DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONS 381b) The sequence {7rn} is convergent in G, and Sis precisely the set of all its
limit points.Conversely, suppose { 7rn} has a limit O'o E G. Then there is a subsequence
{7rn;} with the property that 811'nJ E H(G)* converges to a non-zero element 8 E
H(G)*. In this case 8 must be a sum as in (a), and O'o must belong to S.The theorem does not directly and completely characterjze convergence in G
in terms of characters, but it provides enough information to describe the topology.
Two examples will illustrate what is happening. Consider G = SL(2, IR), and let
O'o be the trivial representation. Let 7rn be the complementary series representation
p(t) with parameter t = 1 - l/n (say for n ;::: 2). Here we think of the comple-
mentary series as parametrized by the open interval (0, 1). The characters of these
representations converge to the character of a reducible limit representation p(l).
The character of this limit representation is the sum of the character of the trivial
representation and two discrete series representations v±:ep(l) = 8ao + 8n+ + 8n-·
The set S in the theorem therefore consists of three representations, and these are
the limits of the sequence.
For a second example, we modify the sequence { 7rn} by replacing half its terms
by 0' 0. This modified sequence still converges to O'o; but the sequence of characters
no longer converges. Rather, it has two limit points: 8p(l)> and the character of
the trivial representation.
For a more complete discussion of Milicic' results, we refer to section 1 of [Tad2].
Theorem 3 Suppose O' is an irreducible unitary representation of G, and that
O' is not an isolated point in the unitary dual of G. Let {7rn} be a sequence of irre-
ducible unitary representations distinct from O' but converging to O'. Then there are
a subsequence { 7r n; }; a parabolic subgroup P = MN of G; an irreducible admissible
representation p of M; and a sequence of one-dimensional characters {Xj} of M,
with the following properties.
a) The characters {Xj} converge to the trivial character of M.
b) The induced representation Ind~ (p 0 Xj) is infinitesimally equivalent to
1fn;.
c) The representation O' is a composition factor of Ind~(p).
Under these circumstances, the characters 87rnJ. converge to the character 8 of
the admissible representati on 7r = Ind~(p). The collection of limit points of the
subsequence { 7r n; } is therefore the set of composition factors of 7r. This result is
almost certainly true exactly as formulated here for reductive groups over any local
field.
Proof. Write 3(g) for the center of the universal enveloping algebra of the Lie
algebra of G. For z E 3(g), write O'(z ) for the scalar by which z acts on the smooth
vectors of O'. By [BD], the sequence 7rn(z ) converges to O'(z) in C.
Write Pm= MmAmNm for a Langlands decomposition of a minimal parabolic
subgroup of G. By Harish-Chandra's subquotient theorem, we can find On E Mm
and Zin E II(A) '.::::'.a;,, so that 7rn is a subquotient of the principal series representa-
tion Ind~m (on 0 vn)· Now it is easy to calculate the infinitesimal characters of the
principal series representations in terms of (the highest weight of) On and Zin. From