A closed signature formula for the Katz-Long-Moody Hermitian form

Abstract: The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible. The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.