Heisenberg Group

The Heisenberg group $H^n$ in $n$ Complex variables is the Group of all $(z,t)$ with $z\in\Bbb{C}^n$ and $t\in\Bbb{R}$ having multiplication

$$(w,t)(z,t')=(w+z,t+t'+\Im[w^{\rm T} z])$$

where $w^{\rm T}$ is the conjugate transpose. The Heisenberg group is Isomorphic to the group of Matrices

$$\left[{\matrix{1 & z^{\rm T} & {\textstyle{1\over 2}}\vert z\vert^2+it\cr 0 & 1 & z\cr 0 & 0 & 1\cr}}\right],$$

and satisfies

$$(z,t)^{-1}=(-z,-t).$$

Every finite-dimensional unitary representation is trivial on $Z$ and therefore factors to a Representation of the quotient $\Bbb{C}^n$.

See also Nil Geometry

References

Knapp, A. W. ``Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.