Information Technology Reference
In-Depth Information
A.4
Formulas
We are left to explain how to write explicit formulas (matrices) for the operators
involved in the construction of
s
O
.
First, we recall that the group
Sp
admits a Bruhat decomposition
Sp
=
S
B
∪
B
w
B,
where
B
is the Borel subgroup and w denotes the Weyl element
w=
01
−
.
10
Furthermore, the Borel subgroup
B
can be written as a product
B
=
AU
=
UA
,where
A
is the standard diagonal torus and
U
is the standard unipotent
group
U
=
1
u
01
:
u
∈
F
p
.
UA
w
U
.
Second, we give an explicit description of operators in the Weil representation,
associated with different types of elements in
Sp
. The operators are specified up
to a unitary scalar, which is enough for our needs.
-
The standard torus
A
acts by (normalized) scaling: An element
a
=
a
Therefore, we can write the Bruhat decomposition also as
Sp
=
UA
∪
0
a
−
1
,
0
acts by
S
a
[
f
](
t
)=
σ
(
a
)
f
a
−
1
t
,
is the Legendre character,
σ
(
a
)=
a
p−
2
(
mod p
).
-
The standard unipotent group
U
acts by quadratic characters (chirps): An
F
p
where
σ
:
→{±
1
}
element
u
=
1
u
01
,actsby
M
u
[
f
](
t
)=
ψ
(
u
2
t
2
)
f
(
t
)
,
F
p
→
C
×
is the character
ψ
(
t
)=
e
2
π
p
t
.
-
The Weyl element w acts by discrete Fourier transform
where
ψ
:
1
√
p
F
[
f
](
w
)=
ψ
(
wt
)
f
(
t
)
.
t
∈
F
p
Hence, we conclude that every operator
ρ
(
g
), where
g
∈
Sp
, can be written
either in the form
ρ
(
g
)=
M
u
◦
S
a
or in the form
ρ
(
g
)=
M
u
2
◦
S
a
◦
F
◦
M
u
1
.
Example 1.
For
g
∈
R
,with
c
= 0, the Bruhat decomposition of
g
is given
explicitly by
g
=
1
1+
bc
−
c
01
−
1
c
01
,
0
c
01
0
−
c
10
and
ρ
(
g
)=
M
1+
bc
c
◦
S
−
c
◦
F
◦
M
c
.
