Pseudokähler forms on complex Lie groups
Let \(G\) be a semisimple complex Lie group and \(G_{\mathbf R}\) its real form. Define \(f: i\mathfrak g_{\mathbf R}\times G_{\mathbf R}\to G\) by \(f(iX,r) = \exp(iX)r\) and denote by \(M\subset i\mathfrak g_{\mathbf R}\times G_{\mathbf R}\) the subset of nonsingular points of \(df\). Then \(M\) is a complex manifold, with complex structure \(J\) induced by the complex structure on \(G\). The form \(\omega = -i\partial\bar\partial\phi\) on \(M\), where \(\phi(iX,r) = \kappa(X,X)\), \(\kappa\) being the Killing form on \(\mathfrak g\), is considered. It is proved that \((A,B)\mapsto\omega(JA,B) + i\omega(A,B), A,B\in T_mM\), is a pseudo-Kähler form on \(M\) that is invariant under the \(G_{\mathbf R}\times G_{\mathbf R}\)-action on \(M\) given by \((r_1,r_2)(iX,r) = (i\text{Ad}r_1(X),r_1rr_2^{-1})\). Using \(f\), one can regard it as a \(G_{\mathbf R}\times G_{\mathbf R}\)-invariant pseudo-Kähler form on a neighborhood of the identity in \(G\). The signature of this form is equal to that of the Hermitian form \((Z,W)\mapsto\kappa(Z,\overline W)\). The moment map of the symplectic form \(\omega\) is also calculated.
scientific article; zbMATH DE number 1537285