Holomorphic sections of prequantum line bundles on \(G/N\)
Let \(K\) be a compact semisimple Lie group and \(G\) its complexification with a chosen Iwasawa decomposition \(G=KAN\). Then \(G/N\) is a complex manifold and may be equipped with a \(K\)-invariant Kähler structure. The Kähler form gives rise to a `prequantum' line bundle \({\mathbf L}\) in the usual way. As a smooth line bundle \({\mathbf L}\) is trivial. However, it is also naturally a holomorphic line bundle and, as such, generally not trivial. The space \(H({\mathbf L})\) provides a representation of \(K\) and in joint work with \textit{V. Guillemin} [Contemp. Math. 154, 181-195 (1993; Zbl 0809.53060)], the author shows that if the Kähler structure on \(G/N\) is \(T\)-invariant, where \(T\) is the centralizer of \(A\) in~\(K\), then every finite-dimensional representation of \(K\) occurs inside \(H({\mathbf L})\) with multiplicity one. In this article the author shows that in all other cases, \(H({\mathbf L})\) vanishes. This is a striking dichotomy reminiscent of the Borel-Weil theorem. The case \(K={\text{SU}}(2)\) is already interesting and not as easy as one might think. The space \(G/N\) is easily identified as \({\mathbb C}^2\setminus\{0\}\). For a \(K\)-invariant Kähler form one can take \(\partial\alpha+\bar\partial\bar\alpha\) where \(\alpha=(\bar z_1d\bar z_2-\bar z_2 d\bar z_1)/(z_1\bar z_1+z_2\bar z_2)^2\). In this case, after some work, it turns out that \(H({\mathbf L})=0\). In fact this is the key to the rank one case and then the general.
scientific article; zbMATH DE number 1548269