A simplified proof of the reduction point crossing sign formula for Verma modules
The Unitary Dual Problem is one of the most important open problems in mathematics: classify the irreducible unitary representations of a group. That is, classify all irreducible representations admitting a definite invariant Hermitian form. Signatures of invariant Hermitian forms on Verma modules...
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| Date: | 2020 |
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Lugansk National Taras Shevchenko University
2020
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admjournalluguniveduua-article-12182020-02-10T19:12:26Z A simplified proof of the reduction point crossing sign formula for Verma modules St. Denis, Matthew Yee, Wai Ling unitary representations 22E50, 05E10 The Unitary Dual Problem is one of the most important open problems in mathematics: classify the irreducible unitary representations of a group. That is, classify all irreducible representations admitting a definite invariant Hermitian form. Signatures of invariant Hermitian forms on Verma modules are important to finding the unitary dual of a real reductive Lie group. By a philosophy of Vogan introduced in [Vog84], signatures of invariant Hermitian forms on irreducible Verma modules may be computed by varying the highest weight and tracking how signatures change at reducibility points (see [Yee05]). At each reducibility point there is a sign \(\varepsilon\) governing how the signature changes. A formula for \(\varepsilon\) was first determined in [Yee05] and simplified in [Yee19]. The proof of the simplification was complicated. We simplify the proof in this note. Lugansk National Taras Shevchenko University 2020-02-10 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1218 Algebra and Discrete Mathematics; Vol 28, No 2 (2019) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1218/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1218/393 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1218/394 Copyright (c) 2020 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2020-02-10T19:12:26Z |
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| language |
English |
| topic |
unitary representations 22E50 05E10 |
| spellingShingle |
unitary representations 22E50 05E10 St. Denis, Matthew Yee, Wai Ling A simplified proof of the reduction point crossing sign formula for Verma modules |
| topic_facet |
unitary representations 22E50 05E10 |
| format |
Article |
| author |
St. Denis, Matthew Yee, Wai Ling |
| author_facet |
St. Denis, Matthew Yee, Wai Ling |
| author_sort |
St. Denis, Matthew |
| title |
A simplified proof of the reduction point crossing sign formula for Verma modules |
| title_short |
A simplified proof of the reduction point crossing sign formula for Verma modules |
| title_full |
A simplified proof of the reduction point crossing sign formula for Verma modules |
| title_fullStr |
A simplified proof of the reduction point crossing sign formula for Verma modules |
| title_full_unstemmed |
A simplified proof of the reduction point crossing sign formula for Verma modules |
| title_sort |
simplified proof of the reduction point crossing sign formula for verma modules |
| description |
The Unitary Dual Problem is one of the most important open problems in mathematics: classify the irreducible unitary representations of a group. That is, classify all irreducible representations admitting a definite invariant Hermitian form. Signatures of invariant Hermitian forms on Verma modules are important to finding the unitary dual of a real reductive Lie group. By a philosophy of Vogan introduced in [Vog84], signatures of invariant Hermitian forms on irreducible Verma modules may be computed by varying the highest weight and tracking how signatures change at reducibility points (see [Yee05]). At each reducibility point there is a sign \(\varepsilon\) governing how the signature changes. A formula for \(\varepsilon\) was first determined in [Yee05] and simplified in [Yee19]. The proof of the simplification was complicated. We simplify the proof in this note. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2020 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1218 |
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2025-12-02T15:34:36Z |
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2025-12-02T15:34:36Z |
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