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 …
This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define a Whittaker module corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker module is simple given certain constraints on the corresponding algebra homomorphism.
We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted D-modules on the associated flag variety using Beilinson-Bernstein localization. The main result of this paper is the development of a geometric algorithm for computing the composition multiplicities of standard Whittaker modules. This algorithm establishes that these multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan-Lusztig polynomials. In the case of trivial nilpotent character, this algorithm specializes to the usual algorithm fo…
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 classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category $\mathcal{O}$ to the Miličić--Soergel category $\mathcal{N}$. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard …
Studied are the composition series of the standard Whittaker (g,K)-modules. For a generic infinitesimal character, the structures of these modules are completely understood, but if the infinitesimal character is integral, then there are not so many cases in which the structures of them are known. In this paper, as an example of the integral case, we determine the socle filtrations of the standard Whittaker (g,K)-modules when G is the group Spin(r,1) and the infinitesimal character is regular integral.
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker $\mathfrak{g}$-modules $Y(χ, η)$ introduced by Kostant. We prove that the set of all contravariant forms on $Y(χ, η)$ forms a vector space whose dimension is given by the cardinality of the Weyl group of $\mathfrak{g}$. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the deg…
Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_{1}$ (\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced and studied by Smith in \cite{S}. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained in \cite{S} and \cite{Ku}. But it seems that not much has been done on the part of nonweight modules. In this note, we generalize Kostant's results in \cite{K} on the Whittaker …
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…
We define Whittaker modules for the twisted Heisenberg-Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters.
In this paper, the property and the classification the simple Whittaker modules for the schrödinger algebra are studied. A quasi-central element plays an important role in the study of Whittaker modules of level zero. For the Whittaker modules of nonzero level, our arguments use the Casimir element of semisimple Lie algebra $sl_2$ and the description of simple modules over conformal Galilei algebras by R. Lü, V. Mazorchuk and K. Zhao.
Let $G$ be a reductive group over a non archimedean local field, and $ψ$ a non-degenerate character of the unipotent radical $U_0$ of a minimal parabolic subgroup $P_0=M_0U_0$. For $P=MU\supseteq P_0$, we show that the descent to the Jacquet module $J_P(\mathcal{W}(G,ψ))$ of Delorme's constant term map from the space $\mathcal{W}(G,ψ)$ of generalized Whittaker functions on $G$ to $\mathcal{W}(M,ψ_{|U_0\cap M})$ is the dual map of the inverse of the isomorphism of Bushnell and Henniart from $J_{P^-}(\mathcal{W}_c(G,ψ^{-1}))$ to $\mathcal{W}_c(M,ψ_{|U_0\cap M}^{-1})$ (in particular the constant …
In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra $\mathcal{D}$, including Whittaker modules, $\mathcal{U}(\mathbb{C} d_0)$-free modules, and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all $\mathcal{D}$-module structures on $\mathcal{U}(\mathbb{C} d_0)$, and find the necessary and sufficient conditions for these modules to be irreducible. At last we determine the necessary and sufficient conditions for the tensor products of Whittaker modules an…
In this paper, we construct a new class of modules for the Schrödinger algebra $\mS$, called quasi-Whittaker module. Different from \cite{[ZC]}, the quasi-Whittaker module is not induced by the Borel subalgebra of the Schrödinger algebra related with the triangular decomposition, but its Heisenberg subalgebra $\mH$. We prove that, for a simple $\mS$-module $V$, $V$ is a quasi-Whittaker module if and only if $V$ is a locally finite $\mH$-module; Furthermore, we classify the simple quasi-Whittaker modules by the elements with the action similar to the center elements in $U(\mS)$ and their quasi-…
For any positive integer $n$, let $A_n=\mathbb{C}[t_1,\dots,t_n]$, $W_n=\text{Der}(A_n)$ and $Δ_n=\text{Span}\{\frac{\partial}{\partial{t_1}},\dots,\frac{\partial}{\partial{t_n}}\}$. Then $(W_n, Δ_n)$ is a Whittaker pair. A $W_n$-module $M$ on which $Δ_n$ operates locally finite is called a Whittaker module. We show that each block $Ω_{\mathbf{a}}^{\widetilde{W}}$ of the category of $(A_n,W_n)$-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over $L_n$, where $L_n$ is the Lie subalgebra of $W_n$ consisting of vector …
Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of the q-Toda integrable integrable system (defined by means of the quantum group version of Kostant-Whittaker reduction due to Etingof and Sevostyanov). All the proofs are algebro-geometric and rely on our previous work which interprets the universal eigen-function of the q-Toda system in terms of rings of functions on the spaces of based quasi-maps from P^1 t…
In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for irreducibility of these modules.
We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra $\widehat{sl_2}$ of type $A_1^{(1)}$ with noncritical level which are also irreducible Whittaker modules over $\widetilde{sl_2} =\widehat{sl_2} + {\Bbb C} d $ with the same Whittaker function and central charge. We have to modulo a central character for ${sl_2}$ to obtain irreducible degenerate Whittaker $\widehat{sl_2} $-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker $\widehat{sl_2…
In this paper, we study intertwining operators between subregular Whittaker modules of $\gl_N$ generalizing, on the one hand, the classical exchange construction of dynamical quantum groups, on the other hand, earlier results for principal W-algebras. We explicitly construct them using the generators of W-algebras introduced by Brundan-Kleshchev. We interpret the fusion on intertwining operators in terms of categorical actions and compute the semi-classical limit of the corresponding monoidal isomorphisms which turn out to depend on dynamical-like parameters.
For any $n\in \mathbb{Z}_{\geq 2}$, let $\mathfrak{m}_n$ be the subalgebra of $\mathfrak{sp}_{2n}$ spanned by all long negative root vectors $X_{-2ε_i}$, $i=1,\dots,n$. An $\mathfrak{sp}_{2n}$-module $M$ is called a Whittaker module with respect to the Whittaker pair $(\mathfrak{sp}_{2n},\mathfrak{m}_n)$ if the action of $\mathfrak{m}_n$ on $M$ is locally finite, according to a definition of Batra and Mazorchuk. This kind of modules are more general than the classical Whittaker modules defined by Kostant. In this paper, we show that each non-singular block $\mathcal{WH}_{\mathbf{a}}^μ$ with fi…
In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this paper we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defin…
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…
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 extend the Dong-Mason theorem on the irreducibility of modules for orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)] 305-321] for the category of weak modules. Let $V$ be a vertex operator algebra, $g$ an automorphism of order $p$. Let $W$ be an irreducible weak $V$--module such that $W,W\circ g,\dots,W\circ g^{p-1}$ are inequivalent irreducible modules. We prove that $W$ is an irreducible weak $V^{\left\langle g\right\rangle }$-module. This result can be applied on irreducible modules of certain Lie algebra $\mathfrak L$ such that $W,W\circ g,\dots,W\circ g^{p-1}$ …
