Products of reflections in the quaternionic unitary group

Abstract

Let Sp( n) denote the unitary group of a positive definite hermitian form on an n-dimensional right quaternionic vector space. Thus Sp( n) is the compact real form of the complex symplectic group Sp(2 n, C). For each A ϵ Sp( n), n ⩾ 2, we determine the smallest number m such that A can be written as a product of m reflections in Sp( n). In particular, we show that if n ⩾ 4 then every A ϵ Sp( n) is a product of n, n + 1, or n + 2 reflections. In the projective group PSp( n), n > 4, every element is a product of n reflections.