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

TL;DR

This paper investigates conditions for the existence of invariant non-degenerate bilinear forms under linear maps, characterizes real elements in general linear groups, and classifies indecomposable subspaces and unipotent levels in isometry groups.

Contribution

It provides new criteria for invariant bilinear forms, characterizes real elements in linear groups, and classifies indecomposable subspaces and unipotent levels in isometry groups.

Findings

Criteria for the existence of invariant bilinear forms under linear maps.

Characterization of real elements in general linear groups.

Classification of indecomposable subspaces and unipotent levels in isometry groups.

Abstract

Let $\V$ be a vector space over a field $\F$. Assume that the characteristic of $\F$ is \emph{large}, i.e. $char(\F)>\dim \V$. Let $T: \V \to \V$ 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-Zuckerman \cite{fz}, 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 $\G(\V, \F)$. As a consequence of the answers to the above question, we offer a characterization of the real elements in $\G(V, \F)$. Suppose $\V$ is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form $B$. Let $S$ be an element in the isometry group $I(\V, B)$. 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 $(T-I)^k=0$. We also classify the levels of unipotents in $I(\V, B)$.