LIMITS OF CHARACTERS OF WREATH PRODUCTS Gn (T) OF A COMPACT GROUP T WITH THE SYMMETRIC GROUPS AND CHARACTERS OF G∞(T), I

In the first half of this paper, all the limits of irreducible Characters of G(n) = G(n) (T) as n -> infinity are calculated. The set of all continuous limit functions on G = G(infinity)(T) is exactly equal to the set of all characters of G determined in [HH6]. We give a necessary and sufficient condition for a series of irreducible characters of G, to have a continuous limit and also such a condition to have a discontinuous limit. In the second half, we study the limits of characters of certain induced representations of G(n) which are usually reducible. The limits turn out to be characters of G, and we analyse which of irreducible components are responsible to these limits.