In his seminal paper [51], Atle Selberg introduced fundamental new ideas into the
classical theory of automorphic forms, a theory whose origins lie in the works of
Riemann, Klein, and Poincare. These ideas are connected with an extension of the
earlier notion of an automorphic function (or form). Instead of an analytic automorphic
function, Selberg considered a mapping which is automorphic relative to a
given finite-dimensional unitary representation of a discrete group and is an eigenfunction
for a commutative ring of elliptic differential operators. At that time Hans
Maass' article [36] had appeared, containing similar nonanalytic automorphic " wave"
functions defined in a special situation; however, it was Selberg who first took a
serious look at Maass' work. In order to implement the new ideas, certain new
techniques, not normally used in the classical theory of automorphic functions, were
invoked: first, methods from the theory of selfadjoint operators in Hilbert space;
then, methods from group representation theory over various fields, methods which
turned out to be more natural in spaces of rank greater than one. It was the
subsequent global development of Selberg's ideas in the setting of the representation
theory of Lie groups which determined the true place of the classical theory of
automorphic functions-in both its function theoretic and number theoretic aspects
-in the new more general theory, and also clarified the interaction between the old
and new theories.