Showing papers on "Representation theory published in 1999"

Geometric Models for Noncommutative Algebras

Abstract: UNIVERSAL ENVELOPING ALGEBRAS Algebraic constructions The Poincare-Birkhoff-Witt theorem POISSON GEOMETRY Poisson structures Normal forms Local Poisson geometry POISSON CATEGORY Poisson maps Hamiltonian actions DUAL PAIRS Operator algebras Dual pairs in Poisson geometry Examples of symplectic realizations GENERALIZED FUNCTIONS Group algebras Densities GROUPOIDS Groupoids Groupoid algebras Extended groupoid algebras ALGEBROIDS Lie algebroids Examples of Lie algebroids Differential geometry for Lie algebroids DEFORMATIONS OF ALGEBRAS OF FUNCTIONS Algebraic deformation theory Weyl algebras Deformation quantization.

Double Bruhat cells and total positivity

Abstract: We study intersections of opposite Bruhat cells in a semisimple complex Lie group, and associated totally nonnegative varieties.

Quasi-Hopf twistors for elliptic quantum groups

Abstract: The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al. [FIJKMY1], Felder [Fe]). Fronsdal [Fr1, Fr2] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebraU q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universalR matrix ofU q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. This construction entails that, for generic values of the deformation parameters, the representation theory forU q(g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebraA q,p ( $$\widehat{\mathfrak{s}\mathfrak{l}}_2 $$ ).

Semisimple Types in GLn

Abstract: This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.

Affine sl_p controls the representation theory of the symmetric group and related Hecke algebras

Abstract: In this paper we prove theorems that describe how the representation theory of the affine Hecke algebra of type A and of related algebras such as the group algebra of the symmetric group are controlled by integrable highest weight representations of the characteristic zero affine Lie algebra \hat{sl}_l. In particular we parameterise the representations of these algebras by the nodes of the crystal graph, and give various Hecke theoretic descriptions of the edges. As a consequence we find for each prime p a basis of the integrable representations of \hat{sl}_l which shares many of the remarkable properties, such as positivity, of the global crystal basis/canonical basis of Lusztig and Kashiwara. This {\it $p$-canonical basis} is the usual one when p = 0, and the crystal of the p-canonical basis is always the usual one. The paper is self-contained, and our techniques are elementary (no perverse sheaves or algebraic geometry is invoked).

Exchange Dynamical Quantum Groups

Abstract: For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group Uq(?). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of Uq(?), and is an algebraic structure standing behind these relations.

Whittaker functions on quantum groups and q-deformed Toda operators

Abstract: In this paper we q-deform a construction of Kazhdan and Kostant from 1970's which produces quantum Toda Hamiltonians by considering the action of Casimirs of a simple Lie algebra on Whittaker functions on the corresponding Lie group. We also give the affine analog of this generalization. This is done by extending the notion of a Whittaker function to quantum groups and quantum affine algebras. We compute the q-deformed Toda Hamiltonians for Lie algebras of type A and show that they coincide with those known in the theory of integrable systems.

General linear and functor cohomology over finite fields

Abstract: In recent years, there has been considerable success in computing Extgroups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories [F-L-S], [F-S]. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p. The reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of Ext-groups. For example, the existence of nonzero Ext-groups of positive degree is equivalent to the existence of objects which are not “direct sums” of simple objects. Indeed, a knowledge of Ext-groups provides considerable knowledge of compound objects. In the study of modular representation theory of finite Chevalley groups such as GLn(Fq), Ext-groups play an even more central role: it has been shown in [CPS] that a knowledge of certain Ext-groups is sufficient to prove Lusztig’s Conjecture concerning the dimension and characters of irreducible representations. We consider two different categories of functors, the category F(Fq) of all functors from finite dimensional Fq-vector spaces to Fq-vector spaces, where Fq is the finite field of cardinality q, and the category P(Fq) of strict polynomial functors of finite degree as defined in [F-S]. The category P(Fq) presents several advantages over the category F(Fq) from the point of view of computing Extgroups. These are the accessibility of injectives and projectives, the existence of a base change, and an even easier access to Ext-groups of tensor products. This explains the usefulness of our comparison in Theorem 3.10 of Ext-groups in the category P(Fq) with Ext-groups in the category F(Fq). Weaker forms of this theorem have been known to us since 1995 and to S. Betley independently

The projective geometry of Freudenthal's magic square

Abstract: We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial.

Classification of irreducible holonomies of torsion-free affine connections

Abstract: The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original lists. Some remarks involving an interpretation of some of the examples in terms of supersymmetric constructions are also included.

A Uniqueness theorem for constraint quantization

Abstract: This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called `rigging maps' associated with refined algebraic quantization, a particular realization of the Dirac scheme. Our main result is to provide a condition under which the rigging map is unique, in which case we also show that it is given by group-averaging techniques. Our results comprise all cases where the gauge group is a finite-dimensional Lie group.

Cellular algebras: inflations and morita equivalences

Abstract: Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra.The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper.The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

From endomorphisms to automorphisms and back: dilations and full corners

Abstract: When S is a discrete subsemigroup of a discrete group G such that G = S^{-1} S, it is possible to extend circle-valued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost-Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product corresponding to the multiplicative action of the positive rationals on the additive group of finite adeles.

Algebraic nature of shape-invariant and self-similar potentials

Abstract: Self-similar potentials generalize the concept of shape invariance which was originally introduced to explore exactly solvable potentials in quantum mechanics. In this paper it is shown that the previously introduced algebraic approach to the latter can be generalized to the former. The infinite Lie algebras introduced in this context are shown to be closely related to the q-algebras. The associated coherent states are investigated.

Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer

Abstract: Some 19th-century algebra and number theory. Frobenius and the invention of character theory. Burnside: Representations and structure of finite groups. Schur: A new beginning. Polynomial representations of GL_n(\Bbb{C}). Richard Brauer and Emmy Noether: 1926-1933. Modular representation theory. Bibliography. Index

Geometric Eisenstein series

Abstract: The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic, whose existence is non-obvious from the point of view of classical tools.

Block-compatible metaplectic cocycles

Abstract: has cardinality r ≥ 1. Let G be the F-rational points of a simple Chevalley group defined over F. In his thesis, Matsumoto [5] gave a beautiful construction for the metaplectic cover G of G, a central extension of G by μr(F) whose existence is intimately connected with the deep properties of the r-th order Hilbert symbol (·, ·)F : F × × F → μr(F). Metaplectic groups figure prominently in the study of number theory, representation theory, and physics, arising naturally in the theory of theta functions, dual pair correspondences, Weil representations, and spin geometry. In this paper we study the class of central extensions of a simple Chevalley group over an arbitrary infinite field, of which the metaplectic groups form an important subclass. Metaplectic groups were constructed quite explicitly in Weil’s memoir [10] in the case that G is symplectic. In [3] and [4], Kubota gave the construction of the r-fold metaplectic cover of GL2(F). Moreover, he described an explicit 2-cocycle σK on GL2(F) that represents the second cohomology class of the extension (cf. §3 Corollary 8), which makes it possible to deal quite rapidly with many concrete problems in this setting. Steinberg [9] and Moore [7] considered the algebraic problem of determining the central extensions of a simple Chevelley group over an arbitrary field; they were also led to the metaplectic groups. This line of investigation was completed by Matsumoto [5], whose work forms the foundation of the present paper. To summarize our results, let F be an infinite field, G the F-rational points of a simple Chevalley group defined over F, A an abelian group, and c : F × F → A a Steinberg symbol that is bilinear if G is not symplectic (cf. §1). In this paper we describe an explicit 2-cocycle σG in Z (G;A) that represents the cohomology class in H(G;A) of the central extension G of G by A constructed by Matsumoto [5]

Representations of restricted Lie algebras and families of associative ℒ-algebras

Abstract: Abstract Let ℒ be an n-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on ℒ and a scalar λ ∈ K, we introduce an associative algebra Uξ,λ (ℒ) of dimension pn over K. The algebra Uξ,1 (ℒ) is isomorphic to the reduced enveloping algebra Uξ (ℒ), while the algebra Uξ,0 (ℒ) is nothing but the reduced symmetric algebra Sξ (ℒ). Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple ℒ-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple ℒ-modules for all ℒ having a toral stabiliser of a linear function.

Noncommutative Gröbner Bases, and Projective Resolutions

Abstract: These notes consist of five sections. The aim of these notes is to provide a summary of the theory of noncommutative Grobner bases and how to apply this theory in representation theory; most notably, in constructing projective resolutions.

Homogeneous nilmanifolds attached to representations of compact Lie groups

Abstract: For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive We study some geometric aspects of these manifolds, finding many parallels with H-type groups We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

Abstract: We provide an explicit quantization of dynamical r-matrices for semisimple Lie algebras, classified earlier by the third author, which includes the Belavin-Drinfeld r-matrices. We do so by constructing an appropriate (dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group U_q(g), which twists the R-matrix of U_q(g) into the desired quatization. The construction of this twist is based on the method stemming from the work of Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on defining the twist as a unique solution of a suitable difference equation. This yields a simple closed formula for the twist. This construction allows one to confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture (about quantization of Belavin-Drinfeld r-matrices for sl(n) in the vector representation), which was stated earlier by the second author on the basis of computer evidence. It also allows one to define new quantum groups associated to semisimple Lie algebras. We expect them to have a rich structure and interesting representation theory.

Contractions of Lie algebras and separation of variables. The n -dimensional sphere

Abstract: Inonu–Wigner contractions from the rotation group O(n+1) to the Euclidean group E(n) are used to relate the separation of variables in Laplace–Beltrami operators on n-dimensional spheres and Euclidean spaces, respectively. In this article we consider all subgroup type coordinates corresponding to different chains of subgroups of O(n+1) and E(n), respectively. In particular, the contractions relate the graphical formalism of “trees” on spheres to the “clusters” on Euclidean spaces (introduced in this article). The contractions are considered analytically on several levels: the vector fields realizing the Lie algebras, the complete sets of commuting operators characterizing separable coordinate systems, the coordinate systems themselves and the separated eigenfunctions.

Characteristic cycles for the loop Grassmannian and nilpotent orbits

Abstract: α (X), which is a linear combination of closures of conormal bundles to submanifolds of X. Intuitively, the microlocal multiplicities cα() me asurethesingularity of at α. In settings related to representation theory, a group G acts on X, is G-equivariant, and the microlocal multiplicities play a significant but only partially understood role in representation theory (see (Ro), (SV), (ABV), and (KaSa), e.g.). In this paper, we compute microlocal multiplicities for certain cases of interest in representation theory. Let G be a connected reductive group with loop group LG, and let P bethesubgroup of LG consisting of loops with positive Fourier coefficients. Then P -orbits λ on theloop Grassmannian LG/P = correspond to the irreducible representations L(λ) of thedual re ductivegroup ˇ G. For dominant weights µ and λ of a torus of ˇ

Phantom Maps and Purity in Modular Representation Theory, II

Abstract: In the second part of this paper, we use the theory described in the first part to construct an example of a counterintuitive phenomenon. We show how to produce examples of filtered systems in the stable module category stmod(kG) which do not lift to filtered systems in the module category mod(kG). Our main tool is the Extended Milnor Sequence, a five-term exact sequence which specializes to the classical Milnor sequence under certain countability conditions.

Parabolic Kazhdan-Lusztig polynomials and Schubert varieties

Abstract: We shall give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by Deodhar.