1. INTRODUCTION 5

the Hilbert space

L2(ω)

=

⎧

⎨

⎩

φ : G(A) → C

φ(zγg) = ω(z)φ(g) (z ∈ Z(A),γ ∈ G(Q)),

Z(A)G(Q)\G(A)

|φ|2 ∞,

where G = GL2, Z is the center, and ω is a finite order Hecke character. Rela-

tive to the right regular representation R of G(A) on

L2(ω),

there is a spectral

decomposition

L2(ω)

= Ldisc(ω)

2

⊕ Lcont(ω).

2

The classical cusp forms correspond

to certain elements in the discrete part, while the continuous part is essentially a

direct integral of certain principal series representations H(it) of G(A). We begin

Section 6 by describing this in detail, following Gelbart and Jacquet [GJ]. For a

function f ∈

L1(ω)

attached to a classical Hecke operator, we then investigate the

kernel

(1.9) K(x, y) =

γ∈Z(Q)\G(Q)

f(x−1γy)

of the operator R(f). We assume that f∞ is bi-invariant under SO(2), compactly

supported in G(R)+, and suﬃciently differentiable. Then letting φ range through

an orthonormal basis for the subspace of vectors in H(0) of weight 0 and level N,

the main result of the section is a proof that the spectral expansion

K(x, y) = δω,1

3

π

G(A)

f(g)dg +

ϕ∈F(N)

R(f)ϕ(x)ϕ(y)

ϕ 2

+

1

4π

φ

∞

−∞

E(πit(f)φit,x)E(φit,y)dt,

is absolutely convergent and valid for all x, y. These are, respectively, the residual,

cuspidal, and continuous components of the kernel.

In Section 5, we discuss the Eisenstein series. We give an explicit description of

the finite set of Eisenstein series E(φs,g) that contribute to the above expression for

K(x, y). Their Fourier coeﬃcients involve generalized divisor sums and Dirichlet

L-values on the right edge of the critical strip, directly generalizing (1.4). We derive

bounds for these Fourier coeﬃcients, which are useful for both the convergence and

applications of the Kuznetsov formula. For this purpose we require lower bounds

for Dirichlet L-functions on the right edge of the critical strip, reviewed in Section 2.

(We note that more generally, in establishing absolute convergence of the spectral

side of Jacquet’s GL(n) relative trace formula, Lapid makes use of lower bounds for

Rankin-Selberg L-functions on the right edge of the critical strip due to Brumley,

[Lap], [Brum].)

In Section 7 we integrate each variable of K(x, y) against a character over the

unipotent group N(Q)\N(A). Using the geometric form (1.9) of the kernel, we

obtain the geometric side of the Kuznetsov formula as a sum of orbital integrals

whose finite parts evaluate to generalized twisted Kloosterman sums, defined by

Sω (m2,m1; n; c) =

dd ≡n mod c

ω

(d)e2πi(dm2+d m1)/c

(for N|c),

where ω is the Dirichlet character of modulus N attached to ω. These sums also

arise in the generalized Petersson formula of [KL1]. After an extra averaging at the

archimedean place, we obtain the J-Bessel integrals as in (1.6). Using the spectral