Products of Reflections in the Group SO*(2n)

Let SO*(2n) be the group of quaternionic n × n matrices A satisfying A*JA = J, where J is a fixed skew-hermitian invertible matrix. An element R ∊ SO*(2n) is called a reflection if R2 = In and R — In has rank one. We assume that n ≧ 2, in which case S*(2n) is generated by reflections. The length of A ∊ SO*(2n) is the smallest integer k(≧0) such that A can be written as A = R1R2 … Rk where R1, …, Rk are reflections. In this paper, for each A ∊ SO*(2n), we compute its length l(A). Set r(A) = rank (A — In). Already in Section 3 we are able to show that the difference δ = l(A) – r(A) can take only three values 0, 1, or 2. The remainder of the paper deals with the problem of separating these three possibilities. The main results are stated in Section 4 and proved in Section 6. The intermediate Section 5 consists of a sequence of lemmas which are needed for the proof. Clearly l(A) depends only on the conjugacy class of A and the main results in Section 4 are stated in terms of conjugacy classes. For the description of conjugacy classes in SO*(2n) we refer the reader to [1]. The present paper relies heavily on our previous paper [5] where the analogous problem was solved for the groups U(p, q). It is worth remarking that only the various lemmas from that paper were used but not the main theorem.