380 DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONSif G is the group of real points of a connected reductive algebraic group defined
over R We fix a maximal compact subgroup K of G , with corresponding Cartan
involution B. By Harish-Chandra's subquotient theorem, K is a "large compact
subgroup of G" in the sense that a fixed irreducible representation T of K occurs
in irreducible unitary representations of G with multiplicity bounded by a constant
depending only on T.
Definition 1. The Hecke algebra of G is the convolution algebra H( G) of com-
pactly supported complex-valued measures μ on G having the following properties.
a) The measureμ is a smooth multiple of Haar measure on G: μ = f(g)dg
for some compactly supported smooth function f.
b) The measureμ is left and right K-finite; that is, its left and right translates
by K span a finite- dimensional space. (It is equivalent to require this of
the function f in (a).)
Suppose (n, H'Tr) is an irreducible admissible Hilbert space representation of G and
μ E H(G). Definen(μ) = fc n(g)dμ(g),
an operator on H'Tr. As a consequence of (b), the operator n(μ) is zero on all but
finitely many of the K-isotypic subspaces of H7r. In particular, it is of finite rank,
and therefore certainly of trace class. The character of n is the linear functional
87r E H ( G) * defined by
87r(μ) = tr(n(μ)).
Every continuous function f on G defines an element of H ( G) * (by sending μ to
f cf dμ). We therefore refer to elements of H( G)* as "generalized functions." This
is a slight abuse of terminology: a generalized function usually means a continuous
linear functional on the larger space of all test densities on G (in which H ( G) is
dense). Harish-Chandra showed that characters of irreducible admissible represen-
tations are actually generalized functions in this stronger sense, but we will not
make explicit use of this fact.
It is not hard to see that the map from n to 87r defines an embedding of the
unitary dual G in H ( G) *. The point of [Mil] is to describe the topology on G in
terms of this embedding. We can give H(G) the topology of weak convergence:
a sequence 81 E H ( G) converges to 8 if and only if 8 j (μ) converges to 8 (μ)
for every μ E H(G). Now points in G are closed, and the closure of any subset
of G is the set of all limits of convergent sequences in the subset. The following
theorem therefore provides the desired description of the topology on G in terms
of characters.
Theorem 2 ([Mil]). Suppose { 7r n} is a sequence of irreducible unitary repre-
sentations of G. Assume that the sequence of characters 87rn E H(G)* converges
to a non-zero element 8 E H ( G) *. Then
a) The generalized function 8 is a finite sum of characters of irreducibleunitary representations. That is, there are a non-empty finite set S C G,
and positive integers { n 17 ICT E S}, so thate = L nO'eO'.
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