Isotropy of unitary involutions

TL;DR

This paper proves the Unitary Isotropy Theorem, establishing conditions for isotropy of unitary involutions, and relates it to known results on orthogonal and symplectic involutions, with a detailed study of quasi-split unitary grassmannians.

Contribution

It introduces the Unitary Isotropy Theorem, unifying and extending previous results on involution isotropy and hyperbolicity across different types.

Findings

Proves the Unitary Isotropy Theorem.

Shows how known isotropy and hyperbolicity results follow from this theorem.

Provides a detailed analysis of quasi-split unitary grassmannians.

Abstract

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.