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

Matthew St. Denis, Wai Ling Yee


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.


unitary representations

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