On the existence of an invariant non-degenerate bilinear form under a linear map

Abstract

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F) > dim V. Let T V > V be an invertible linear map. We answer the following question in this paper. When does V admit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form ?We also answer the infinitesimal version of this question. Following Feit and Zuckerman 121, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V, F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V, F). Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group 1(V, B). A non-degenerate S-invariant subspace W of (V, B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is nontrivial for the unipotent elements in I(V, B). The level of a unipotent T is the least integer k such that (T - 1)(k) = 0. We also classify the levels of unipotents in I(V,B). (C) 2010 Elsevier Inc. All rights reserved.