The Orbit Method for Finite Groups of Nilpotency Class Two of Odd Order

Abstract

First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to construct adjoint and coadjoint representations as usual. For a finite group G of nilpotency class 2 of odd order, I construct a basis in its group algebra C[G], parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G]. That construction gives us a one-to-one correspondence between G-orbits in g* and classes of equivalence of irreducible unitary representations of G, implying a very simple character formula. The properties of that correspondence are similar to the properties of the analogous correspondence given by Kirillov's orbit method for nilpotent connected and simply connected Lie groups. The diagram method introduced in my article 'Diagrams of Representations, math.RT/9803079 (1998)' and my thesis 'A Combinatorial Approach to Representations of Lie Groups and Algebras, University of Pennsylvania (1998)' gives us a convenient way to study normal forms on the orbits and corresponding representations.