Scott H. Murray (University of Chicago)

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 ${\mbox GL}_n(k)$ 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. 

Current status:  Preprint.  Submitted to Journal of Algebra.  There is no dvi file available.