On invariant Kählerian structures on non-compact homogeneous manifolds admitting the action of a semisimple Lie group (Q1074140)

Let G be a connected non-compact real semisimple Lie group, L be a connected compact subgroup which is a centralizer of a torus in G and K be the maximal compact subgroup such that \(K\supset L\). In this paper the author gives a necesary and sufficient condition to decide whether there exists a corresponding G-invariant Kählerian structure for a given G- invariant complex structure G/L. As a corollary it is shown that if there exist G-invariant Kählerian structures on G/L and \(L\subsetneqq K\) then the number of G-invariant Kählerian structures on G/L is twice the number of G-invariant complex structures on K/L. Finally as an example it is shown that for \(G=SO(2\ell -1,2)\) the number of G-invariant Kählerian structures on G/L is \(2^{\ell -k}(\ell -k-1)!\). Also a table of such numbers of Kählerian structures (if exist) for all G is given except that G is of type EVII with \(| \pi_ 0| =2\) or 3. For these two cases it states that it is too complicated to list the details in the paper.