I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated.
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My original question was ambiguous. Let me explain and give a few further details.
I want to understand some combinatorial properties (expansion of some type) of the group $GL_{\mathbb{F}}(n)$, where the $\mathbb{F}$ is a finite field. One possible approach for doing that (that was successful in, e.g. understanding similar aspects of the permutation group) is through unitary representations of that group. As far as I can tell, most of the texts cover $GL(n)$ over fields of characteristic zero, which are not what I'm interested in. So I'm asking for sources for unitary representations of the linear group over finite fields..
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To clarify further, by unitary representations I mean homomorphisms of GL(n) of a finite fields, into the group of finite-dimensional unitary matrices over $\mathbb C$. As you might guess, I'm a cs/combinatorics person, and far from expert on representation theory — please excuse my lack of verbal skills in this area and otherwise..
Best Answer
All finite dimensional complex representations of finite groups are equivalent to unitary representations, so the requirement that the representations be unitary is not really a restriction.
The 1955 work of J.A. Green gives the definitive description of the complex characters of the groups ${\rm GL}(n,q)$ for q a prime power: see
http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0072878-2/S0002-9947-1955-0072878-2.pdf
Later work of Deligne-Lusztig studied the complex characters of other finite classical groups (and finite groups of Lie type), and the book of Digne and Michel mentioned by Neil Strickland is a good source of information.