  We reelaborate on the basic properties of lossless multilayers. We show that
the transfer matrices for these multilayers have essentially the same algebraic
properties as the Lorentz group SO(2,1) in a (2+1)-dimensional spacetime, as
well as the group SL(2,R) underlying the structure of the ABCD law in
geometrical optics. By resorting to the Iwasawa decomposition, we represent the
action of any multilayer as the product of three matrices of simple
interpretation. This group-theoretical structure allows us to introduce
bilinear transformations in the complex plane. The concept of multilayer
transfer function naturally emerges and its corresponding properties in the
unit disc are studied. We show that the Iwasawa decomposition reflects at this
geometrical level in three simple actions that can be considered the basic
pieces for a deeper undestanding of the multilayer behavior. We use the method
to analyze in detail a simple practical example.
