Classifying triples of Lagrangians in a Hermitian vector space

The purpose of this paper is to study the diagonal action of the unitary group U(n) on triples of Lagrangian subspaces of Cn. The notion of angle of Lagrangian subspaces is presented here, and we show how pairs (L1,L2) of Lagrangian subspaces are classified by the eigenvalues of the unitary map σL2∘σL1 obtained by composing the Lagrangian involutions relative to L1 and L2. We then study the case of triples of Lagrangian subspaces of C2 and give a complete description of the orbit space. Our main result is Theorem 3.5. As an application of the methods presented here, we give a way of computing the inertia index of a triple (L1,L2,L3) of Lagrangian subspaces of Cn from the measures of the angles (Lj,Lk), and relate the classification of Lagrangian triples of C2 to the classification of two-dimensional unitary representations of the fundamental group π1(S2⧹{s1,s2,s3}).