  In this paper we give a complete invariant of the action of GL_n(F)\times
GL_m(F) acting on the Euclidean building B[GL_n(F)], where F is a
non-archimedian field. We then use this invariant to give a natural metric on
the resulting quotient space. In the special case of the torus acting on
B[GL_2(F)] this gives a method of calculating of any vertex to any fixed
apartment.
