A factorization in GSp(V)

Let V be a vector space over the field F such that char(F) not equal 2, and let V have a skew-symmetric nondegenerate bilinear form. Wonenburger proved that any element g of Sp (V) is the product of two skew-symplectic involutions. Let GSp (V) be the group of general similitudes with similitude character mu. We give a generalization of Wonenburger's result in the following form. Let gis an element ofGSp (V) with mu(g)=beta. Then g=t(1)t(2) such that t(1) is a skew-symplectic involution, and t(2) is such that t(2)(2)=betaI and mu (t(2))=-beta. One application that follows from this result is a necessary and sufficient condition for an element of GL (V) to be conjugate to a scalar multiple of its inverse. Another result is that we find an extension of the group Sp(2n,F-q) , for qequivalent to3(mod 4), all of whose complex representations have real-valued characters.