1549380232-Automorphic_Forms_and_Applications__Sarnak_

DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONS 389

a) The representation μLi is a lambda-lowest £ 1 n K-type; so it has multi-

plicity one, and nLi belongs to II(L 1 ) (μLi).

b) The representation μ has multiplicity one in R~i ( nLi), and is a lambda-

lowest K -type. Consequently R~i ( nLi) has a unique irreducible composi-

tion factor Jqi (nLi) in II(G)(μ).

c) The correspondence Jqi of (b) defines a bijection

Suppose that q = [ + u is another B-stable parabolic subalgebra containing q 1. Then

there is a unique irreducible representation μL of L n K with the property that

(q 1 nC, μLi) is the set of classification data attached to Land μL. Puts= dim(unt),

So= dim(u1 n [ n t). Suppose 1["L E II(L)(μL).

d) The cohomological induction functor R~i is naturally equivalent to the

composi •t e 'D"'q S o 'D" 'qinr· SQ

e) The representation μ has multiplicity one in R~ ( nL), and is a lambda-

lowest K-type. Consequently R~(nL) has a unique irreducible composition

factor Jq(nL) in II(G)(μ).

f) The correspondence Jq of (e) defines the top row in a commutative diagram

in which all maps are bijections. (The other two arrows arise from (b)
applied to (L,μL) and to (G,μ).)
The main classification theorem in [Green] includes essentially parts (a)-(c) of
this theorem. The remainder is a consequence of induction by stages (see [Green],
Proposition 6.3.6; one needs also Proposition 6.3.21 to control the spectral se-
quence). In order to apply this result in our setting, we need to know that the
functor of Theorem 7 is a special case of t he correspondence in ( e) of Theorem 11.
This is the content of the following lemma.

Lemma 12. In the setting of Theorem 7, suppose nL is an irreducible (C, LnK)-
module of infinitesimal character,\ - p(u), and μL is a lambda-lowest L n K-type
of nL. Write (q 0 ,μLi) for a set of classification data for μL (Theorem 11), with
qo = C1 + uo. Define q1 = qo + u, a B-stable parabolic subalgebra of g. Then there
is a lambda-lowest K-type μ ofR(nL) for which (q 1 ,μLi) is a set of classification
data.

Proof. Choose a maximal torus Y8 in L n K, and write H e = ye Ac for the
Cartan decomposition of its centralizer in£. This is a fundamental Cartan subgroup
of L and of G. Fix also a set of positive roots for tc in [ n t We may then speak
of the highest weights of a representation of L n K; these are characters of ye. By
abuse of notation, we also write μL E W) for the differential of a highest weight of
μL. The proof of Theorem 11 ([Green], Proposition 5.3.3) attaches to μL another
weight >.f E W). The parabolic subalgebra q 0 in the classification data may be
chosen to contain f)c, and is then defined by