84 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMSconsider the action of we on L~(r\G(JE.)/ K 00 ). Then the spectrum ofwc is bounded
below by Ca.In general, the choice of H will give a lower bound for Cc (see [15c]). On the
other hand, Theorem 3.9 implies that certain representations of G occur and gives
an upper bound.
4.3 ... and beyond
This last §describes common work with N. Bergeron.
From the previous §we know that: (i) the T conjecture is true (for any prime v),
and (ii) for v = oo, it amounts to the existence of a uniform lower bound (for G/Q
fixed) on non-zero eigenvalues of the Laplacian acting on r \ X = r \G/K 00 - G =
G(JE.), ran arbitrary congruence subgroup relative to G/Q.
The Laplacian here is we, acting on functions, so on L^2 (f\X). Now fix i E
[O, dimX] and consider the space of i-forms A~(f\X). Note that:
(a) i-forms are still naturally functions on r \ G - transforming on the right
under K 00 according to the action of K 00 on the i-th exterior power Ai(fl/e 00 ),
fl= Lie(ec), e = Lie(K 00 ). So we are still essentially in C^00 (f\G), although not in
C^00 (I'\X).
(b) We still have a natural Laplacian wi on Ai, given by Hodge theory:wi = dd + dd, the adjoint being given by the invariant metric on I'\X.
It is then natural to ask:
Question 4.6. Are there uniform bounds (for varying I') on the eigenvalues ,\ =f. 0
of wi?
For simplicity we suppose that G is anisotropic, i.e., I'\G is compact. For
geometric reasons a natural, weaker question is the following. We can consider
A~=o' the space of forms such that dcx = 0. (Consider C^00 -forms, or L^2 -forms at
the cost of a little operator theory).
Question 4. 7. Are there uniform bounds on the eigenvalues,\ =f. 0 of wi on A~=O?Before stating a positive result, a few remarks. The kernel of wi on Ai is the
space JHli(r / X) of harmonic forms - finite-dimensional, isomorphic to Hi(I'\X, q.
By Matsushima theory, this is described as follows. Assume (7r, 1i) is an irreducible
unitary representation of G, and assume
(i) 7r c L^2 (f\G)
(ii) HomK 00 (Ai(fl/ e 00 ) ,7r) =/. 0
(iii) The Casimir operator of fl acts trivially on 7r (or 1i).
For the meaning of (iii) we must refer the reader to Borel-Wallach [9, Ch. 2].
Then JHli(7r) = HomK 00 (Ai,7r) injects into JHli(I'\X), and JHli (I'\X) is the sum
of these contributions over all 7r verifying (i)-(iii) -7r being orthogonal summands
in L^2.
By the way, for arbitrary G, unitary representations 7r verifying (ii)-(iii) are
classified by the work of Kumaresan and Vogan-Zuckerman [58].