x+iy,b = In y and exp(iA z,b) = y1 ; the other eigenfunctions are
translates of y1 by isometries. Ve then define Op(r) as the unique operator
satisfying: Op(0-) exp iX z,b = r(z,b) exp(iA z,b). Uniqueness follows
from the Fourier inversion formula on h [He]. Also, we are regarding a
function on G as a function on h x B, using the natural identification between
these spaces via the action of G ([He], [Z.6]).
A simple but important property of Op(r) is that it commutes with
translation T by an isometry if and only if 0"(g«z,g-b) = r(z,b). Hence a r €
C,(r\G) defines an operator on T\h. Unbounded functions (or, symbols) a can
also define operators, although it requires some technicalities to prove it
When a is an automorphic form of weight m, Op(r) can be described in a
very simple way: on an eigenfunction u, (e.g.), Op(r)ui = a iTu,, where IT is
the (normalized) lowering operator of sl2(R)-theory. Thus, L'u, is a form of
weight (-m) and, after multiplication by a it goes back to weight 0 [Z.3,
§le]. All the matrix coefficients appearing in this paper involve such
automorphic forms; the reader may prefer to ignore the pseudo-differential
aspect of Qp(r).
When a is orthogonal to 1, the matrix coefficient 0p(r)l,l = 0. In
that case, fp(r,T) is of exponentially lower order than fp(T):
(0.5) Theorem ([6.1]) If a ± 1 and if the eigenvalue parameter s of a
satisfies \ Re 1, then: fr(r,T) = 0(e X ) + 0(e_1/20T).
This theorem leads to a proof of the equidistribution theorem /jm — / i on
C,(r\h) (6.4-6.5). It also gives exponential error terms for a certain class
of functions a. The problem of obtaining such error terms is also considered