Scale Functions on Linear Groups Over Local Skew Fields

LetGbe a general or special linear group over a local skew field. ThenGis a totally disconnected, locally compact group, to which G. Willis (Math. Ann.300, 1994, 341–363) associates its scale functions : G → . We computeson the subset of diagonalizable matrices. We also consider the projective situation, and we discuss scale functions on groups of upper triangular matrices. The latter can be computed completely, provided that the underlying field is commutative. The data computed suffice to determine the modular functions on the groups considered, and they facilitate an easy proof of the fact that general or special linear groups over local skew fields with distinct modules cannot be isomorphic as topological groups.