Geodesic
A simplified proof of the reduction point crossing sign formula for Verma modules
Algebra and discrete mathematics, Tome 28 (2019) no. 2, pp. 195-202

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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.
Keywords: unitary representations. 
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     title = {A simplified proof of the reduction point crossing sign formula for {Verma} modules},
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Matthew St. Denis; Wai Ling Yee. A simplified proof of the reduction point crossing sign formula for Verma modules. Algebra and discrete mathematics, Tome 28 (2019) no. 2, pp. 195-202. http://geodesic.mathdoc.fr/item/ADM_2019_28_2_a4/