1549380232-Automorphic_Forms_and_Applications__Sarnak_

3. FIRST PROPERTIES OF AUTOMORPHIC FORMS 9

2.2. Remark. The notion of moderate growth (but not the exponent m) is

independent of the embedding. One can also define a canonical Hilbert-Schmidt

norm as follows: Ong, let K(x, y) =tr (adxoady) be the Killing form, and let e be

the Cartan involution of G with respect to K. Then the form (x,y) = -K(Bx,y)

is positive definite on g, and the associated Hilbert-Schmidt norm on the adjoint

group is 119112 =tr (Ade9-^1 .Ad9) (Exercise).

2.3. Relation with classical automorphic forms on the upper half plane.

Here G = SL2(1R), K = S02, and X = {z E Cl 8'z > 0 }, the action of G being

defined by

(~ ~) .z= :::~,(zEX).
Let (cz + d)m = μ(9, z). It is an automorphy factor, i.e.
(4) μ(9.9^1 ,z) = μ(9,9'.z)μ(9',z).
K is the isotropy group of i EX. Equation (4) gives fork, k' EK, and z = i
(5) μ(kk', i) = μ(k, i)μ(k', i),
i.e. k ~ μ(k, i) is a character Xm of K.
Let r be a subgroup of finite index in SL 2 (Z). An automorphic form f on X
of weight m is a function satisfying
(Al') f(r.z) = μ(r, z)f(z) (r Er, z EX)
(A2') f is holomorphic
(A3') f is regular at the cusps.
Let J be the function on G defined by
f (9) = μ(9, i)-l f(9.i).
Then (Al') for f implies (Al) and (A2) for J by a simple computation using (5).
Note in particular
(6) ](9.k) = ](9)x_m(k).
The condition (A2') implies that J is an eigenfunction of the Casimir operator C.
As C generates Z(g), this yields (A3). Consider the cusp at oo. In the inverse image
of the "Siegel set" lxl ::::; c and y > t, where c and t are positive constants, it is
easily seen that 11911 ;::::: y~, hence moderate growth means-< ym for some m. On the
other hand, r contains a translation x ~ x + p, for some non-zero integer p. Since
f is invariant under this translation in the x direction, f admits a development in
a Laurent series 2..'::n an exp(^2 11'~nz ). For f to be bounded by ym in the Siegel set, it
is necessary and sufficient that an = 0 for n < 0. This is the regularity condition
(A3'). (cf. [6], 5.14)

3. First properties of automorphic forms

In this section, G, K, X, and rare as in 1.1, and f is an automorphic form for r.

3.1. f is analytic. For this it suffices, by a regularity theorem, to show that it is
annihilated by an analytic elliptic operator.

Let g = e E17 p be the Cartan decomposition of g, where p is the orthogonal

complement of e with respect to the Killing form. The latter is negative (resp.

positive) definite on e (resp. p ). Let {xi} (resp. {yj}) be an orthonormal basis of