Dirichlet series and automorphic forms on unitary groups
Authors: Orloff Tobias 
Published: June 2010

In a special case our unitary group takes the form
\[

G
=
{
g
∈

GL

(
p
+
2
,

C

)

|

t

g
¯

R
g
=
R
}
.

G = \{ g \in {\text {GL}}(p + 2,{\mathbf {C}}){|^t}\bar gRg = R\} .

\]
Here
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals Start 3 By 3 Matrix 1st Row 1st Column upper S 2nd Column a m p semicolon 0 3rd Column a m p semicolon 0 2nd Row 1st Column 0 2nd Column a m p semicolon 0 3rd Column a m p semicolon 1 3rd Row 1st Column 0 2nd Column a m p semicolon negative 1 3rd Column a m p semicolon 0 EndMatrix">

R
=

(

S

a
m
p
;
0

a
m
p
;
0

0

a
m
p
;
0

a
m
p
;
1

0

a
m
p
;

−
1

a
m
p
;
0

)

R = \left ( {\begin {array}{*{20}{c}} S & 0 & 0 \\ 0 & 0 & 1 \\ 0 & { - 1} & 0 \\ \end {array} } \right )

\]
is a skew-Hermitian matrix with entries in an imaginary quadratic number field

K
K

. We suppose that

−
i
R

- iR

has signature

(
p
+
1
,
1
)

(p + 1,1)

. This group acts naturally on the symmetric domain
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D equals StartSet w element-of bold upper C Superscript p Baseline comma z element-of bold upper C vertical-bar upper I m left-parenthesis z right-parenthesis greater-than minus one half Superscript t Baseline w overbar upper S w EndSet period">

D
=

{

w
∈

C

p

,
z
∈

C

|

Im
⁡
(
z
)
>
−

1
2

t

w
¯

S
w

}

.

D = \left \{ {w \in {{\mathbf {C}}^p},z \in {\mathbf {C}}|\operatorname {Im} (z) > - {{\frac {1}{2}}^t}\bar wSw} \right \}.

\]
If

Γ
=
G
∩

SL

(
p
+
2
,

O

K

)

\Gamma = G \cap {\text {SL}}(p + 2,{\mathcal {O}_K})

with

O

K

{\mathcal {O}_K}

the ring of integers in

K
K

, then an automorphic form

f
(
w
,
z
)

f(w,z)

with respect to

Γ
\Gamma

has an expansion

Σ
r

g
r

(
w
)
⋅

e

2
π
i
r
z

{\Sigma _r}{g_r}(w) \cdot {e^{2\pi irz}}

. The functions

g
r

(
w
)

{g_r}(w)

are theta functions. Given another automorphic form

g
(
w
,
z
)

g(w,z)

with an expansion

Σ
s

h
s

(
w
)
⋅

e

2
π
i
s
z

{\Sigma _s}{h_s}(w) \cdot {e^{2\pi isz}}

we define a Dirichlet series

Σ
r

⟨

g
r

,

h
r

⟩

r

−
s

{\Sigma _r}\langle {g_r},{h_r}\rangle {r^{ - s}}

. Here

⟨

g
r

,

h
r

⟩

\langle {g_r},{h_r}\rangle

is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper P Subscript normal upper Gamma Baseline minus upper D Endscripts f g overbar dot left-parenthesis upper I m left-parenthesis z right-parenthesis plus one half i Superscript t Baseline w overbar upper S w right-parenthesis Superscript s d w d w overbar d z d z overbar">

∫

P
Γ

∖
D

f

g
¯

⋅

(

Im
⁡
(
z
)
+

1
2

i
t

w
¯

S
w

)

s

d
w

d

w
¯

d
z

d

z
¯

{\int _{{P_\Gamma }\backslash D}}f\bar g\cdot {\left ( {\operatorname {Im} (z) + \frac {1} {2}{i^t}\bar wSw} \right )^s}dw\;d\bar w\;dz\;d\bar z

\]
with

P
Γ

⊆
Γ

{P_\Gamma } \subseteq \Gamma

a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over

Γ
∖
D

\Gamma \backslash D

. In the case

p
=
1

p = 1

our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.