The Orbit Method for Finite Groups of Nilpotency Class 2 of Odd Order

Aleksandrs Mihailovs 
Department of Mathematics 
Shepherd College
Shepherdstown, WV 25443 
http://www.mihailovs.com/Alec/

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 thesis, gives us a 
convenient way to study normal forms on the orbits and 
corresponding representations. 

Preprint (January 20, 2000)