TITLE: are irreducible representations with large fixed subspaces trivial?
QUESTION [21 upvotes]: 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?

REPLY [15 votes]: 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