This construction arises when constructing the Szego projector.

Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, *i.e* $\overline{D}$ = $\{$ $v$ $\in$$L^*$ : $\|L^*\|$ $\leq$$1$ $\}$.

Let $H^2$($\overline{D}$) = $Ker$ $\bar\partial$ $\bigcap$ $L^2$($\overline{D}$). Now $S^1$ acts on $H^2$($\overline{D}$): given a funtion $f\in$$H^2$($\overline{D}$) and $\lambda\in S^1$ the action gives the function $(\lambda,f)\in H^2$($\overline{D}$) such that $(\lambda,f)(p,v)=f(p,\lambda v) $ with $(p,v)\in \overline{D}, v\in \pi^{-1}(p)$

We can check easily that we have a unitary representation of $S^1$ on $H^2$($\overline{D}$): $\lambda$ $\longmapsto$ $T_{\lambda}$ $\in\mathbb{U} $($H^2$($\overline{D}$)) (unitary operators), such that $T_{\lambda}(f)=(\lambda,f)$

Using basic representation theory of Abelian groups we know that this representation decompose in irreducible representations indexed by $\mathbb{Z}$ (the dual group of $S^1$). This decompose $H^2$($\overline{D}$) in subspaces of dimension 1 (because $S^1$ is Abelian). We can easily check that the subspace corresponding to $k\geq0$ is given by those functions which are k-linear in the direction of the fibers $i.e$ $H^2_k$($\overline{D}$) = $\{$ $f\in$$H^2$($\overline{D}$): $f(p,\lambda v)=\lambda^k f(p, v)$ $\forall (p, v)\in \overline{D}$, $\lambda\in S^1 \}$

Thinking about the subspaces $H^2_k$ I wonder how it is possible that those have dimension 1. What is a basis for each subspace $H^2_k?$ I know the decomposition of the well-know space $L^2(S^1)$ in one dimesional subspaces but I am strugling to use this in the dual disc bundle case. Did I misunderstood something in the explaniation I wrote?

Thanks for your answers, if you could also give some references with more details about the Szego projector in the dual disc bundle case I would be very grateful.