Cycle Space Constructions for Exhaustions of Flag Domains

TL;DR

This paper introduces a new incidence-based method to construct canonical exhaustion functions on cycle spaces of flag domains, leading to potential advances in cohomology and representation theory of real reductive Lie groups.

Contribution

It presents a novel incidence method for constructing canonical plurisubharmonic exhaustion functions on cycle spaces, reversing previous approaches and enabling new applications.

Findings

Constructed a canonical plurisubharmonic exhaustion function on cycle spaces.

Derived a q-convex exhaustion function on flag domains from the cycle space.

Potential implications for cohomology vanishing and representation theory.

Abstract

In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space $\mathcal M_D$ of a flag domain $D$ is a Stein manifold. That fact has a long history. The earliest approach relied on construction of a strictly plurisubharmonic function on $\mathcal M_D$, 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 $D$ required that $D$ be measurable. In that case the exhaustion on $D$ was transferred to $\mathcal M_D$, using a special case of a method due to Barlet. Here we do the opposite: we use an incidence method to construct a canonical plurisubharmonic exhaustion function on $\mathcal M_D$ 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.