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In the same fashion like in the split case, restricting the Weil representation
to a non-split torus
T
yields a decomposition into character spaces
=
H
χ
.
H
T
∨
we obtain an orthonormal
Choosing a unit vector
ϕ
χ
∈H
χ
for every
χ
∈
B
T
. Overall, we constructed a system of signals
basis
n
O
=
S
{
ϕ
∈B
T
:
T
⊂
Sp
non-split
}
.
n
O
will be referred to as the
non-split oscillator
system
.
The con-
struction of the system
S
The system
n
O
together with the formulation of some
of its properties are the main contribution of this survey.
S
O
=
S
s
O
∪
S
3.5
Behavior under Fourier Transform
The oscillator system is closed under the operation of Fourier transform, i.e.,
for every
ϕ
∈
S
O
we have
ϕ
∈
S
O
.
The Fourier transform on the space
C
(
F
p
)
appears as a specific operator
ρ
(w) in the Weil representation, where
w=
01
−
∈
Sp.
10
Given a signal
ϕ
∈B
T
⊂
S
O
, its Fourier transform
ϕ
=
ρ
(w)
ϕ
is, up to a
where
T
=w
T
w
−
1
.Infact,
unitary scalar, a signal in
S
O
is closed under
all the operators in the Weil representation! Given an element
g
B
T
∈
Sp
and a signal
ϕ
∈B
T
,where
T
=
gTg
−
1
.
In addition, the Weyl element w is an element in some maximal torus
T
w
(the split type of
T
w
depends on the characteristic
p
of the field) and as a result
signals
ϕ
∈B
T
we have, up to a unitary scalar, that
ρ
(
g
)
ϕ
∈B
T
w
are, in particular, eigenvectors of the Fourier transform. As a
consequences a signal
ϕ
ϕ
differ by a unitary
constant, therefore are practically the ”same” for all essential matters.
These properties might be relevant for applications to OFDM (Orthogonal
Frequency Division Multiplexing) [8] where one requires good properties both
from the signal and its Fourier transform.
∈B
T
w
and its Fourier transform
3.6
Relation to the Harmonic Oscillator
Here we give the explanation why functions in the non-split oscillator system
S
ns
O
constitute a finite analogue of the eigenfunctions of the harmonic oscillator
in the real setting. The Weil representation establishes the dictionary between
these two, seemingly, unrelated objects. The argument works as follows.
The one-dimensional harmonic oscillator is given by the differential operator
D
=
∂
2
t
2
. The operator
D
can be exponentiated to give a unitary repre-
sentation of the circle group
ρ
:
SO
(2
,
−
R
)where
ρ
(
θ
)=
e
iθD
.
Eigenfunctions of
D
are naturally identified with character vectors with respect
to
ρ
. The crucial point is that
ρ
is the restriction of the Weil representation of
SL
2
(
U
L
2
(
R
)
−→
).
Summarizing, the eigenfunctions of the harmonic oscillator and functions in
R
) to the maximal non-split torus
SO
(2
,
R
)
⊂
SL
2
(
R
n
O
are governed by the same mechanism, namely both are character vectors
S
