  The representation theory of the group U(1,q) is discussed in detail because
of its possible application in a quaternion version of the Salam-Weinberg
theory.
  As a consequence, from purely group theoretical arguments we demonstrate that
the eigenvalues must be right-eigenvalues and that the only consistent scalar
products are the complex ones. We also define an explicit quaternion tensor
product which leads to a set of additional group representations for integer
``spin''.
