Say that $G$ is a finite group, and $V$ is an irreducible representation of $G$, over an algebraically closed field $k$. Suppose that for every $g \in G$, there is some subspace $W\_g \subset V$ which is (pointwise) fixed by $g$, such that the dimension of $W\_g$ is at least half the dimension of $V$.


If $k$ has characteristic zero, then a simple argument with character tables shows that $V$ must be the trivial representation. (When $k = \mathbb{C}$, then $\sum\_{g \in G} tr\_V(g)$ has positive real part; so appealing to the Lefschetz principle, $\sum\_{g \in G} tr\_V(g)$ is nonzero whenever $k$ has characteristic zero.)


Is this still true when $k$ has positive characteristic?



are irreducible representations with large fixed subspaces trivial?
Unless I have misread either your question or their result, Corollary 1.2 in "Average dimension of fixed point spaces with applications" by Bob Guralnick and Attila Mar\'oti includes a positive answer to the question. You can find this on the arxiv at


<http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.3836v1.pdf>


