Computing the Cousin-Zuckerman Resolution and the Lusztig-Vogan Bijection

TL;DR

This paper characterizes global sections of standard D-modules on flag varieties, computes the Cousin-Zuckerman resolution, and proves the Lusztig-Vogan bijection for specific cases, linking representation classifications.

Contribution

It provides a proof of a key characterization of global sections of standard D-modules and applies it to compute resolutions and bijections in representation theory.

Findings

Computed the Cousin-Zuckerman resolution of the trivial representation.

Proved the Lusztig-Vogan bijection for n=2,3 in the case of GL(n,H).

Determined the lowest K-type map for various orbits.

Abstract

The goal of this article is to give a proof of a result seemingly absent from the literature characterizing global sections of standard $\mathcal{D}$-modules on the flag variety. This characterization yields a mixture of the Langlands Classification of admissible representations with the Knapp-Zuckerman classification of tempered representations of a real reductive group. We use this result to compute the Cousin-Zuckerman resolution of the trivial representation in terms of standard $(\mathfrak{g},K)$-modules. Further, in the case of $GL(n,\mathbb{H})$ we use this to prove the Lusztig-Vogan bijection for $n=2,3$ and compute the lowest $K$-type map for the zero and principal orbits for general $n$ as well as the image of the trivial representation for even orbits.