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 an arbitrary fixed 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
and
It can be seen that each operator generates a one-parameter subgroup:
A general group element then acts on a holomorphic function f(z) as
where is the center of H, the commutator subgroup. The parameter on serves only to remind that every different value of gives rise to a different representation of the action of the group.
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.
Mumford sets the norm as , but in this way is not unitary.
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.
The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of 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