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where
ν
is a normalization constant [6]. The operators
ρ
a
,ρ
T
and
ρ
S
are uni-
tary. Let us consider the subgroup of unitary operators generated by
ρ
a
,ρ
S
and
ρ
T
. This group turns out to be isomorphic to the finite group
Sp
=
SL
2
(
F
p
),
therefore we obtained a homomorphism
ρ
:
Sp
). The representation
ρ
is
called the
Weil representation
[7] and it will play a prominent role in this survey.
→
U
(
H
3.3
Systems Associated with Maximal (Split) Tori
The group
Sp
consists of various types of commutative subgroups. We will be
interested in maximal
diagonalizable
commutative subgroups. A subgroup of
this type is called maximal
split torus.
The standard example is the subgroup
consisting of all diagonal matrices
A
=
a
0
a
−
1
:
a
G
m
,
0
∈
which is called the
standard torus
. The restriction of the Weil representation
toasplittorus
T
⊂
Sp
yields a decomposition of the Hilbert space
H
into a
=
H
χ
,where
χ
runs in the set of charac-
ters
T
∨
. Choosing a unit vector
ϕ
χ
direct sum of character spaces
H
∈H
χ
for every
χ
we obtain a collection
T
∨
,χ
of orthonormal vectors
B
T
=
{
ϕ
χ
:
χ
∈
=
σ
}
. Overall, we constructed a
system
s
O
=
S
{
ϕ
∈B
T
:
T
⊂
Sp
split
}
,
which will be referred to as the
split oscillator system
. We note that our initial
system
B
std
is recovered as
B
std
=
B
A
.
3.4
Systems Associated with Maximal (Non-split) Tori
From the point of view of this survey, the most interesting maximal commutative
subgroups in
Sp
are those which are diagonalizable over an extension field rather
than over the base field
F
p
. A subgroup of this type is called maximal
non-split
torus.
It might be suggestive to first explain the analogue notion in the more
familiar setting of the field
. Here, the standard example of a maximal non-split
torus is the circle group
SO
(2)
R
). Indeed, it is a maximal commutative
subgroup which becomes diagonalizable when considered over the extension field
C
⊂
SL
2
(
R
of complex numbers.
The above analogy suggests a way to construct examples of maximal non-
split tori in the finite field setting as well. Let us assume for simplicity that
−
1
does not admit a square root in
F
p
. The group
Sp
acts naturally on the plane
V
=
F
p
×
F
p
. Consider the symmetric bilinear form
B
on
V
given by
B
((
t, w
)
,
(
t
,w
)) =
tt
+
ww
.
An example of maximal non-split torus is the subgroup
T
ns
⊂
Sp
consisting of
all elements
g
∈
Sp
preserving the form
B
, i.e.,
g
∈
T
ns
if and only if
B
(
gu, gv
)=
B
(
u, v
) for every
u, v
∈
V
.
