# The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

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.

-

It was shown by Zelditch that the spaces $H_k$ with the standard metric on $\mathbb{C}$ are isometric to the spaces $H^0(L^k)$ of holomorphic sections of the $k$-th power of the line bundle $L$, please see arXiv: math-ph/0002009v1 (propositions 6,7).

-
Many thanks for the reference, it was very useful. Let me remark one point that it is still not clear for me: In page 4 of the reference it says: "The S1 action commutes with $\bar\partial$, hence H2(X) =⊕$H^2_k$" I think that the action of S1 commutes with $\bar\partial$ implies that we have a representation of S1 and the decomposition in the direct sum is the decomposition in irreducible representations.However this seems to be a contradiction because S1 is abelian thus dim H^2_k=1 (all irreps have dim=1 when the group is Abelian). So let me ask why H^2 splits in the direct sum? Thanks David –  Josh Oct 10 '13 at 0:54
@josh the Abelian group action implies that the irreducible components are one dimensional, but they can have multiplicities. The space$H_k$ is a direct sum of all one dimensional subspaces of eigenvalue (weight) $k$. –  David Bar Moshe Oct 13 '13 at 7:53