LECTURE 4. APPLICATIONS: CONTROL OF THE SPECTRUM 85Now assume 7r E G occurs in r\G, and 7r verifies (i)- (iii). Assume moreover
that 7r can be deformed in G -that there are arbitrarily close representations 7r^1
in G. For close representations (ii) remains true (an exercise using the notions in
§ 3.2), while the corresponding eigenvalue .A of wi will be close to 0. It will not b e
0 because the set of representations verifying (ii)-(iii) is finite.
So if:
(a) 7r can be deformed; AND(b) Arbitrarily close representations 7r^1 occur in L^2 (r'\G) for some r' c r ,
Question 4 .6 h as a negative answer.
Example 4.8. G = SL(n)/Q, n 2 3. Then G(JR) = G has tempered representa-
t ions verifying (i)-(iii) if i E [q -e, q + e] where q = n^2 + 4 n -^2 , e = ~ ( n - 1 - rn]) > 0.
These can be deformed. Moreover (b) is true by the Corollary to Theorem 3.9.
Thus Question 4.6 will h ave a negative answer in this range. (We co uld replace
SL(n)/Q by a «Ji-anisotropic group G with G(JR) = SL(n, JR).)
We do not know what groups Goffer a n interesting ra nge of degrees i where
Question 4.6 might h ave a positive answer.^4 We are sure, however , that SO(n, 1)
and SU(n, 1) are natural candidates.^5 In fact, we would like to b elieve t hat (the
real group G b eing as follows, and G/Q arbit rary) :
Conjecture 10. -
(a) (4 .6) is true for SU(n, 1) and any i
(b) (4.6) is true for SO(n, 1) (n even), any i, and if n is odd for i "I-n;-^1
(4.7) is true fo r SO(n, 1) (n odd), i:::; n;-^1.
The Burger-Sarnak method of § 3.3 can be extended to prove the following
theorem: we co nsider a true unitary group defined by a totally real number field
F , a CM, quadratic extension E of F , and a n E / F-Hermitian form H over En.
This defines a group G 1 = SU(H) over F and by restriction of scalars a group
G/Q. We assume that G(JR) = SU(n, 1) x SU(n)d-l where d = [F : Q]. Let
X = G(JR)/K 00 ~ IIBn, the unit ball in en. Then:
Theorem 4.9 ([6]). - Under these assumptions, any eigenvalue .A "I- 0 of w1 in
Al(r\X) verifies
.A> lOn - 11- 25
The proof is a non-trivial extension of the methods of § 3.3 to 1-forms. The
big group is G, with G(JR.) (essentially) equal to SU(n, 1). The small group is H
wit h H(JR) (essentially ) equal to SU(2, 1). To bound .A in Thm. 4.9 we show that
it suffices to bound eige nvalues relative to 0-forms and 1-forms on H. Then the
result is due to Harris and Li [25a]; it is important to note t hat t heir proof again
relies on the Luo-Rudnick-Sarnak estimates (see§ 3.1). Note t hat eigenvalues in AO
verify the bound .A ;:::: 2n - 1.
Finally, why is this interesting? The work of Kudla, Millson and others, ex-
tended by Bergeron in his thesis, shows that such a "spectral gap" for i-forms has
immediate consequences for the homology of r\x: it implies that totally geodesic
(^4) Now we do, after reading Vogan's paper included in this volume!
(^5) T he efficient way to see this is , again, to use t he Arthur conjectures (§ 2 .4) at the real prime.
This is too complicated to be d escribed here.