Signed winding-number formula for indefinite unitary groups

Investigate whether, for tuples whose elements and ordered product are elliptic elements of an indefinite unitary group U(p,q), a signed version of the winding number m—weighting each eigenangle by the sign of the indefinite Hermitian form on the corresponding eigenline—computes the signature of the associated Katz–Long–Moody Hermitian form.

Background

The paper develops a closed signature formula for the Katz–Long–Moody Hermitian form when the seed representation preserves a positive-definite Hermitian form. Its proof relies on unitary eigenangles, Hermitian Cayley transforms, and Loewner monotonicity of a normalized Hermitian pencil.

The invariance theory also permits seed representations preserving an indefinite non-degenerate Hermitian form of signature (p,q). In that setting, the Cayley transforms are only H-self-adjoint, the associated pencil is not generally Loewner-monotone, and group elements need not have spectra on the unit circle. For elliptic tuples, eigenangles remain available, motivating the unresolved question of whether an appropriately signed winding-number invariant can still determine the form’s signature.