the Heisenberg group. In particular, this implies that and are isomorphic as H-modules. Let stand for a general group element of In the canonical Weyl representation, for every real number h, there is a representation acting on as for and Here, h is Planck's constant. Each such representation is unitarily inequivalent. The corresponding theta representation is: Discrete subgroup Define the subgroup as The Jacobi theta function is defined as It is an entire function of z that is invariant under This follows from the properties of the theta function: and when a and b are integers. It can be shown that the Jacobi theta is the unique such function. See also Segal–Bargmann space Hardy space References David Mumford, Tata Lectures on Theta I (1983), Birkhäuser, Boston Elliptic functions Theta functions Lie groups Mathematical quantization