I'm interested in an infinite dim'l Heisenberg group associated to the vector space $V = L\mathbb{C}/\mathbb{C}$ = {$f \colon S^1 \to \mathbb{C}$|$f$ smooth}/(const. maps). The group is $\mathbb{C}^\times \times V$ with group law

$(z,f)(z',g) = (zz' e^{\pi i (f,g)}, f+g)$

where $(f,g) = \int fdg$ is a symplectic form.

SOME BACKGROUND

There's a pairing $e(f,g) = e^{2\pi i (f,g)}$. The isotropic subspaces $W \subset V$ are the ones s.t. $e = 1$ on $W \times W$. General theory says for such $W$ you can construct a representation $F(W)$. Whenever you have a Lagrangian (= maximal isotropic) subspace $L \subset V$ you get up to equivalence a unique irreducible representation where $\mathbb{C}^\times$ acts by scalars.

(actually I've heard this for groups that are extensions by $U(1)$ and then then there is a unique *unitary* representation; but I'm guessing it works with $\mathbb{C}^\times$ too by just removing unitary)

One way to describe this representation is as continuous maps $\phi\colon V \to \mathbb{C}$ that satisfy $\phi(v + l) = e^{\pi i (v,\ l)}\phi(v)$ and $\int_{V/L} |\phi|^2 dk < \infty$ where $dk$ is a Haar measure on $V/L$.

QUESTION

For $V = L\mathbb{C}/\mathbb{C}$, $z = e^{i\theta}$ it seems that $z^k$ for $k \in \mathbb{Z} - 0$ forms a basis. Also $L^\pm =$ the vector spaces spanned by the positive/negative powers of $z$ are Lagrangian. `Span' here doesn't mean finite linear combinations but linear combinations where the coefficients form maybe absolutely convergent series?

My question is what is an example of a $\phi \in F(L^+)$?

I think all such $\phi$ should be described as follows. Let $p_\pm \colon V \to L^\pm$ be the projections. Then $\phi(v) = e^{i\pi (p_-(v),\ p_+(v))}\overline{\phi}(p_i(V))$ where $\overline{\phi} \in L^2(L_-;dk)$

Among my difficulties with answering this question is that $L_-$ is still a really big space and I don't know what a Haar measure would be on this a space.

I should say, the answer to this question wont really help me in any research per se; I ask it because morally I feel better talking about $F(L)$ if I could write down at least one of its elements.