Abstract: We study closely the explicit determination of the
compact model of the principal series of representations of the real unimodular
group which is
thereafter shown to be equivalent to the induced picture Our main result shows that the
compact subgroup M, encountered in the
Levy-Langlands decomposition of the parabolic subgroup P,
is the largest possible subgroup of that can be used in the construction
of the unitary family In other words, we prove that M
cannot be further enlarged in K in the
methods of parabolic induction from Since minimal parabolic
subgroups (which are, by definition, the K-conjugates ofare the
chief instruments in parabolic induction, and since a parabolic subgroup is minimal if and only if ([4, p. 536]) we see that our result
is a strengthening of that of Harish-Chandra in [4].
Keywords and phrases: cocycles, parabolic induction, principal series, representations,
