Classifying pairs of Lagrangians a Hermitian vector space (Q1183646)

Let \((L_ 1,L_ 2)\) be a pair of Lagrangian subspaces of a Hermitian linear space \((V,\langle\;,\;\rangle)\) together with orthonormal bases \((e_ 1,\dots,e_ n)\) and \((f_ 1,\dots,f_ n)\) of \(L_ 1\) and \(L_ 2\), respectively. The Souriau matrix \(AA^ t\) where \(A_{ij}=\langle f_ j,e_ i\rangle\) is a basic invariant of the pair \((L_ 1,L_ 2)\). The author proves that pairs of Lagrangian subspaces in \((V,\langle\;,\;\rangle)\) are determined up to unitary equivalence by the characteristic polynomial of their Souriau matrices and so a classification of all pairs of Lagrangians is obtained. Applications of it to Lagrangian pairs in symplectic vector spaces and to vector bundles are pointed out.