On the meromorphic potential for a harmonic surface in a \(k\)-symmetric space
The paper deals with harmonic maps \(\psi: \Omega\to G/K\), where \(\Omega\) is the complex plane or the Poincaré disc and \(G/K\) denotes the homogeneous space of a compact semisimple Lie group \(G\) generated by a periodic automorphism \(\sigma\in \Aut (G)\) of order \(k\geq 2\) (thus, \(G/K\) is a \(k\)-symmetric space in the sense of \textit{O. Kowalski} [`Generalized symmetric spaces' (Springer, Berlin) (1980; Zbl 0431.53042)]). In the paper under review, there are given canonical ways of constructing primitive harmonic maps from holomorphic and meromorphic potentials. For example, the holomorphic potential is defined by means of some loop group \(L_iG^\mathbb{C}_\sigma\) and its Lie algebra \(L_i{\mathfrak g}^\mathbb{C}_\sigma\) (defined in the article) as follows. Let \[ \Lambda_{-1,\infty} =\{X\in L_1 {\mathfrak g}^\mathbb{C}_\sigma \mid \lambda X\quad \text{extends holomorphically into } I_0\} \] \((I_0\) is a disc) and define the holomorphic potential as a holomorphic section \(\xi\) of \(T^*_{1,0} \otimes \Lambda_{-1, \infty}\). In the same fashion, the meromorphic potentials are defined using meromorphic sections of \(T^*_{1,0} \otimes {\mathfrak g}_{-1}\) \(({\mathfrak g}_i\) denote subspaces of the grading determined by \(\sigma)\). Some formulae for meromorphic and holomorphic potentials are given.
scientific article; zbMATH DE number 1019705