In mathematics, the Weil–Brezin map, named after André Weil^[1] and Jonathan Brezin,^[2] 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.^[3][4][5] 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,^[6] 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.

The (continuous) Heisenberg group ${\displaystyle N}$ is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule

${\displaystyle \langle x,y,t\rangle \langle a,b,c\rangle =\langle x+a,y+b,t+c+xb\rangle .}$

The discrete Heisenberg group ${\displaystyle \Gamma }$ is the discrete subgroup of ${\displaystyle N}$ whose elements are represented by the triples of integers. Considering ${\displaystyle \Gamma }$ acts on ${\displaystyle N}$ on the left, the quotient manifold ${\displaystyle \Gamma \backslash N}$ is called the Heisenberg manifold. The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure ${\displaystyle \mu =dx\wedge dy\wedge dt}$ 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:

${\displaystyle L^{2}(\Gamma \backslash N)=\oplus _{n\in \mathbb {Z} }H_{n}}$

where

${\displaystyle H_{n}=\{f\in L^{2}(\Gamma \backslash N)\mid f(\Gamma \langle x,y,t+s\rangle )=\exp(2\pi ins)f(\Gamma \langle x,y,t\rangle )\}}$.

The Weil–Brezin map ${\displaystyle W:L^{2}(\mathbb {R} )\to H_{1}}$ is the unitary transformation given by

${\displaystyle W(\psi )(\Gamma \langle x,y,t\rangle )=\sum _{l\in \mathbb {Z} }\psi (x+l)e^{2\pi ily}e^{2\pi it}}$

for every Schwartz function ${\displaystyle \psi }$, where convergence is pointwise.

The inverse of the Weil–Brezin map ${\displaystyle W^{-1}:H_{1}\to L^{2}(\mathbb {R} )}$ is given by

${\displaystyle (W^{-1}f)(x)=\int _{0}^{1}f(\Gamma \langle x,y,0\rangle )dy}$

for every smooth function ${\displaystyle f}$ on the Heisenberg manifold that is in ${\displaystyle H_{1}}$.

For each real number ${\displaystyle \lambda \neq 0}$, the fundamental unitary representation ${\displaystyle U_{\lambda }}$ of the Heisenberg group is an irreducible unitary representation of ${\displaystyle N}$ on ${\displaystyle L^{2}(\mathbb {R} )}$ defined by

${\displaystyle (U_{\lambda }(\langle a,b,c\rangle )\psi )(x)=e^{2\pi i\lambda (c+bx)}\psi (x+a)}$.

By Stone–von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relation

${\displaystyle U_{\lambda }(\langle a,0,0\rangle )U_{\lambda }(\langle 0,b,0\rangle )=e^{2\pi i\lambda ab}U_{\lambda }(\langle 0,b,0\rangle )U_{\lambda }(\langle a,0,0\rangle )}$.

The fundamental representation ${\displaystyle U=U_{1}}$ of ${\displaystyle N}$ on ${\displaystyle L^{2}(\mathbb {R} )}$ and the right translation ${\displaystyle R}$ of ${\displaystyle N}$ on ${\displaystyle H_{1}\subset L^{2}(\Gamma \backslash N)}$ are intertwined by the Weil–Brezin map

${\displaystyle WU(\langle a,b,c\rangle )=R(\langle a,b,c\rangle )W}$.

In other words, the fundamental representation ${\displaystyle U}$ on ${\displaystyle L^{2}(\mathbb {R} )}$ is unitarily equivalent to the right translation ${\displaystyle R}$ on ${\displaystyle H_{1}}$ through the Wei-Brezin map.

Let ${\displaystyle J:N\to N}$ be the automorphism on the Heisenberg group given by

${\displaystyle J(\langle x,y,t\rangle )=\langle y,-x,t-xy\rangle }$.

It naturally induces a unitary operator ${\displaystyle J^{*}:H_{1}\to H_{1}}$, then the Fourier transform

${\displaystyle {\mathcal {F}}=W^{-1}J^{*}W}$

as a unitary operator on ${\displaystyle L^{2}(\mathbb {R} )}$.

For each ${\displaystyle n\neq 0}$, the subspace ${\displaystyle H_{n}\subset L^{2}(\Gamma \backslash N)}$ can further be decomposed into right-translation-invariant orthogonal subspaces

${\displaystyle H_{n}=\oplus _{m=0}^{|n|-1}H_{n,m}}$

where

${\displaystyle H_{n,m}=\{f\in H_{n}\mid f(\Gamma \langle x,y+{1 \over n},t\rangle )=e^{2\pi im/n}f(\Gamma \langle x,y,t\rangle )\}}$.

The left translation ${\displaystyle L(\langle 0,1/n,0\rangle )}$ is well-defined on ${\displaystyle H_{n}}$, and ${\displaystyle H_{n,0},...,H_{n,|n|-1}}$ are its eigenspaces.

The left translation ${\displaystyle L(\langle m/n,0,0\rangle )}$ is well-defined on ${\displaystyle H_{n}}$, and the map

${\displaystyle L(\langle m/n,0,0\rangle ):H_{n,0}\to H_{n,m}}$

is a unitary transformation.

For each ${\displaystyle n\neq 0}$, and ${\displaystyle m=0,...,|n|-1}$, define the map ${\displaystyle W_{n,m}:L^{2}(\mathbb {R} )\to H_{n,m}}$ by

${\displaystyle W_{n,m}(\psi )(\Gamma \langle x,y,t\rangle )=\sum _{l\in \mathbb {Z} }\psi (x+l+{m \over n})e^{2\pi i(nl+m)y}e^{2\pi int}}$

for every Schwartz function ${\displaystyle \psi }$, where convergence is pointwise.

${\displaystyle W_{n,m}=L(\langle m/n,0,0\rangle )\circ W_{n,0}.}$

The inverse map ${\displaystyle W_{n,m}^{-1}:H_{n,m}\to L^{2}(\mathbb {R} )}$ is given by

${\displaystyle (W_{n,m}^{-1}f)(x)=\int _{0}^{1}e^{-2\pi imy}f(\Gamma \langle x-{m \over n},y,0\rangle )dy}$

for every smooth function ${\displaystyle f}$ on the Heisenberg manifold that is in ${\displaystyle H_{n,m}}$.

Similarly, the fundamental unitary representation ${\displaystyle U_{n}}$ of the Heisenberg group is unitarily equivalent to the right translation on ${\displaystyle H_{n,m}}$ through ${\displaystyle W_{n,m}}$:

${\displaystyle W_{n,m}U_{n}(\langle a,b,c\rangle )=R(\langle a,b,c\rangle )W_{n,m}}$.

For any ${\displaystyle m,m'}$,

${\displaystyle (W_{n,m'}^{-1}J^{*}W_{n,m}\psi )(x)=e^{2\pi im'm/n}{\hat {\psi }}(nx)}$.

For each ${\displaystyle n>0}$, let ${\displaystyle \phi _{n}(x)=(2n)^{1/4}e^{-\pi nx^{2}}}$. Consider the finite dimensional subspace ${\displaystyle K_{n}}$ of ${\displaystyle H_{n}}$ generated by ${\displaystyle \{{\boldsymbol {e}}_{n,0},...,{\boldsymbol {e}}_{n,n-1}\}}$ where

${\displaystyle {\boldsymbol {e}}_{n,m}=W_{n,m}(\phi _{n})\in H_{n,m}.}$

Then the left translations ${\displaystyle L(\langle 1/n,0,0\rangle )}$ and ${\displaystyle L(\langle 0,1/n,0\rangle )}$ act on ${\displaystyle K_{n}}$ and give rise to the irreducible representation of the finite Heisenberg group. The map ${\displaystyle J^{*}}$ acts on ${\displaystyle K_{n}}$ and gives rise to the finite Fourier transform

${\displaystyle J^{*}{\boldsymbol {e}}_{n,m}={1 \over {\sqrt {n}}}\sum _{m'}e^{2\pi im'm/n}{\boldsymbol {e}}_{n,m'}.}$

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^[7] 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.

• Nilmanifold
• Nilpotent group
• Nilpotent Lie algebra
• Weil representation
• Theta representation
• Oscillator representation