Conjugacy Classes in Maximal Parabolic Subgroups of General Linear Groups

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem.” Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GLn(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in q with integral coefficients.