382 DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONS
the convergence of infinitesimal characters explained in the preceding paragraph,
we deduce first that there are only finitely many different 6n, and second that the
sequence vn is bounded in u;',,. After passing to a subsequence, we may therefore
assume that 6n = 6, and that Vn converges to v 0 E u;',,. Because each principal se-
ries representation has only finitely many composition factors, we may also assume
that Vn ::j:. vo for every n.
To go further, we need to understand the reducibility of the induced represen-
tations J(v) =Ind~,,, (6 0 v). Write ~m = ~(g, Um) C u;',, for the set of restricted
roots of Um in g. For each a E ~m, let Mo; be the reductive subgroup of G gen-
erated by MmAm and the root subgroups for multiples of a. We are interested
in the reducibility of the principal series representation Io:(v) = Ind~;nM" (6 0 v).
Because the kernel of a in Am is central in Mo:, this reducibility occurs along a
discrete set of hyperplanes parallel to the kernel of av in u;',,. Define the reducibility
set for a by
R(a) = {z E qio:(v) is reducible whenever (av,v) = z}.
(Of course this set depends on 6.) It is a discrete set of rational numbers. We will
also need
R(a)+ = R(a) u {O}.
It follows from the Langlands classification that if there is no root a E ~m with the
property that (av,v) E R(a)+, then J(v) is irreducible (see [SV], Theorem 3.14).
We need a variant of this result.
Lemma 4. In the setting above, there is for every a E ~m a discrete set
R(a)++ of rational numbers {depending on 6), with the following property. Let
P = MN be a parabolic subgroup of G with MmAm C M. Suppose that every
root a E ~m with the property that (av, v) E R(a)++ is actually a root of Um
in m. Then induction from P to G carries every irreducible composition factor of
Ind~,,,nM(6 0 v) to an irreducible representation of G.
This is a straightforward consequence of the arguments in [SV], particularly
Theorem 1.1 and section 3. Probably it suffices to take R(a)++ = R(a)+, but this
would require a more careful analysis of intertwining operators. In any case we
omit the argument.
After passing to a subsequence, we may assume that each restricted root a
satisfies exactly one of the following conditions: either
(av, vn) €f_ R(a)++for all n, or
(av, vn) = ro: E R(a)++for all n. We call such roots good and bad respectively. Let ~m(m) be the set of all
restricted roots in the rational span of the bad roots. These roots are the restricted
root system of a Levi subgroup M of G containing MmAm; choose a corresponding
parabolic subgroup P =MN of G. By the construction of M, we have
({Jv, Vn) = ({Jv, v) = r13
for every root {3 of Um in m. It follows that vn - v extends from Am uniquely to a
one-dimensional character Xn of M trivial on Mm. Since {vn} converges to v, the
characters Xn converge to the trivial character of M.