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In [[mathematics]], the '''Weil–Brezin map''', named after [[André Weil]]<ref>Weil, André. "Sur certains groupes d'opérateurs unitaires." Acta mathematica 111.1 (1964): 143-211.</ref> and Jonathan Brezin,<ref>Brezin, Jonathan. "Harmonic analysis on nilmanifolds." Transactions of the American Mathematical Society 150.2 (1970): 611-618.</ref> 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]].<ref>Auslander, Louis, and Richard Tolimieri. Abelian harmonic analysis, theta functions and function algebras on a nilmanifold. Springer, 1975.</ref><ref>Auslander, Louis. "Lecture notes on nil-theta functions." Conference Board of the Mathematical Sciences, 1977.</ref><ref>Zhang, D. [https://www.researchgate.net/profile/Dapeng_Zhang3/publication/270341923_Integer_Linear_Canonical_Transforms_Their_Discretization_and_Poisson_Summation_Formulae/links/54a88d7d0cf257a6360be0e1.pdf "Integer Linear Canonical Transforms, Their Discretization, and Poisson Summation Formulae"]</ref> The image of [[Gaussian function]]s under the Weil–Brezin map are nil-theta functions, which are related to [[theta function]]s. The Weil–Brezin map is sometimes referred to as the [[Zak transform]],<ref>{{Cite web|url=http://mathworld.wolfram.com/ZakTransform.html|title = Zak Transform}}</ref> 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 [[Heisenberg group|(continuous) Heisenberg group]] <math>N</math> is the 3-dimensional [[Lie group]] that can be represented by triples of real numbers with multiplication rule
: <math>\langle x,y,t\rangle \langle a,b,c\rangle = \langle x+a, y+b, t+c+xb\rangle.</math>
The [[Heisenberg group|discrete Heisenberg group]] <math>\Gamma</math> is the discrete subgroup of <math>N</math> whose elements are represented by the triples of integers. Considering <math>\Gamma</math> acts on <math>N</math> on the left, the [[quotient manifold]] <math>\Gamma\backslash N</math> is called the '''Heisenberg manifold'''. 
The Heisenberg group acts on the Heisenberg manifold on the right. The [[Haar measure]] <math>\mu = dx \wedge dy \wedge dt</math> on the Heisenberg group induces a right-translation-invariant measure on the Heisenberg manifold. The space of complex-valued [[square-integrable function]]s on the Heisenberg manifold has a right-translation-invariant orthogonal decomposition:
:<math>L^2(\Gamma\backslash N) = \oplus_{n \in \mathbb{Z}}  H_n</math>
where 
:<math>H_n =\{f\in L^2(\Gamma\backslash N)  \mid f(\Gamma \langle x, y, t+s\rangle) = \exp(2\pi i ns) f(\Gamma \langle x,y,t\rangle)\}</math>.

==Definition==
The '''Weil–Brezin map''' <math>W: L^2(\mathbb R) \to H_1</math> is the unitary transformation given by 
: <math>W(\psi) (\Gamma \langle x, y, t \rangle) = \sum_{l\in \mathbb Z} \psi(x + l) e^{2 \pi i l y} e^{2\pi i t} </math>
for every Schwartz function <math>\psi</math>, where convergence is [[Pointwise convergence|pointwise]].

The inverse of the Weil–Brezin map <math>W^{-1}: H_1 \to L^2(\mathbb R)</math> is given by
:<math>(W^{-1}f) (x) = \int_0^{1} f(\Gamma \langle x, y, 0\rangle) dy</math>
for every smooth function <math>f</math> on the Heisenberg manifold that is in <math>H_1</math>.

==Fundamental unitary representation of the Heisenberg group==
For each real number <math>\lambda\ne 0</math>, the '''fundamental unitary representation''' <math>U_{\lambda}</math> of the Heisenberg group is an [[Irreducible representation|irreducible]] [[unitary representation]] of <math>N</math> on <math>L^2(\mathbb{R})</math> defined by
:<math>(U_{\lambda}(\langle a, b, c \rangle) \psi) (x) =  e^{2 \pi i \lambda (c + bx)} \psi(x +a) </math>. 
By [[Stone–von Neumann theorem]], this is the unique irreducible representation up to unitary equivalence satisfying the [[canonical commutation relation]]
:<math>U_{\lambda}(\langle a, 0, 0 \rangle) U_{\lambda}(\langle 0, b, 0 \rangle) =e^{2\pi i \lambda ab} U_{\lambda}(\langle 0, b, 0 \rangle) U_{\lambda}(\langle a, 0, 0 \rangle)</math>.
The fundamental representation <math>U=U_1</math> of <math>N</math> on <math>L^2(\mathbb{R})</math> and the right translation <math>R</math>  of <math>N</math> on <math>H_1 \subset L^2(\Gamma \backslash N)</math> are intertwined by the Weil–Brezin map
:<math>W U(\langle a, b, c \rangle) = R(\langle a, b, c \rangle) W</math>.
In other words, the fundamental representation <math>U</math> on <math>L^2(\mathbb{R})</math> is [[unitary representation|unitarily equivalent]] to the right translation <math>R</math> on <math>H_1</math> through the Weil-Brezin map.

==Relation to Fourier transform==
Let <math>J:  N \to N </math> be the automorphism on the Heisenberg group given by
:<math>J(\langle x, y, t \rangle) =\langle y, -x, t-xy \rangle</math>.
It naturally induces a [[unitary operator]] <math>J^* : H_1 \to H_1</math>, then the [[Fourier transform]] 
:<math>\mathcal F = W^{-1} J^{*} W</math>
as a unitary operator on <math>L^2(\mathbb{R})</math>.

===Plancherel theorem===
The norm-preserving property of <math>W</math> and <math>J^*</math>, 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 <math>\psi</math>,
:<math>\sum_l\psi(l) = W(\psi)(\Gamma\langle 0, 0, 0) \rangle) = (J^*W(\psi))(\Gamma\langle 0, 0, 0) \rangle) = W (\hat{\psi})(\Gamma\langle 0, 0, 0) \rangle)=\sum_l \hat{\psi}(l)</math>.
This is just the Poisson summation formula.

==Relation to the finite Fourier transform ==
For each <math>n\ne 0</math>, the subspace <math>H_n\subset L^2(\Gamma\backslash N)</math> can further be decomposed into right-translation-invariant orthogonal subspaces
:<math>H_n = \oplus_{m=0}^{|n|-1} H_{n,m}</math>
where 
:<math>H_{n,m} =\{f\in H_n \mid f(\Gamma \langle x, y+ {1\over n}, t\rangle) = e^{2\pi i m/n}f(\Gamma \langle x, y, t\rangle)\}</math>.
The left translation <math>L(\langle 0, 1/n, 0\rangle)</math> is well-defined on <math>H_n</math>, and <math>H_{n,0}, ... , H_{n, |n|-1}</math> are its [[eigenspace]]s.

The left translation <math>L(\langle m/n, 0, 0\rangle)</math> is well-defined on <math>H_n</math>, and the map
:<math>L(\langle m/n, 0, 0\rangle) : H_{n,0} \to H_{n,m}</math> 
is a unitary transformation.

For each <math>n\ne 0</math>, and <math>m = 0, ..., |n|-1</math>, define the map <math>W_{n, m}: L^2(\mathbb R) \to H_{n, m}</math> by 
: <math>W_{n,m}(\psi) (\Gamma \langle x, y, t \rangle) = \sum_{l\in \mathbb Z} \psi(x + l + {m \over n}) e^{2 \pi i (nl+m) y} e^{2\pi i n t} </math>
for every Schwartz function <math>\psi</math>, where convergence is pointwise.

:<math>W_{n,m} =L(\langle m/n, 0, 0\rangle) \circ W_{n,0}.</math>

The inverse map <math>W_{n,m}^{-1}: H_{n,m} \to L^2(\mathbb R)</math> is given by 
:<math>(W_{n,m}^{-1}f) (x) = \int_0^{1} e^{-2\pi i m y} f(\Gamma \langle x - {m \over n}, y, 0\rangle) dy</math>
for every smooth function <math>f</math> on the Heisenberg manifold that is in <math>H_{n,m}</math>.

Similarly, the fundamental unitary representation <math>U_n</math> of the Heisenberg group is unitarily equivalent to the right translation on <math>H_{n,m}</math> through <math>W_{n,m}</math>:
:<math>W_{n,m} U_n(\langle a, b, c \rangle) = R(\langle a, b, c \rangle) W_{n,m}</math>.
For any <math>m, m'</math>, 
:<math>(W_{n,m'}^{-1}J^* W_{n,m} \psi ) (x) = e^{2\pi i m'm / n} \hat{\psi}(nx)</math>.

For each <math>n>0</math>, let <math>\phi_n(x) =(2n)^{1/4} e^{- \pi n x^2 }</math>.  Consider the finite dimensional subspace <math>K_n</math> of <math>H_n</math> generated by <math>\{\boldsymbol{e}_{n,0}, ..., \boldsymbol{e}_{n,n-1}\}</math> where
:<math>\boldsymbol{e}_{n,m} = W_{n,m} ( \phi_n ) \in H_{n,m}.</math> 
Then the left translations  <math>L(\langle 1/n, 0, 0\rangle)</math> and <math>L(\langle 0, 1/n, 0\rangle)</math> act on <math>K_n</math> and give rise to the irreducible representation of the finite Heisenberg group.  The map <math>J^*</math> acts on <math>K_n</math> and gives rise to the [[Finite Fourier transform (disambiguation)|finite Fourier transform]]
:<math>J^* \boldsymbol{e}_{n,m} = {1 \over \sqrt{n}}\sum_{m'}  e^{2\pi i m'm / n} \boldsymbol{e}_{n,m'}.</math>

==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<ref>Auslander, L., and R. Tolimieri. "Algebraic structures for⨁Σ _ {𝑛≥ 1} 𝐿<sup>2</sup> (𝑍/𝑛) compatible with the finite Fourier transform." Transactions of the American Mathematical Society 244 (1978): 263-272.</ref> 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 <math>\mathfrak{n}</math> be the complexified [[Lie algebra]] of the Heisenberg group <math>N</math>. A basis of <math>\mathfrak{n}</math> is given by the left-invariant vector fields <math>X, Y, T</math> on <math>N</math>:
:<math>X(x,y,t) = {\partial \over \partial x},</math>
:<math>Y(x,y,t) = {\partial \over \partial y} + x {\partial \over \partial t},</math>
:<math>T(x,y,t) = {\partial \over \partial t}.</math>
These vector fields are well-defined on the Heisenberg manifold <math>\Gamma \backslash N</math>.

Introduce the notation <math>V_{-i} = X-i Y</math>. For each <math>n>0</math>, the vector field <math>V_{-i}</math> on the Heisenberg manifold can be thought of as a [[differential operator]] on <math>C^{\infty} (\Gamma \backslash N) \cap H_{n,m}</math> with the [[Kernel (linear algebra)|kernel]] generated by <math>\boldsymbol{e}_{n,m}</math>.

We call
:<math>\ker(V_{-i}: C^{\infty} (\Gamma \backslash N) \cap H_n \to H_n) = \left\{ \begin{array}{lr} K_n, & n>0 \\ \mathbb{C}, & n=0  \end{array} \right. </math>
the space of '''nil-theta functions''' of degree <math>n</math>.

===Algebra structure of nil-theta functions===
The nil-theta functions with pointwise multiplication on <math>\Gamma \backslash N</math> form a [[graded ring|graded algebra]] <math>\oplus_{n\ge 0} K_n</math> (here <math>K_0 = \mathbb{C}</math>).

Auslander and Tolimieri showed that this graded algebra is [[isomorphic]] to 
:<math>\mathbb{C}[x_1, x_2^2, x_3^3]/(x_3^6 + x_1^4 x_2^2 + x_2^6)</math>,
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 <math>\vartheta(z; \tau) = \sum_{l =-\infty}^\infty \exp (\pi i l^2 \tau + 2 \pi i l z) </math> be the [[Jacobi theta function]]. Then
:<math>\vartheta(n(x+iy); ni) = (2n)^{-1/4} e^{\pi n y^2} \boldsymbol{e}_{n,0}(\Gamma\langle y, x, 0 \rangle)</math>.

===Higher order theta functions with characteristics===
An [[entire function]] <math>f</math> on <math>\mathbb{C}</math> is called a '''theta function of order''' <math>n</math>, '''period''' <math>\tau</math> (<math>\mathrm{Im}(\tau)>0</math>) and '''characteristic''' <math>[^a_b]</math> if it satisfies the following equations:
#<math>f(z+1) = \exp(\pi i a ) f(z)</math>,
#<math>f(z+\tau) =\exp(\pi i b) \exp(-\pi i n (2z +\tau)) f(z)</math>.
The space of theta functions of order <math>n</math>, period <math>\tau</math> and characteristic <math>[^a_b]</math> is denoted by <math>\Theta_n[^a_b](\tau, A)</math>. 
:<math>\dim \Theta_n[^a_b](\tau, A) =n</math>. 
A basis of <math>\Theta_n[^0_0](i, A)</math> is
:<math>\theta_{n,m}(z) = \sum_{l\in \mathbb{Z}} \exp [ -\pi n (l+{m \over n} )^2 + 2 \pi i (l n+ m)  z ) ]</math>.
These higher order theta functions are related to the nil-theta functions by
:<math>\theta_{n,m}(x+ iy) = (2n)^{-1/4} e^{\pi n y^2} \boldsymbol{e}_{n,m} (\Gamma \langle y, x, 0 \rangle )</math>.

==See also==
*[[Nilmanifold]]
*[[Nilpotent group]]
*[[Nilpotent Lie algebra]]
*[[Weil representation]]
*[[Theta representation]]
*[[Oscillator representation]]

==References==
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[[Category:Harmonic analysis]]
[[Category:Representation theory]]