Abstract: Guelfand [4], Godement [3] and Harish-Chandra
in the fifties up to Toure and Kangni [6, 7] today, among many others have
studied spherical functions relatively to the unitary dual of compact groups.In this paper, we
define -spherical functions
relatively to the coadjoint representation of a complex Lie group and examine
the nilpotent case.
Keywords and phrases: coadjoint
representation, -spherical function, quasi-bounded function.