COADJOINT ORBITS FOR THE CENTRAL EXTENSION OF Diff^＋（S^1） AND THEIR REPRESENTATIVES

According to Kirillov′s idea, the irreducible unitary representations of a Liegroup G roughly correspond to the coadjoint orbits O. In the forward direction one ap-plies the methods of geometric quantization to produce a representation, and in the reversedirection one computes a transform of the character of a representation, to obtain a coad-joint orbit. The method of orbits in the representations of Lie groups suggests the detailedstudy of coadjoint orbits of a Lie group G in the space g* dual to the Lie algebra g of G.In this paper, two primary goals are achieved: one is to completely classify the smoothcoadjoint orbits of Virasoro group for nonzero central charge c; the other is to find repre-sentatives for coadjoint orbits. These questions have been considered previously by Segal,Kirillov, and Witten, but their results are not quite complete. To accomplish this, theauthors start by describing the coadjoint action of D-the Lie group of all orientation pre-serving diffeomorphisms on the circle S^1, and its central extension D~, then the authors willgive a complete classification of smooth coadjoint orbits. In fact, they can be parameterizedby a subspace Of conjugacy classes of PSU~(1,1). Finally, the authors will show how to findrepresentatives of coadjoint orbits by analyzing the vector fields stabilizing the orbits, anddescribe the amazing connection between the characteristic (trace) of conjugacy classes of PSU~(1, 1) and that of vector fields stabilizing orbits.     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Toward an Infinite-Component Field Theory with a Double Symmetry: Free Fields

We begin a study of possibilities of describing hadrons in terms of monolocal fields that transform under proper Lorentz group representations that are infinite direct sums of finite-dimensional irreducible representations. The additional requirement that the free-field Lagrangians be invariant under the secondary symmetry transformations generated by the polar or the axial four-vector representation of the orthochronous Lorentz group provides an effective mechanism for selecting the class of representations considered and eliminating an infinite number of arbitrary parameters allowed by the relativistic invariance of the Lagrangians.     

Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

We consider a specific class of coadjoint orbits of maximal unipotent subgroups in classical groups over a finite field; namely, orbits associated with orthogonal subsets in root systems. We derive a formula for the dimension of these orbits in terms of the Weyl group and construct polarizations for canonical forms on the orbits. As a consequence, we describe all possible dimensions of irreducible representations of such unipotent groups.     

Binary polyhedral groups and Euclidean diagrams

Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.     

Affinely Prime Dynamical Systems(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Contractions ofSU(1, n) andSU(n+1) via Berezin quantization

We establish contractions of discrete series representations ofSU(1,n) and of unitary irreducible representations ofSU(n+1) to the unitary irreducible representations of the (2n+1)-dimensional Heisenberg group by use of the Berezin calculus on the coadjoint orbits associated to these representations by the Kirillov-Kostant method of orbits.     

Quantization and the method of orbits

A survey is given of the method of orbits which makes it possible to construct irreducible unitary representations of an arbitrary Lie group proceeding from mechanical considerations. After a brief introduction to symplectic geometry, a construction of a representation associated with an orbit of a group in the dual space of its Lie algebra is given. Various generalizations of this construction are discussed.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 37–58, 1984.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Geometric parameters corresponding to representations of the classical groups with integral infinitesimal characters

In this paper we calculate the orbits of flag manifolds of the complex classical groups under the action of the sets of fixed points of Cartan involutions, and determine all the geometric parameters corresponding to representations of the classical groups with integral infinitesimal characters, which are used to discuss Arthur conjecture and the Langlands classification of the irreducible admissible representations of real classical groups(see[1])Supported by the National Natural Science Foundation of China.The second author is supported by the Post-Doctor's Foundation of China.     

Speed’s Orbit Problem: Variations on Themes of Burnside

The paper offers a solution to T. P. Speed’s Orbit Problem, describing the orbit decomposition of direct powers of a transitive finite group action. The solution is given in terms of the Burnside algebra of the group and the right action of an incidence algebra of the subgroup lattice on a module of finitary functions. As a corollary, it is shown that almost all orbits in powers of a faithful action are regular. This gives one possible ‘GF(1)’ version of Burnside’s original theorem that each irreducible complex representation of a finite group appears in a tensor power of a fixed faithful representation.     

Toward an Infinite-Component Field Theory with a Double Symmetry: Interaction of Fields

We complete the first stage of constructing a theory of fields not investigated before; these fields transform according to Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. We consider only those theories that initially have a double symmetry: relativistic invariance and the invariance under the transformations of a secondary symmetry generated by the polar or the axial four-vector representation of the orthochronous Lorentz group. The high symmetry of the theory results in an infinite degeneracy of the particle mass spectrum with respect to spin. To eliminate this degeneracy, we postulate a spontaneous secondary-symmetry breaking and then solve the problems on the existence and the structure of nontrivial interaction Lagrangians.     

Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

Young’s orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young’s orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.     

Characters of wreath products and some applications to representation theory and combinatorics

The salient point arising out of a consideration of some seemingly independent topics in representation theory, combinatorics and the theory of numerical polynomials turns out to be a result involving characters of representations of wreath products. The topics are: symmetrized inner products of representations, irreducible characters of wreath products, Frobenius' formula for the irreducible ordinary characters of symmetric groups, the Pólya-Redfield theory of enumeration under group action in combinatorics and results of Rudvalis and Snapper that certain polynomials arising from generalized cycleindices of permutation groups are numerical.     

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl–Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.     

Polar Symplectic Representations

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.