We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach–Lie groups and some monothetic subgroups of the unitary group of a separable Hilbert space.

The infinite-dimensional Clifford algebra has a maze of inequivalent irreducible unitary representations. Here we determine their type -real, complex or quaternionic. Some, related to the Fermi-Fock representations, do not admit any real or quaternionic structures. But there are many on L of the circle that do and which seem to have analytic meaning. Table of contents.

Let G = Aut(T ) be the group of automorphisms of a homogeneous tree T and let π be the tensor product of two spherical irreducible unitary representations of G. We complete the explicit decomposition of π commenced in part I of this paper, by describing the discrete series representations of G which appear as subrepresentations of π.

In this report for the course “Lie algebras and quantum groups” at KTH I discuss the origin of the Virasoro algebra, give the physical motivation for studying its unitary irreducible highest weight representations, and examine the necessary and sufficient conditions for such representations to exist.

Multidimensional contractions of irreducible representations of the Cayley-Klein unitary algebras in the Gel’fand-Zetlin basis are considered. Contracted over different parameters, algebras can turn out to be isomorphic. In this case method of transitions describes the same reducible representations in different basises, say, discrete and continuous ones.

We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteris...

The basic formulas for the unitary principal series representations of group SL(2, ℂ) are given, and decomposition a tensor product two into irreducible is considered. A simple proof completeness 3j-symbols given.

Let G be a unitary, symplectic or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.