Applications of Bruhat decompositions to complex hyperbolic geometry (Q1130091)

The double coset space \(A\setminus GL(n,\mathbb{C})/ U(n-1,1)\) is studied, where \(A\) consists of the diagonal matrices in \(GL(n,\mathbb{C})\). This space naturally arises in the harmonic analysis on the Hermitian symmetric space \(GL(n,\mathbb{C})/ U(n-1,1)\). It is shown here that these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants. In particular, it is indicated how the invariants may be used to describe the holomorphic isometry classes of \(n+1\) points in complex hyperbolic \(n\)-space.