1549380232-Automorphic_Forms_and_Applications__Sarnak_

DAVID A. VOGAN, JR, ISOLATED UNITARY REPRESENTATIONS 395

Now the eigenvalue of the Casimir operator for G in 7r (and therefore a lso in

T, by the assumed existence of the extension of n by T) is

CQ = (.\A) - (p, p).

Lemma 17 and (18)(c) therefore imply that

ll(A -p) + P - [L /3i + 2pn(C) + L C¥jJll

2

~ ll>-.11^2 · (18)(d)

This inequality can be analyzed as in [VZ], proof of Lemma 4.5. One sees that the

left side is actually less than or equal to the right, with equality only if there is an

element w' E W(g, fJ) with the following properties:

• w' commutes with B;


• w' fixes ).. - p; and


• w'p = p-[l::/3i + 2pn(C) + I:aj]·
  The third condition is equivalent to
  {'y E .6. +if tt w' .6. +} = {/)i} U .6. + (C n p, tc) U { l¥j } , (19)(a)
  as well as to the two equations

w'pc =Pc - La-j, (19) (b)

We can rewrite (18)(c) as

uμ' - Pn + WPc = (>-. - P) + w' Pc + w' Pn· (19)(c)
The left side here is dominant and regular for w.6. + (e, tc), and the right side for
w' .6. + (e, tc). These two positive systems therefore coincide: WPc = w' Pc· Combining
this with the first equation in (19)(b) and the formula wpc = upc-I: aj used earlier,
we find that upc =Pc· Therefore u = 1. Now (19)(c) can be written as
μ' = (>-. - p) + Pn + w' Pn· (19)(d)
(We could also conclude for example that x = 1, so that {/3i} consists of a single
root /) 1. I do not see how to use this to simplify the rest of the argument, however.)
This is precisely the hypothesis for Proposition 5.16 of [VZ] (which is a mild gen-
eralization of Kumaresan's second main result in [Kum]). That result provides the
B-stable parabolic q' required by Lemma 13. The rest of Lemma 13 follows from
Proposition 6.1 in [VZ]. Q.E.D.
Proof of Lemma 14. We need to show how to recover q and nL (up to conju-
gation by K) from n = Rq ( nL). One way to do this is using the classification of
representations by characters of Cartan subgroups ([Green], Definition 6.6.1). We
will first describe how to extract the Cartan subgroup and character from q and nL.
Let H =TA be a maximally split Cartan subgroup of L, with L = (L n K)ANL
a corresponding Iwasawa decomposition. Write p(NL) E a for half the sum of
the restricted roots with multiplicities. Let ML be the centralizer of A in L n K,
and write p(ML) E t for half the sum of a set .6. +(mL, t) of positive roots of t
in mL. Let _6.+(C,fJ) be the "Iwasawa positive system" containing .6.+(mL,t) and
compatible with NL. The corresponding half sum of positive roots is
p(C) = (p(ML), p(NL)) Et x a. (20)(a)
After replacing the Cartan subalgebra of Theorem 10 by an Ad(C)-conjugate, we
may assume that it is the one just described. Similarly, we may assume that the