Representations Of Simple Lie Groups Research Articles

The classification of thick representations of simple Lie groups

We characterize finite-dimensional thick representations over $\mathbf{C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets. Moreover, using this characterization, we give the classification of thick representations over $\mathbf{C}$ of connected complex simple Lie groups.

Cubature Formulas of Multivariate Polynomials Arising from Symmetric Orbit Functions

The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.

Polynomial Realisations of Lie (Super)Algebras and Bessel Operators

We study realisations of Lie (super)algebras in Weyl (super)algebras and connections with minimal representations. The main result is the construction of small realisations of Lie superalgebras, which we apply for two distinct purposes. Firstly it naturally introduces, and generalises, the Bessel operators for Jordan algebras in the study of minimal representations of simple Lie groups. These have already been applied very successfully by several authors, however an easy direct explanation for their relevance seemed still to be missing. Secondly, we work out the theoretical realisation concretely for the exceptional Lie superalgebra D(2,1;a), giving a useful hands-on realisation.

Minimal representations via Bessel operators

We construct an L^2-model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V) of simple Jordan algebras V. If $V$ is split and G is not of type A_n, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n,1)_0. A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand--Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schroedinger models in L^2-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.

Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Abstract We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start froma pair (V,Q), where V is a complex vector space and Q a homogeneous polynomial of degree 4 on V. The manifold is an orbit of a covering of Conf(V,Q), the conformal group of the pair (V,Q), in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra 𝔤, and furthermore a real form 𝔤ℝ. The connected and simply connected Lie group Gℝ with Lie(Gℝ) = 𝔤ℝ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold .

Low degree representations of simple Lie groups

We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple compact Lie group. We use this prove that the number of conjugacy classes of maximal subgroups of a compact simple Lie group is O(r) where r is the Lie rank. We also give a short proof that the dimensions of the weight spaces of a maximal torus are small relative to the dimension of an irreducible module.

Domains of holomorphy for irreducible unitary representations of simple Lie groups

We classify the domains of holomorphy of all Harish-Chandra modules of irreducible unitary representations of simple non-compact Lie groups.

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ?h and ?t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, Uh(?). In particular, the algebra ?t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr;t,0.

The place of the adjoint representation in the Kronecker square of irreducible representations of simple Lie groups

The multiplicity of occurrence of the adjoint representation in the decomposition of the square of any finite-dimensional irreducible representation of any compact simple Lie group is shown to be equal to the number of non-vanishing components of the Dynkin label of . The resolution of this multiplicity into contributions to the symmetric and antisymmetric squares of is discussed, with complete results being found for all of the classical and some of the exceptional simple Lie groups, and partial results culminating in conjectures for the remaining exceptional groups.

Modular forms and group representation

This paper considers the restriction of adjoint representations of simple Lie groups whose algebras are of even rank to finite subgroups in which each element has a rational characteristic polynomial. The nature of the virtual characters that appear is explained.

Global gauge anomaly and James numbers of Stiefel manifolds.

The global anomaly coefficient of locally anomaly-free representations of simple Lie groups is intimately connected with the James number of Stiefel manifolds via some exact short homotopic sequences. We will prove some propositions concerning this problem and apply them for calculations of global anomalies of simple Lie groups in even dimensions.

Supersymmetric exceptional gauge unification

We discuss the accommodation of quark-lepton generations, classified by SU(5), in the adjoint representations of simple Lie groups. We find SO(11), E/sub 6/, E/sub 7/, and E/sub 8/ as the only possible embedding groups, with the respective contents of one, one, three, and five conventional generations, together with their (V+A) conjugates and other particles. SU(4) supersymmetric unification models based on these gauge groups and which unify, via one coupling constant, the interactions of one vector boson, four spin-1/2 fermion, and six Higgs scalar multiplets, all being in the adjoint representation, are considered. Attention is focused on E/sub 7/ and E/sub 8/. The latter algebras are represented in the familiar SU(8) and SU(9) basis. We discuss quark-lepton assignments and propose patterns of symmetry breaking which can be triggered by the adjoint Higgs scalars, and which are compatible with the observed values of the strong and the weak couplings, as well as the weak mixing angle. Remarks are given with regard to the breaking of supersymmetry and the possible role of radiative corrections and renormalization effects in generating the gauge hierarchy.

Representations of simple Lie groups with a free module of covariants

Representations of simple Lie groups with a free module of covariants

Characters of Irreducible Representations of the Simple Groups. II. Application to Classical Groups

The general formulas found in a preceding paper for the characters of irreducible representations of simple Lie groups are developed in the case of the classical groups: the four series Al, Bl, Cl, Dl,plus the exceptional case G2. Groups of rank 1 and 2, plug A3 ≃ D3 are studied in detail.