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. Let G 0 bearealsemisimpleLiegroup,let R beaparabolicsubgroupofthecomplexication G of G 0 , let D beanopen G 0 -orbitinthe complex ag manifold X = G/R , andlet Y be a maximal compactlinearsubvarietyof D . First,anexplicitparabolicsubgroup Q ⊂ R ⊂ G isconstructedsothattheopen G 0 -orbitson W = G/Q aremeasurableandonesuchorbit D = G 0 ( w ) ⊂ W mapsonto D withanebre. Second,itisshownthat D is( s +1)-completeinthesenseofAndreottiandGrauert, s =dim C Y ;thuscohomologies H q ( D ; F )=0for q>s whenever F→D isacoherentanalyticsheaf. Thiswasknown[7]forthecaseofmeasurableopenorbits,andtheproofusesthatresulton D . Third,itisshownthatthespace M D ofcompactlinearsubvarietiesof D isaSteinmanifold. Forthat,astrictlyplurisubharmonicexhaustionfunctionisconstructedasintheargument[9]forthecaseofmeasurableopenorbits. 1. BackgroundandstatementofresultsLet G 0 beaconnectedreductiverealLiegroup,g 0 itsrealLiealgebra,andg =g 0 ⊗ R C thecomplexication.Asusual,Int(g)denotesthecomplexconnectedsemisimpleLiegroupofallinnerautomorphismsofg,consistingoftheAd(

Exhaustion Functions and Cohomology Vanishing Theorems for Open Orbits on Complex Flag Manifolds