1549380232-Automorphic_Forms_and_Applications__Sarnak_

DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONS 393

([Green], Corollary 6.3.4; here r =dim u n p ). Since the limit is non-zero in degree

1, there are two possibilities: either

Ext~,LnK(Hr(u,T),nL) -=I-0, (15)(a)

or

(15) (b)

(Actually we need also to know that the spectral sequence is first quadrant. This

fact becomes clear in the course of the proof of Theorem 7; we will not stop here

to give the argument.)

In case (a), write TL= Hr(u, T).>--p(u)> a non-zero representation of L. (Recall

that the subscript indicates the direct summand of the cohomology on which 3(r)

acts by the infinitesimal character >. -p( u) .) Theorem 7 says that T is isomorphic to

R.~(TL), and therefore that TL is irreducible and unitary (since Tis). The hypothesis

(15)(a) says that TL is "cohomological" (recall that nL is a unitary character). By

Theorem 8, we can find a Cl-stable parabolic q 0 = (' + u 0 c ( and so on, and a

one-dimensional unitary character TL', so that TL= n:~(TL'). Set q' = q(i + u, a

Cl-stable parabolic subalgebra in g. By induction by stages, we get

T = 'R.~('R.:~(TL

1

)) = R.~'.(TL'),

as we wished to show; the positivity condition on the weight X is also easy to verify.

(In the application to Theorem 10, case (a) corresponds to the easy case that the

lambda-norm of T exceeds that of n .)

Suppose then that (15)(b) holds. Choose re and He= r e Ac as in the proof of

Lemma 12. We may assume that the weight ,\ of Theorem 7 belongs to ([Jc)*. Now

nL is one-dimensional and has infinitesimal character,\ - p(u). It follows that the

differential of nL is

(16)(a)

Consequently the restriction of>. to [Jc n [f, r] must be equal to the half sum p(r) of

.6. + (f, [Jc). After replacing ,\ by an Ad(r) conjugate, we may assume that .6. + (r, [Jc)

is preserved by B. Because of the positivity hypothesis on ,\ in Theorem 7, the

positive system .6. + defined in Theorem 10 is

.6. + (g, [Jc) = .6. + (r, [Jc) U .6. + (u, [Jc). (16)(b)

This is preserved by B, so

Plac = p(r)lac = 0. (16)(c)

Since Lis assumed to have compact center, dnL must also vanish on ac, so (16)(a)

gives

Alac = 0.

After restriction to r e, these roots include a positive system

.6. +(£, tc) = .6. +(r n £, tc) u .6. +(u n £, tc).

(16)(d)

We may therefore speak of highest weights of representations of Kor L n K. Write
Pc for half the sum of the roots in .6. +(£, tc), and Pn = p - Pc·
These definitions allow us to formulate Parthasarathy's Dirac operator inequal-
ity. Here is the statement.

Lemma 17 ([P], (2.26); [BW], Lemma II.6.11; or [VZ], Lemma 4.2). Suppose

.6. + (g, [Jc) is a B-stable system of positive roots for the fundamental Cartan subgroup

fiC =re A c of G; define .6. + (£, tc), Pc, and Pn as above. Suppose T is an irreducible