Bireflectionality of the weak orthogonal and the weak symplectic groups

Abstract

Let V be a finite-dimensional vector space, Q a quadratic form and fa = f the bilinear form associated with Q. We assume (V,&) is regular. Then the orthogonal group O(V) is bireflectional, i.e., every isometry in O(V) is a product of two involutory isometries in O(V). This has been shown in [8] if the field of scalars K has characteristic distinct from 2 and in [4] and [5] if char K = 2. The latter papers also establish the bireflectionality for the symplectic group Sp( V), again under the assumption that (V,f) is regular and char K = 2. For char K # 2 the symplectic group is not bireflectional (see [31). We shall extend the results just mentioned. We shall drop the assumptions that V is finite dimensional and that (V,f) is regular, i.e., the vector space V may be infinite dimensional and the radical of V may be distinct from zero. We shall use the notation and the concepts in [2]. For every 7c E Hom(V, V) we define F(n)= {UE V;?m=u} and B(n)= (vz u; ZJ E V}. The spaces F(r) and B(n) are called fix and path of z, respectively. The groups O*(V) = {n E O(V); rad VC F(z) and dim B(n) < co} and Sp *( V) = { 71 E Sp( V); rad V c F(n) and dim B(n) < co ) are called the weak orthogonal and the weak symplectic group, respectively.