Lecture 4. Applications: control of the spectrum
In this lecture we will try to describe some results (beyond those of§ 3.1) which
verify certain consequences of Arthur's conjectures.
4.1. Bounds on the spectrum of Hecke operators [19a, 17]
The restriction principle of Burger-Sarnak (Theorem 3.10) was introduced in
order to solve the following kind of questions. Assume G is a semi-simple Q-
group and v a place (finite or infinite) of Q. Assume an irreducible representation
1fv of G(Qv) is automorphic and non-trivial. How close can 1fv be to the trivial
representation - of course in the topology of G(«i")?
If v is real, this is tantamount to looking for non-trivial lower bounds for
the eigenvalues of the Laplacian acting on L6(f\G(JR)/ K 00 ) where r c G(Q) is a
congruence subgroup and K 00 C G(JR) is a maximal compact subgroup. Here L6
denotes as usual the space of functions of zero integral.
If v is p-adic we seek upper bounds for the eigenvalues (or the L^2 -norm)
of a Hecke operator (§ 3.3) acting on L6(f\G(JR)) and relative to a compact-open
subgroup Kv C G(Qp) (again, see § 3.3). We would like this bound to be as small
as possible compared to the eigenvalue "on" the constants, i.e., the degree of the
operator.
Assume G x Qp is split, and let A c G = G(Qp) be a maximal split torus,
and K = G(Zp)· (We assume A and K compatibly chosen, see [12]). A typical
Hecke operator is given by the double coset Ka K for same a E A. Denote by p
the half-sum of the roots of (A, B) where B is a Borel subgroup. Assume a E A+
where
A+= {a EA J Ja°'J ~ 1 for any root of (A, B)}.
Write Ta= K aK. The degree of Ta is equivalent to Ja^2 PJ for a___, oo (a EA+).
This is a simple translation of the fact that the Hecke matrix of the trivial represen-
tation is equal to 'ljJ (P
112P_ 112 ) where 'ljJ : SL(2, <C) ___, G is the largest possible
representation. (Precisely, 'ljJ is for the trivial representation defined as follows: A
representation SL(2) ___, G is, by the Jakobson-Morozov theorem, uniquely defined
by a (conjugacy class of) unipotent elements of G. In G, there is a well-defined
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