1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 213

where r a denotes the stabilizer of the cusp a, and that a cusp a is singular whenever

X(CTa ( l i ) CT;;-^1 ) = 1, or (-l)k.

Eisenstein series Ea(z, s) are eigenfunction of 6.k with eigenvalues >-(s) = s(l - s),
6.kEa(z, s) + s(l - s)Ea(z, s) = 0.
Selberg proved that they have a meromorphic continuation (ins) to the complex
plane with no pole in the domain Wes ~ 1/2, except for a simple pole at s = 1
when k = 0 and x(q) is trivial, and that they satisfy a functional equation relating
their value at s and 1 - s. It turns out that the analytically continued Eisenstein
series {Ea(z, 1/2 +it), t E R}a realize the (continuous) spectral decomposition of
the subspace Ek(q, x) of £k(q, x) generated by the incomplete Eisenstein series,

Ea(zl'!/J) = L x(r)jcr;;-1,y(z)-k'1/;(~m(CT;;-^1 1z)),

,,Er. \r

where '1jJ ranges over the smooth compactly supported functions on R+. The or-
thogonal complement of Ek(q, x) is called the cuspidal subspace and we denote it
by Ck(q, x): it turns out that the spectrum of 6.k 1 Ck(q, x) (the cuspidal spectrum) is
discrete and has a basis composed of real analytic, square integrable eigenfunctions
of 6.k (such functions are called Maass cusp forms). It follows that any f E £k(q, x)
admits the following spectral decomposition (see [14] for the proof when k = O):

(2.5) f(z) = L:U, u1)u1(z) + L

4

~ 1 (f, Ea(*, 1 /2+it))Ea(*, 1 /2 + it)dt,

j~l a R

where { u 1 (z) } 1 ~ 1 denotes an orthonormal basis of Ck(q, x) composed of Maass cusp
forms.
We describe in more detail the structure of Ck(q, x). Fort E C, we denote by
Sk(q, x, it) the 6.k-eigenspace associated to the eigenvalue

>-=>-(it):= (1/2 + it)(l/2 - it).

From (2.4), we already know that>-~ ~(1 - ~) and this lower bound is in fact
attained, because of the following map

(2.6) F(z) f--t ykf^2 F(z),

which maps Sk(q, x) isometrically onto Sk(q, x, ~(1 - ~)). More generally, for any
k' = k(2), k' ~ k, the map

IT

I k' -1

F(z) E Sk'(q, x) f--t Ri[yk 12 F(z)] E Sk(q, x, -

2

-)

k'~l<k

l:=k(2)

defines, up to multiplication by some explicit scalar, an isometric isomorphism be-
tween both spaces.
On the other hand, if it is not of the form k' 21 for any k' = k(2), k' ~ k, then

-(it) ~ O (i.e. t E [-i/2, i/2] u R). This follows from the fact that if k = K:(2) for
K: = 0 or 1, the map

f(z) f--+ IT Rtf(z)

t<~l<k

l:=k(2)