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

Abstract

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.