Weil–Brezin Map
In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line  to a smooth function  on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.

Heisenberg manifold
The (continuous) Heisenberg group  is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule
 
The discrete Heisenberg group  is the discrete subgroup of  whose elements are represented by the triples of integers. Considering  acts on  on the left, the quotient manifold  is called the Heisenberg manifold. 
The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure  on the Heisenberg group induces a right-translation-invariant measure on the Heisenberg manifold. The space of complex-valued square-integrable functions on the Heisenberg manifold has a right-translation-invariant orthogonal decomposition:

where 
.

Definition
The Weil–Brezin map  is the unitary transformation given by 
 
for every Schwartz function , where convergence is pointwise.

The inverse of the Weil–Brezin map  is given by

for every smooth function  on the Heisenberg manifold that is in .

Fundamental unitary representation of the Heisenberg group
For each real number , the fundamental unitary representation  of the Heisenberg group is an irreducible unitary representation of  on  defined by
. 
By Stone–von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relation
.
The fundamental representation  of  on  and the right translation   of  on  are intertwined by the Weil–Brezin map
.
In other words, the fundamental representation  on  is unitarily equivalent to the right translation  on  through the Wei-Brezin map.

Relation to Fourier transform
Let  be the automorphism on the Heisenberg group given by
.
It naturally induces a unitary operator , then the Fourier transform 

as a unitary operator on .

Plancherel theorem
The norm-preserving property of  and , which is easily seen, yields the norm-preserving property of the Fourier transform, which is referred to as the Plancherel theorem.

Poisson summation formula
For any Schwartz function ,
.
This is just the Poisson summation formula.

Relation to the finite Fourier transform 
For each , the subspace  can further be decomposed into right-translation-invariant orthogonal subspaces

where 
.
The left translation  is well-defined on , and  are its eigenspaces.

The left translation  is well-defined on , and the map
 
is a unitary transformation.

For each , and , define the map  by 
 
for every Schwartz function , where convergence is pointwise.

The inverse map  is given by 

for every smooth function  on the Heisenberg manifold that is in .

Similarly, the fundamental unitary representation  of the Heisenberg group is unitarily equivalent to the right translation on  through :
.
For any , 
.

For each , let .  Consider the finite dimensional subspace  of  generated by  where
 
Then the left translations   and  act on  and give rise to the irreducible representation of the finite Heisenberg group.  The map  acts on  and gives rise to the finite Fourier transform

Nil-theta functions
Nil-theta functions are functions on the Heisenberg manifold that are analogous to the theta functions on the complex plane. The image of Gaussian functions under the Weil–Brezin Map are nil-theta functions. There is a model of the finite Fourier transform defined with nil-theta functions, and the nice property of the model is that the finite Fourier transform is compatible with the algebra structure of the space of nil-theta functions.

Definition of nil-theta functions
Let  be the complexified Lie algebra of the Heisenberg group . A basis of  is given by the left-invariant vector fields  on :

These vector fields are well-defined on the Heisenberg manifold .

Introduce the notation . For each , the vector field  on the Heisenberg manifold can be thought of as a differential operator on  with the kernel generated by .

We call

the space of nil-theta functions of degree .

Algebra structure of nil-theta functions
The nil-theta functions with pointwise multiplication on  form a graded algebra  (here ).

Auslander and Tolimieri showed that this graded algebra is isomorphic to 
,
and that the finite Fourier transform (see the preceding section #Relation to the finite Fourier transform) is an automorphism of the graded algebra.

Relation to Jacobi theta functions
Let  be the Jacobi theta function. Then
.

Higher order theta functions with characteristics
An entire function  on  is called a theta function of order , period  () and characteristic  if it satisfies the following equations:
,
.
The space of theta functions of order , period  and characteristic  is denoted by . 
. 
A basis of  is
.
These higher order theta functions are related to the nil-theta functions by
.

See also
Nilmanifold
Nilpotent group
Nilpotent Lie algebra
Weil representation
Theta representation
Oscillator representation

References

Harmonic analysis
Representation theory