The theta representation is a representation of the continuous Heisenberg group over the field of the real numbers. In this representation, the group elements act on a particular Hilbert space. The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of the isomorphism to the usual representations.
Let f(z) be a holomorphic function, let a and b be real numbers, and let be fixed, but arbitrary complex number in the upper half-plane; that is, so that the imaginary part of is positive. Define the operators Sa and Tb such that they act on holomorphic functions as
It can be seen that each operator generates a one-parameter subgroup:
The action of the group elements is unitary and irreducible on a certain Hilbert space of functions. For a fixed value of τ, define a norm on entire functions of the complex plane as
Here, is the imaginary part of and the domain of integration is the entire complex plane. Let be the set of entire functions f with finite norm. The subscript is used only to indicate that the space depends on the choice of parameter . This forms a Hilbert space. The action of given above is unitary on , that is, preserves the norm on this space. Finally, the action of on is irreducible.