!AS/Park City Mathematics Series
Volume 12, 2002Spectral Theory of Automorphic Forms
L. Clozel
Foreword
Assume G is a semi-simple Lie group, r a congruence subgroup of G (so discrete,
with finite co-volume), and consider the space of (L^2 ) automorphic forms
Ac= L^2 (r\C).
On this space, we have naturally defined C-invariant differential operators - or
rather, their closure; we have a representation of G by right translation; and we
have Hecke operators, of arithmetic significance.
I will try to describe the conjectural properties of the spectra of these operators,
and what is known about them in particular cases, as well as general properties
which are now known for essentially all groups.
The lectures center around two themes:
(1) Very deep conjectures of J. Arthur give us an a priori understanding of
what the spectrum should be. Although they were formulated more than 20 years
ago these conjectures are not widely known, and difficult of access to many since
they are couched in Langlands's theory of the dual group. This cannot be totally
avoided, but I have tried to state the conjectures (for Hecke operators) using the
elements of the dual group contained in Shahidi's lectures and easy representation
theory.
(2) These conjectures are a very sure guide to the spectral properties, but
they (or their spectral consequences) are totally out of reach at the present time.
For instance they include the Ramanujan conjecture for cusp forms on GL(n) - see
Lecture 3. I have tried to describe proven approximations of the conjectures.
This includes approximations to the Ramanujan conjecture (for the simplest
group, GL(n)); for other groups, upper bounds (for Hecke operators) or lower
bounds (for the Laplacian) acting on the space A~ of functions of zero mean.
In dealing with general groups, a very useful method has been introduced by
Burger, Li and Sarnak in order to relate spectral estimates for a group G and
(^1) Universite de Paris-Sud, 91405 ORSAY CEDEX, FRANCE.
E-mail address: laurent. clozel©math. u-psud. fr.
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@ 2007 Laurent Clozel