  We consider GL_n(F_q)-analogues of certain factorization problems in the
symmetric group S_n: rather than counting factorizations of the long cycle (1,
2, ..., n) given the number of cycles of each factor, we count factorizations
of a regular elliptic element given the fixed space dimension of each factor.
We show that, as in S_n, the generating function counting these factorizations
has attractive coefficients after an appropriate change of basis. Our work
generalizes several recent results on factorizations in GL_n(F_q) and also uses
a character-based approach.
  As an application of our results, we compute the asymptotic growth rate of
the number of factorizations of fixed genus of a regular elliptic element in
GL_n(F_q) into two factors as n goes to infinity. We end with a number of open
questions.
