Computing the Cousin-Zuckerman Resolution and the Lusztig-Vogan Bijection
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917426180915200 |
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| author | Cook, Jack A. |
| author_facet | Cook, Jack A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19554 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Computing the Cousin-Zuckerman Resolution and the Lusztig-Vogan Bijection Cook, Jack A. Representation Theory 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. |
| title | Computing the Cousin-Zuckerman Resolution and the Lusztig-Vogan Bijection |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2604.19554 |