The unitary representations of the generalized Lorentz groups

Abstract:

For $n \geqslant 2$, let $G(n)$ denote the two-fold covering group of ${\text {SO} _e}(1,n)$. In case $n \geqslant 3,G(n)$ is isomorphic to $\operatorname {Spin} (1,n)$ and is simply connected. In a previous paper we determined all the irreducible quasi-simple representations of these groups, up to infinitesimal equivalence. The main purpose of the present paper is to determine which of these representations are unitarizable. Thus, with the aid of some results of Harish-Chandra and Nelson we determine all the irreducible unitary representations of $G(n)$, up to unitary equivalence. One by-product of our analysis is the explicit construction of the infinitesimal equivalences, which are known to exist from our previous work, between the various subquotient representations and certain subrepresentations in the nonirreducible cases of the nonunitary principal series representations of $G(n)$.