1549380232-Automorphic_Forms_and_Applications__Sarnak_

394 DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONS

unitary representation of G, andμ E (tc)* is an extremal weight of a representation

of K occurring in 7f. Let w be an element of the Weyl group of K with the property

thatμ - Pn is dominant for w6. + (£, tc). Finally let co denote the eigenvalu e of the

Casimir operator of G in T. Then

ffμ - Pn + WPcff

2

~Co+ ffpff

2

.

Generally one requires not only t hat μ be an extremal weight, but also that it be

dominant for 6. (£, tc). But replacing μ by the conjugate dominant weight can only

decrease the left side of the inequality we want ([VZ], Lemma 4.3); so this version

of the inequality follows.

The representation 7fL contains a unique representation 7fLnK of L n K, of

highest weight d1fL = ,\-p. (Equations (16)(c) and (d) allow us to regard A and p

as weights in W)* .) Equation (15)(b) says that this representation of L n K must

appear in Hr-^1 (u, T). A Hochschild-Serre spectral sequence ([Green], Theorem

5.2.2) then produces an irreducible representation TK of K appearing in T, and

non-negative integers x and y with the properties that 7fLnK occurs in

!\r-x(u n p) 0 HY(u n e, TK),

and that x - y = 1. We can analyze this condition as in the proof of Proposition

5.4.2 in [Green]. The conclusion is that there are

• a highest weight μ' of TK;


• x distinct weights {,8 1 , ... , .Bx} of Tc in u n p;


• an element () E Wl of length y;

with the property that

dμ' +Pc) - Pc - 2p(un P) + 2-.:.Bi = ,-p. (18)(a)

Here We^1 is Kostant's cross section for the cosets of the Weyl group Wrne (of tc in

( n £)in We, and the right side is just the differential of t he weight 7fLnK_ (It is at

this point that the hypothesis that L has compact center is used: it guarantees that

,\vanishes on ac. In the general case we would have to put >-ftc in (18)(a). In the

Dirac operator inequality ((18)(d) below) it would still be>-appearing on the right,

and we would no longer be able to draw strong conclusions from the inequality.)

We want to apply Lemma 1 7 to T and the extremal weight () μ'. For this purpose

we define an element w E We by the requirement that

w6. +(e, tc) ={a E 6.(£, tc)f (a,()μ' -Pn) < O} U {a E -6. +(e, tc)f (a, ()μ^1 - Pn) = O}.

From (18)(a) we calculate

(18)(b)

The last term in parentheses is minus the sum of the roots { aj} of tc in e that are

positive for () 6. + (£, tc) but negative for w6. + (£, tc). It is an easy consequence of the

definition of w that these roots must all belong to 6. + (£, tc). Therefore

(18)(c)

The last term in brackets is a sum of distinct positive roots of tc in g.