# References for analysis on the Heisenberg group(s)

$\newcommand{\Real}{{\mathbb R}}$
*Under construction.*

Perhaps put some of this on the nLab?

## Particular to the Heisenberg group

[Article] Fabec, TAMS 1991

Starts off using Fock space model rather than Schrödinger picture

Uses unpolarized form

Proves things about operators of the form $\pi_\lambda(f)$ where $f$ is a Schwartz function on the group.

Section 4 has some Fourier inversion

#### [Article] D. Geller, JFA 1980

#### [Thesis] D. Rottensteiner

Foundations of harmonic analysis on the Heisenberg group
[URL]

Uses the unpolarized/symmetric form

Explicit description of Schrödinger representations: equation (2.28) is
\[
\pi_h(t,q,p) f(x) = e^{iht + iqx+ ihpq/2}f(x+hp)
\]

Is the "twisted convolution" on $L^1(\Real^2)$ (Defn 2.32) related to what we are looking at? I guess not.

Should see what references he uses for Bochner integral, measurability, etc
#### [Book] Thangavelu

Polarized or unpolarized? He seems to start by saying he works on the polarized version, but then when he defines the Schr&oumml;dinger reps the notation suggests these are reps of the unpolarized form

Schrödinger rep: in Equation (1.2.1)
\[
\pi_\lambda(x,y,t)\varphi(\xi)= e^{i\lambda t} e^{i\lambda(x\cdot\xi+\frac{1}{2}x\cdot y)}\varphi(\xi+y)
\qquad(\varphi\in L^2(\Real^n))
\]

## More general

#### [Book] E. Kaniuth, K. F. Taylor

Seems to use the polarized/asymmetric form

[Book] G. B. Folland, A Course in Abstract Harmonic Analysis

Section 7.6 does several examples, and Example 4 is the polarized form of the Heisenberg group (2n+1 dimensions).
\[
[\rho_h (x,\xi,t)f](y) = e^{2\pi i ht + \pi h \xi\cdot x} e^{-2\pi i h\xi\cdot y} f(y-x)
\]
Folland also says:

(These differ from the representations called $\rho_h$ in Folland [40] but are equivalent to them via the map $Uf(y)=|h|^{1/2}f(-hy)$.)

An explicit description of the integral kernel corresponding to $\widehat{f}(\rho_h)$ can be found just before (7.51)
#### [Book] H. Führ

At start of Section 6.1, uses the unpolarized form

At start of Section 6.1, the Schrödinger representation is given as
\[
[\rho_h (p,q,t)f](x) = e^{2\pi i ht} e^{2\pi i qx} e^{\pi i hpq} f(x+hp)
\]