Cycle space constructions for exhaustions of flag domains

In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space MD of a flag domain D is a Stein manifold. That fact has a long history; see [4]. The earliest approach ([9], [11]) relied on construction of a strictly plurisubharmonic exhaustion function on MD, starting with a q–convex exhaustion function on D, where q is the dimension of a particular maximal compact subvariety of D (we use the normalization that 0–convex means Stein). Construction of that exhaustion function on D [8] required that D be measurable [10]. In that case the exhaustion on D was transferred to MD, using a special case of a method due to Barlet [2]. Here we do the opposite: we use the incidence method of [4] to construct a canonical plurisubharmonic exhaustion function on MD and use it in turn to construct a canonical q–convex exhaustion function on D. This promises to have strong consequences for cohomology vanishing theorems and the construction of admissible representations of real reductive Lie groups. It also completes the argument of [12].