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A good reference for the character theory of the finite general linear groups is "Hall Functions and Symmetric Polynomials," by MacDonald, which I'm guessing you're familiar with. Thiem and Vinroot describe the analogous construction for the finite unitary groups in "On the characteristic map of finite unitary groups."
Thanks for pointing that out, I should have just included that in the original question as it may make it easier. I've edited to question to reflect your comment.
What sort of properties of these matrices are you looking for? I would think that, despite perhaps having an infinite number of nonzero entries in some columns, these matrices would in many ways behave like finite dimensional matrices. Also, another interesting infinite matrix setting is upper-triangular matrices. If you simply require that all matrices are upper-triangular, you remove any convergence issues.
Thanks for the response! I still have to work out why $GL(2,q)$ does not have such a Hall subgroup for most $q$, but this reduction of the problem is very helpful.
