A Unified Approach to the Minimal Unitary Realizations of Noncompact Groups and Supergroups

TL;DR

This paper presents a unified framework for constructing minimal unitary representations of various noncompact groups and supergroups, extending to supergroups and providing explicit realizations for many cases.

Contribution

It introduces a unified formalism for minimal unitary representations of noncompact groups and supergroups, including explicit constructions and extensions to supergroups.

Findings

Unified construction for minimal unitary representations of noncompact groups.

Explicit realizations for groups like SU(m,n), SO(m,n), Sp(2n,R).

Extension of formalism to supergroups and specific superalgebras.

Abstract

We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized light-cones, defined by a quartic norm, invariant and have maximal rank subgroups of the form H X SL(2,R) such that G/H X SL(2,R) are para-quaternionic symmetric spaces. We give a unified formulation of the minimal unitary representations of simple non-compact groups of type A_2, G_2, D_4, F_4, E_6, E_7, E_8 and Sp(2n,R). The minimal UIRs of Sp(2n,R) are simply the singleton representations and correspond to a degenerate limit of the unified construction. The minimal unitary representations of the other noncompact groups SU(m,n), SO(m,n), SO*(2n) and SL(m,R) are also given explicitly. We extend our formalism to define and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H X SL(2,R). If H is noncompact then the supergroup G does not admit any unitary representations, in general. The unified construction with H simple or Abelian leads to the minimal representations of G(3), F(4) and OSp(n|2,R) (in the degenerate limit). The minimal unitary representations of OSp(n|2,R) with the even subgroup SO(n) X SL(2,R) are the singleton representations. We also give the minimal realization of one parameter family of Lie superalgebras D(2,1;\sigma).