We show that for $p$small highest weight $λ$, 1) there is a $\Z_p$-integral version of the Bernstein-Gelfand-Gelfand complex, still a direct summand subcomplex of the standard complex for $V(λ)$ 2) Similarly, a $\Z_p$-integral (as well as a mod. p) version of Kostant formula holds true. This paper is a companion paper to the one by Mokrane-Tilouine (AG, subm. 12/12/00), where these results are requested.
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 consider analogues of the Bernstein-Gelfand-Gelfand resolution in a highest weight category $\mathscr{P}$. We prove the resulting category of complexes is a chain-level lift of the heart of the constructible $t$-structure on its bounded derived category $D^b(\mathscr{P})$. In particular, an object admits a Bernstein-Gelfand-Gelfand resolution if and only if a certain $\operatorname{Ext}$ vanishing with costandard objects holds. These results appear to be new even for Category $\mathscr{O}$.
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.
We study blocks of category O for the Cherednik algebra having the property that every irreducible module in the block admits a BGG resolution, and as a consequence prove a character formula conjectured by Oblomkov-Yun.
Let g be a complex simple Lie algebra and h a Cartan subalgebra. The Clifford algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of the simple 2m+1-dimensional module of the principal s-triple obtained from the Langlands dual. Bazlov settled this conjecture positively in type A. The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of the Harish-Chandra map for the enveloping algebra U(g) composed with …
In this paper we construct resolutions of finite dimensional irreducible gl(m|n)-modules in terms of generalized Verma modules. The resolutions are determined by the Kostant cohomology groups and extend the strong (Lepowsky-)Bernstein-Gelfand-Gelfand resolutions to the setting of Lie superalgebras. It is known that such resolutions for finite dimensional representations of Lie superalgebras do not exist in general. Thus far they have only been discovered for gl(m|n) in case the parabolic subalgebra has reductive part equal to gl(m)+gl(n) and for tensor modules. In the current paper we prove th…
We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential cup product on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A-infinity algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of hel…
Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs $(K,N)$ where the $K$-orbits in the center of $N$ have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set $Δ(K,N)$ of bounded $K$-spherical functions on $N$ an…
Let $G$ be a $p$-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra---the endomorphism algebra of a pro-generator of the given component. Using Heiermann's construction of these algebras, we describe the Bernstein components of the Gelfand--Graev representation for $G=$SO$(2n+1)$.
It has been shown that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L^1(N)^K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)^K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L^1(N)^K which extend to Schwartz functions on Rd, once Σ is suitably embedded in Rd. We call this property (S). We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once t…
We study invariant distributions on the tangent space to a symmetric space. We prove that an invariant distribution with the property that both its support and the support of its Fourier transform are contained in the set of non-distinguished nilpotent orbits, must vanish. We deduce, using recent developments in the theory of invariant distributions on symmetric spaces that the symmetric pair (GL(2n,R),Sp(2n,R)) is a Gelfand pair. More precisely, we show that for any irreducible smooth admissible Frechet representation $(π,E)$ of GL(2n,R) the space of continuous functionals $Hom_{Sp_{2n}(R)}(E…
This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's cond…
We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resol…
Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of left-invariant and K-invariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N). Fixing a system D of d formally self-adjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilpot…
The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases o…
We prove a singular version of Beilinson-Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case done by \cite{BMR2}. We apply this theory to translation functors, singular blocks in the Bernstein-Gelfand-Gelfand category $\BGGcat$ and Whittaker modules.
Let $F(n)$ be a connected and simply connected free 2-step nilpotent lie group and $K$ be a compact subgroup of Aut($F(n)$). We say that $(K,F(n))$ is a Gelfand pair when the set of integrable $K$-invariant functions on $F(n)$ forms an abelian algebra under convolution. In this paper, we consider the case when $K=O(n)$. In [1], we know the only possible Galfand pairs for $(K,F(n))$ is $(O(n),F(n))$, $(SO(n),F(n))$. So we just consider the case $(O(n),F(n))$, the other case can be obtained in the similar way.We study the natural topology on $Δ(O(n),F(n))$ given by uniform convergence on compact…
Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\frak g$ over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand-Kirillov conjecture holds for such a W-algebra then it holds for the universal enveloping algebra $\mathrm U(\frak g)$. This together with a result of A. Premet implies that the analogue of the Gelfand-Kirillov conjecture fails for some $W$-algebras attached to some nilpotent orbits in Lie algebras of types $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.
We discuss how the Hochschild cohomology of a dg category can be computed as the trace of its Serre functor. Applying this approach to the principal block of the Bernstein--Gelfand--Gelfand category $\mathcal{O}$, we obtain its Hochschild cohomology as the compactly supported cohomology of an associated space. Equivalently, writing $\mathcal{O}$ as modules over the endomorphism algebra $A$ of a minimal projective generator, this is the Hochschild cohomology of $A$. In particular our computation gives the Euler characteristic of the Hochschild cohomology of $\mathcal{O}$ in type A.
Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector.
Let $F(n)$ be a connected and simply connected free 2-step nilpotent lie group and $K$ be a compact subgroup of Aut($F(n)$). We say that $(K,F(n))$ is a Gelfand pair when the set of integrable $K$-invariant functions on $F(n)$ forms an abelian algebra under convolution. In this paper, we consider the case when $K=O(n)$. In this case, the Gelfand space $(O(n),F(n)$ is equipped with the Godement-Plancherel measure, and the spherical transform $\land:L_{O(n)}^{2}(F(n))\rightarrow L^{2}(Δ(O(n),F(n)))$ is an isometry. I will prove the Gelfand space $Δ(O(n),F(n))$ is equipped with the Godement-Planc…
We construct explicit bases of simple modules and Bernstein-Gelfand-Gelfand (BGG) resolutions of all simple modules of the (graded) Temperley-Lieb algebra of type B over a field of characteristic zero.
Using the Penrose transform, we construct analogues of the BGG (Bernstein-Gelfand-Gelfand) resolutions in certain singular infinitesimal characters, in the holomorphic geometric setting, over the Lagrangian Grassmannian. We prove the exactness of the constructed complex over the big affine cell.
We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1 and include in particular all admissible simple highest weight modules and all simple modules induced from sl_2. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.
In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i…
We study the Iwahori-component of the Gelfand-Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro-$p$ level. Thus to begin we study the structure of a genuine pro-$p$ Iwahori-Hecke algebra, establishing Iwahori-Matsumoto and Bernstein presentations. With this structure theory we first describe the pro-$p$ part of the Gelfand-Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand-Graev representation to the metaplectic represe…
The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this paper, we define a family of categories which generalizes the BGG category, and we classify the simple modules for a subfamily. As a consequence, we show that some of the obtained categories are semisimple.
For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over $\…
We calculate the Plancherel formula for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. As a consequence we obtain a concrete description of their associated reduced group $ C^* $-algebras. The main ingredients in our proof are the Bernstein-Gelfand-Gelfand complex and the Hopf trace formula.
We associate to an arbitrary positive root $α$ of a complex semisimple finite-dimensional Lie algebra $\mfrak{g}$ a twisting endofunctor $T_α$ of the category of $\mfrak{g}$-modules. We apply this functor to generalized Verma modules in the category $\mcal{O}(\mfrak{g})$ and construct a family of $α$-Gelfand--Tsetlin modules with finite $Γ_α$-multiplicities, where $Γ_α$ is a commutative $\C$-subalgebra of the universal enveloping algebra of $\mfrak{g}$ generated by a Cartan subalgebra of $\mfrak{g}$ and by the Casimir element of the $\mfrak{sl}(2)$-subalgebra corresponding to the root $α$. Thi…
This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that $GL(m|n)$ has non-conjugate Borel subgroups and we show how Serganova's odd reflections give…
Gabriel's Theorem, and the work of Bernstein, Gelfand and Ponomarev established a connection between the theory of quiver representations and the theory of simple Lie algebras. Lie superalgebras have been studied from many perspectives, and many results about Lie algebras have analogues for Lie superalgebras. In this paper, the notion of a super-representation of a quiver is introduced, as well as the notion of reflection functors for odd roots. These ideas are then used to give a categorical construction of the root system A(n,m) by establishing a version of Gabriel's Theorem and modifying th…
I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver, and so it is considered as hopeless. We give a simple normal form of the matrices of (A,B) if the intersection of kernels of A and B is one-dimensional. We prove that this form is canonical if the Jordan matrix of A is a direct sum of Jordan blocks of the same size and the field is of zero c…
Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $ψ$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}ψ)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reduction…
Given a nilpotent Lie group $N$, a compact subgroup $K$ of automorphisms of $N$ and an irreducible unitary representation $(τ,W_τ)$ of $K$, we study conditions on $τ$ for the commutativity of the algebra of $\mathrm{End}(W_τ)$-valued integrable functions on $N$, with an additional property that generalizes the notion of $K$-invariance. A necessary condition, proved by F. Ricci and A. Samanta, is that $(K\ltimes N,K)$ must be a Gelfand pair. In this article we determine all the commutative algebras from a particular class of Gelfand pairs constructed by J. Lauret.
In this paper we construct Bernstein--Gelfand--Gelfand type resolution of simple highest weight modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p'}=3\left(1-\frac{2p'}{p}\right)$ by means of the Kazama--Suzuki coset construction. As an application, we compute the twisted Zhu algebras of the simple $\mathcal{N}=2$ vertex operator superalgebra. We also compute the Frenkel--Zhu bimodule structure associated with a certain simple highest weight module of central charge $c_{3,2}=-1$.
We construct exact sequences of invariant differential operators acting on sections of certain homogeneous vector bundles in singular infinitesimal character, over the isotropic $2$-Grassmannian. This space is equal to $G/P$, where $G$ is $\operatorname{Sp}(2n,\mathbb{C})$, and $P$ its standard parabolic subgroup having the Levi factor $\operatorname{GL}(2,\mathbb{C}) \times \operatorname{Sp}(2n-4,\mathbb{C})$. The constructed sequences are analogues of the Bernstein-Gelfand-Gelfand resolutions. We do this by considering the Penrose transform over an appropriate double fibration. The result di…
In this paper we deal with Radon transforms for generalized flag manifolds in the framework of quasi-equivariant D-modules. We shall follow the method employed by Baston-Eastwood and analyze the Radon transform using the Bernstein-Gelfand-Gelfand resolution and the Borel-Weil-Bott theorem. We shall determine the transform completely on the level of the Grothendieck groups. Moreover, we point out a vanishing criterion and give a sufficient condition in order that a D-module associated to an equivariant locally free O-module is transformed into an object of the same type. The case of maximal par…
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…
Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. Let $B$ be a minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. Let $ψ$ be a non-degenerate character of $U(F)$ and $λ$ a character of $T(F)$. Let $(K,ρ)$ be a Bushnell-Kutzko type associated to the Bernstein block of $G(F)$ determined by the pair $(T,λ)$. We study the $ρ$-isotypical component $(c\text{-ind}_{U(F)}^{G(F)}ψ)^ρ$ of the induced space $c\text{-ind}_{U(F)}^{G(F)}ψ$ of functions compactly supported mod $U(F)$. We show that $(c\text{-ind}_{U(F)}^{G(F)}ψ)^ρ$ is cyclic mod…
In this work, we consider a family of Gelfand pairs $(K \ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost all these 2-step nilpotent Lie group have square integrable representations. In this cases, following Moore-Wolf's theory, we find an explicit expression for the inversion formula of $N$, and as a consequence, we decompose the regular action of $K \ltimes N$ on $L^{2}(N)$. This result completes the analysis carried out by Wolf, where the inversion formula i…
Regularity and the Gelfand Property for Complex Symmetric Pairs
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra of type A of admissible level k. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl(2)-induced modules in these orbits. In particular, simple admissible sl(2)-induced modules are fully classified for the principal nilpotent orbit.
Let $G$ be a $p$-adic reductive group and $\mathfrak{g}$ its Lie algebra. We construct a functor from the extension closure of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ associated to $\mathfrak{g}$ into the category of locally analytic representations of $G$, thereby expanding on an earlier construction of Orlik-Strauch. A key role in this new construction is played by $p$-adic logarithms on tori. This functor is shown to be exact with image in the subcategory of admissible representations in the sense of Schneider and Teitelbaum. En route, we establish some basic results in the the…
We define a generalization of Shalika models for $GL_{n+m}(F)$ and prove that they are multiplicity-free, where $F$ is either a non-Archimedean local field or a finite field and $n,m$ are any natural numbers. In particular, we give new proof for the case of $n=m$. We also show that the Bernstein-Zelevinsky product of an irreducible representation of $GL_n(F)$ and the trivial representation of $GL_m(F)$ is multiplicity-free. We relate the two results by a conjecture about twisted parabolic induction of Gelfand pairs.
In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extension…
We study highest weight representations of shifted Yangians over an algebraically closed field of characteristic 0. In particular, we classify the finite dimensional irreducible representations and explain how to compute their Gelfand-Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. Our approach exploits the relationship between shifted Yangians and the finite W-algebras associated to nilpotent orbits in general linear Lie algebras.
Given a Lie group $G$, a compact subgroup $K$ and a representation $τ\in\hat K$, we assume that the algebra of $\text{End}(V_τ)$-valued, bi-$τ$-equivariant, integrable functions on $G$ is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of $G$-invariant differential operators on the homogeneous bundle $E_τ$ over $G/K$. In particular, we observe that, under the above assumptions, $(G,K)$ is a Gelfand pair and show that the Gelfand spectrum for the triple $(G,K,τ)$ admits homeomorphic embeddings in $\mathbb C^n$. In…
