On the Existence of an Invariant Non-degenerate Bilinear Form under a Linear Map

Let V be a vector space over a field F. The characteristic p of F is large if either p = 0 or p > dim V. When F is of large characteristic, the following questions, as well as their infinitesimal versions, are answered in this paper. Given an invertible linear map T : V → V, when does the vector space V admit a T -invariant non-degenerate symmetric (resp. skew-symmetric) biliner form? An element g in a group G is called real if it is conjugate in G to its own inverse. As a consequence of the answers to the above questions, we characterize the real elements in GL(n, F) which admit an invariant symmetric (resp. skew-symmetric) bilinear form. We have also given a bound for the level of a unipotent in the orthogonal and the symplectic groups.