A simplified proof of the reduction point crossing sign formula for Verma modules

• Published in: Algebra and Discrete Mathematics
• Date: 2019
• Main Authors: Denis, M.St., Yee, W.L.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/188488
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: A simplified proof of the reduction point crossing sign formula for Verma modules / M.St. Denis, W.L. Yee // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 195–202. — Бібліогр.: 7 назв. — англ.

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 ε governing how the signature changes. A formula for ε was first determined in [Yee05] and simplified in [Yee19]. The proof of the simplification was complicated. We simplify the proof in this note.