## *Visible actions and multiplicity-free theorems*.

colloquium. Université de Reims, France, 23 September 2008.

The structure "multiplicity-freeness" (each building block is used no more
than once) is sometimes hidden in classical mathematical methods such as
the Taylor series and the Fourier transform.
On the other hand, multiplicity-free property is fairly rare in the general setting of representations.

Then, how to find such a structure systematically ?

In this talk, I discuss the original theory of "visible actions" on complex manifolds, and explain a trick that multiplicity-free structure propagates from fibers to sections. Thus, we get the machinery that makes new out of old, and big out of small.

Systematic and synthetic applications of this idea to branching problems
(the description of "breaking symmetries") are illusrated on both finite
and infinite dimensional cases (possibly with continuous spectrum).

© Toshiyuki Kobayashi