The subjects of C*-algebras and of unitary representations of locally compact groups are both approximately 50 years old. While it was known from the start that these subjects are related, it was noL originally appreciated just how close the relationship is, especially in the case of Mackey's theory of induced representations and of representations of group extensions. U G is a (locally compact...

For a real or p-adic connected reductive group G, Harish-Chandra introduced the Plancherel measure on the tempered dual Ĝtemp and founded the Plancherel formula [7, 18] relating functions on G to functions on Ĝtemp. While the Plancherel measure is equal to the formal degree for square-integrable representations, there has been no similar interpretation for tempered but nonsquare-integrable repr...

The Schwinger oscillator operator representation of SU(3) is analysed with particular reference to the problem of multiplicity of irreducible representations. It is shown that with the use of an Sp(2, R) unitary representation commuting with the SU(3) representation, the infinity of occurrences of each SU(3) irreducible representation can be handled in complete detail. A natural ‘generating rep...

The representation theory of the unitary groups is of fundamental significance in many areas of physics and chemistry. In order to label states in a physical system with unitary symmetry, it is necessary to have explicit bases for the irreducible representations. One systematic way of obtaining bases is to generalize the ladder operator approach to the representations of SU(2) by using the form...

The Langlands classiication theorem describes all admissible representations of a reductive group G in terms of the tempered representations of Levi subgroups of G. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of G in terms of certain very special unitary representations of Levi subgroups. x1. Introduction Sup...

We show that there are precisely two, up to conjugation, anti-involutions σ± of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D̂± of the Lie subalgebra of this algebra fixed by −σ±, and find the unitary ones. We realize them in terms of highest weight represen...

We study quantum chromodynamics (QCD) on a finite lattice Λ in the Hamiltonian approach. First, we present the field algebra AΛ as comprising a gluonic part, with basic building block being the crossed product C∗-algebra C(G) ⊗α G, and a fermionic (CAR-algebra) part generated by the quark fields. By classical arguments, AΛ has a unique (up to unitary equivalence) irreducible representation. Nex...

We investigate the quantum aspects of three-dimensional gravity with a positive cosmological constant. The reduced phase space of the three-dimensional de Sitter gravity is obtained as the space which consists of the Kerr-de Sitter space-times and their Virasoro deformations. A quantization of the phase space is carried out by the geometric quantization of the coadjoint orbits of the asymptotic...

Let ? be an irreducible unitary representation of a finitely generated non-abelian free group ?; suppose is weakly contained in the regular representation. In 2001 first and third authors conjectured that such must either odd or monotonous duplicitous. 2004 they introduced class multiplicative representations: this large representations obtained by looking at action ? on its Cayley graph. secon...

Abstract As a continuation of [9], this paper studies scattered representations $SO(2n+1, {{\mathbb{C}}})$, $Sp(2n, and $SO(2n, {{\mathbb{C}}})$. We describe the Zhelobenko parameters these representations, count their cardinality, determine spin-lowest $K$-types. also disprove conjecture raised in 2015 asserting that unitary dual can be obtained via parabolic induction from irreducible with no...