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 \(\varepsilon\) 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.