A family of adapted complexifications for noncompact semisimple Lie   groups
  The maximal complexifications adapted to the Levi Civita connection for a distinguished one-parameter family of left-invariant metrics on a real semisimple Lie group G are determined. For G = SL(2,R) their realizations as invariant Riemann domains into SL(2,C) are carried out and their complex-geometric properties are investigated. 
S. Halverscheid, A. Iannuzzi